Abstract

A simple and useful formula governing the skew rays propagating through a multimode fiber taper has been derived by using geometrical optics. Using this formula, the transmission properties of the fiber taper are studied. The total light transmitted and the effective numerical aperture for both the meridional rays and the skew rays are compared with those of a uniform fiber.

© 1985 Optical Society of America

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References

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  1. T. K. Lim, B. K. Garside, J. P. Marton, “An analysis of optical waveguide tapers,” Appl. Phys. 18, 53–62 (1979).
    [CrossRef]
  2. B. K. Garside, T. K. Lim, J. P. Marton, “Ray projections in optical fiber tapered sections,” Appl. Opt. 17, 3670–3674 (1978).
    [CrossRef] [PubMed]
  3. T. Ozeki, B. S. Kawasaki, “Mode behaviour in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (1976).
    [CrossRef]
  4. T. Ozeki, T. Ito, T. Tamura, “Tapered sections of multimode cladded fibers as mode filters and mode analyzers,” Appl. Phys. Lett. 26, 386–388 (1975).
    [CrossRef]
  5. J. E. Midwinter, “The prism-taper coupler for the excitation of single modes in optical transmission fibers,” Opt. Quantum Electron. 7, 297–303 (1975).
    [CrossRef]
  6. S. E. Miller, E. A. J. Marcatili, T. Li, “Research towards optical fiber transmission systems. Pt. II,” Proc. IEEE 61, 1726–1751 (1973).
    [CrossRef]
  7. T. Ozeki, B. S. Kawasaki, “Optical directional coupler using tapered sections in multimode fibers,” Appl. Phys. Lett. 28, 528–529 (1976).
    [CrossRef]
  8. F. Szarka, A. Lightstone, J. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integ. Opt. 3, 285–298 (1980).
    [CrossRef]
  9. T. Okoshi, Optical Fibers (Academic, New York, 1982), p. 41.
  10. N. S. Kapany, Fiber Optics (Academic, New York, 1967), Chap. 2.
  11. R. J. Potter, “Transmission properties of optical fibers,”J. Opt. Soc. Am. 51, 1079–1089 (1961).
    [CrossRef]
  12. R. J. Potter, E. Donath, R. Tynan, “Light-collecting properties of a perfect circular optical fiber,”J. Opt. Soc. Am. 53, 256–260 (1963).
    [CrossRef]

1980 (1)

F. Szarka, A. Lightstone, J. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integ. Opt. 3, 285–298 (1980).
[CrossRef]

1979 (1)

T. K. Lim, B. K. Garside, J. P. Marton, “An analysis of optical waveguide tapers,” Appl. Phys. 18, 53–62 (1979).
[CrossRef]

1978 (1)

1976 (2)

T. Ozeki, B. S. Kawasaki, “Mode behaviour in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (1976).
[CrossRef]

T. Ozeki, B. S. Kawasaki, “Optical directional coupler using tapered sections in multimode fibers,” Appl. Phys. Lett. 28, 528–529 (1976).
[CrossRef]

1975 (2)

T. Ozeki, T. Ito, T. Tamura, “Tapered sections of multimode cladded fibers as mode filters and mode analyzers,” Appl. Phys. Lett. 26, 386–388 (1975).
[CrossRef]

J. E. Midwinter, “The prism-taper coupler for the excitation of single modes in optical transmission fibers,” Opt. Quantum Electron. 7, 297–303 (1975).
[CrossRef]

1973 (1)

S. E. Miller, E. A. J. Marcatili, T. Li, “Research towards optical fiber transmission systems. Pt. II,” Proc. IEEE 61, 1726–1751 (1973).
[CrossRef]

1963 (1)

1961 (1)

Donath, E.

Garside, B. K.

T. K. Lim, B. K. Garside, J. P. Marton, “An analysis of optical waveguide tapers,” Appl. Phys. 18, 53–62 (1979).
[CrossRef]

B. K. Garside, T. K. Lim, J. P. Marton, “Ray projections in optical fiber tapered sections,” Appl. Opt. 17, 3670–3674 (1978).
[CrossRef] [PubMed]

Hughes, R.

F. Szarka, A. Lightstone, J. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integ. Opt. 3, 285–298 (1980).
[CrossRef]

Ito, T.

T. Ozeki, T. Ito, T. Tamura, “Tapered sections of multimode cladded fibers as mode filters and mode analyzers,” Appl. Phys. Lett. 26, 386–388 (1975).
[CrossRef]

Kapany, N. S.

N. S. Kapany, Fiber Optics (Academic, New York, 1967), Chap. 2.

Kawasaki, B. S.

T. Ozeki, B. S. Kawasaki, “Mode behaviour in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (1976).
[CrossRef]

T. Ozeki, B. S. Kawasaki, “Optical directional coupler using tapered sections in multimode fibers,” Appl. Phys. Lett. 28, 528–529 (1976).
[CrossRef]

Li, T.

S. E. Miller, E. A. J. Marcatili, T. Li, “Research towards optical fiber transmission systems. Pt. II,” Proc. IEEE 61, 1726–1751 (1973).
[CrossRef]

Lightstone, A.

F. Szarka, A. Lightstone, J. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integ. Opt. 3, 285–298 (1980).
[CrossRef]

Lim, T. K.

T. K. Lim, B. K. Garside, J. P. Marton, “An analysis of optical waveguide tapers,” Appl. Phys. 18, 53–62 (1979).
[CrossRef]

B. K. Garside, T. K. Lim, J. P. Marton, “Ray projections in optical fiber tapered sections,” Appl. Opt. 17, 3670–3674 (1978).
[CrossRef] [PubMed]

Lit, J.

F. Szarka, A. Lightstone, J. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integ. Opt. 3, 285–298 (1980).
[CrossRef]

Marcatili, E. A. J.

S. E. Miller, E. A. J. Marcatili, T. Li, “Research towards optical fiber transmission systems. Pt. II,” Proc. IEEE 61, 1726–1751 (1973).
[CrossRef]

Marton, J. P.

T. K. Lim, B. K. Garside, J. P. Marton, “An analysis of optical waveguide tapers,” Appl. Phys. 18, 53–62 (1979).
[CrossRef]

B. K. Garside, T. K. Lim, J. P. Marton, “Ray projections in optical fiber tapered sections,” Appl. Opt. 17, 3670–3674 (1978).
[CrossRef] [PubMed]

Midwinter, J. E.

J. E. Midwinter, “The prism-taper coupler for the excitation of single modes in optical transmission fibers,” Opt. Quantum Electron. 7, 297–303 (1975).
[CrossRef]

Miller, S. E.

S. E. Miller, E. A. J. Marcatili, T. Li, “Research towards optical fiber transmission systems. Pt. II,” Proc. IEEE 61, 1726–1751 (1973).
[CrossRef]

Okoshi, T.

T. Okoshi, Optical Fibers (Academic, New York, 1982), p. 41.

Ozeki, T.

T. Ozeki, B. S. Kawasaki, “Mode behaviour in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (1976).
[CrossRef]

T. Ozeki, B. S. Kawasaki, “Optical directional coupler using tapered sections in multimode fibers,” Appl. Phys. Lett. 28, 528–529 (1976).
[CrossRef]

T. Ozeki, T. Ito, T. Tamura, “Tapered sections of multimode cladded fibers as mode filters and mode analyzers,” Appl. Phys. Lett. 26, 386–388 (1975).
[CrossRef]

Potter, R. J.

Szarka, F.

F. Szarka, A. Lightstone, J. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integ. Opt. 3, 285–298 (1980).
[CrossRef]

Tamura, T.

T. Ozeki, T. Ito, T. Tamura, “Tapered sections of multimode cladded fibers as mode filters and mode analyzers,” Appl. Phys. Lett. 26, 386–388 (1975).
[CrossRef]

Tynan, R.

Appl. Opt. (1)

Appl. Phys. (1)

T. K. Lim, B. K. Garside, J. P. Marton, “An analysis of optical waveguide tapers,” Appl. Phys. 18, 53–62 (1979).
[CrossRef]

Appl. Phys. Lett. (2)

T. Ozeki, T. Ito, T. Tamura, “Tapered sections of multimode cladded fibers as mode filters and mode analyzers,” Appl. Phys. Lett. 26, 386–388 (1975).
[CrossRef]

T. Ozeki, B. S. Kawasaki, “Optical directional coupler using tapered sections in multimode fibers,” Appl. Phys. Lett. 28, 528–529 (1976).
[CrossRef]

Electron. Lett. (1)

T. Ozeki, B. S. Kawasaki, “Mode behaviour in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (1976).
[CrossRef]

Fiber Integ. Opt. (1)

F. Szarka, A. Lightstone, J. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integ. Opt. 3, 285–298 (1980).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Quantum Electron. (1)

J. E. Midwinter, “The prism-taper coupler for the excitation of single modes in optical transmission fibers,” Opt. Quantum Electron. 7, 297–303 (1975).
[CrossRef]

Proc. IEEE (1)

S. E. Miller, E. A. J. Marcatili, T. Li, “Research towards optical fiber transmission systems. Pt. II,” Proc. IEEE 61, 1726–1751 (1973).
[CrossRef]

Other (2)

T. Okoshi, Optical Fibers (Academic, New York, 1982), p. 41.

N. S. Kapany, Fiber Optics (Academic, New York, 1967), Chap. 2.

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Figures (11)

Fig. 1
Fig. 1

Geometry and coordinate system for tracing ray down a taper.

Fig. 2
Fig. 2

Geometry and coordinate system for tracing ray down tapered section.

Fig. 3
Fig. 3

Geometry for tracing ray in a taper that follows a uniform section. Both sections have the same refractive index.

Fig. 4
Fig. 4

Geometry for a group of parallel rays entering the taper. Both sections have the same refractive index.

Fig. 5
Fig. 5

Cross section in the entrance plane of taper.

Fig. 6
Fig. 6

Geometry for tracing ray in a uniform fiber. The corresponding angles in the media specified by indices of refraction n′ and n″ are θ′ and θ″, respectively. The fiber core of index n1 has a cladding of index n2.

Fig. 7
Fig. 7

The total light (in normalized units) transmitted by either a perfect uniform fiber or a perfect taper as a function of the incident angle. n1 = 1.75, n2 = 1.52, n′ = 1.0, R = rb/ra cos Ω = 0.5, θM′ = 60.1°, θMt′ = 25.7°.

Fig. 8
Fig. 8

The total light transmitted by a perfect taper as a function of the taper ratio R. n1 = 1.75, n2 = 1.52, n′ = 1.0, θN = θc.

Fig. 9
Fig. 9

The normal and the effective numerical apertures of a perfect uniform fiber and a perfect taper as a function of the cladding index n2. n1 = 1.70, n′ = 1.0, R = 0.5.

Fig. 10
Fig. 10

The effective numerical aperture of a taper as a function of R. n1 = 1.70, n2 = 1.40, n′ = 1.0.

Fig. 11
Fig. 11

The ratio of the numerical aperture for skew rays to the numerical aperture for meridional rays and the ratio of the total light transmitted by the skew rays to the total light transmitted by the meridional rays of a perfect taper as a function of R. n1 = 1.70, n2 = 1.40, n′ = 1.0.

Equations (85)

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x 2 r a 2 + y 2 r a 2 = z 2 L 2 ,
x 2 + y 2 = z 2 tan 2 Ω ,
r a = L tan Ω .
x 1 - x 0 L 1 = y 1 - y 0 M 1 = z 1 - z 0 N 1 ,
x 1 = [ ( M 1 2 - N 1 2 tan 2 Ω ) x 0 - L 1 M 1 y 0 + L 1 N 1 z 0 tan 2 Ω ] ± L 1 ζ 1 1 / 2 η 1 ,
y 1 = [ ( L 1 2 - N 1 2 tan 2 Ω ) y 0 - L 1 M 1 x 0 + M 1 N 1 z 0 tan 2 Ω ] ± M 1 ζ 1 1 / 2 η 1 ,
z 1 = [ ( L 1 2 + M 1 2 ) z 0 - L 1 N 1 x 0 - M 1 N 1 y 0 ] ± N 1 ζ 1 1 / 2 η 1 ,
ζ 1 = ( N 1 x 0 - L 1 z 0 ) 2 tan 2 Ω + ( N 1 y 0 - M 1 z 0 ) 2 tan 2 Ω - ( M 1 x 0 - L 1 y 0 ) 2 ,
η 1 = L 1 2 + M 1 2 - N 1 2 tan 2 Ω .
N ^ 1 = ( x 1 cos Ω r 1 , y 1 cos Ω r 1 , - sin Ω ) .
cos ϕ 1 = N ^ 1 · S 1 = ( L 1 x 1 + M 1 y 1 ) cos Ω r 1 - N ^ 1 sin Ω .
S 2 = S 1 - 2 ( S 1 · N ^ 1 ) N ^ 1 .
L 2 = L 1 - 2 x 1 cos Ω r 1 ( S 1 · N ^ 1 ) ,
M 2 = M 1 - 2 y 1 cos Ω r 1 ( S 1 · N ^ 1 ) ,
N 2 = N 1 + 2 sin Ω ( S 1 · N ^ 1 ) .
cos ϕ 1 = cos 2 Ω sin Ω · ζ 1 1 / 2 z 1 .
cos ϕ n = S n · N ^ n = cos 2 Ω sin Ω · ζ n 1 / 2 z n .
ζ 2 = ζ 1 ,
ζ n = ζ 1 .
cos ϕ n = cos 2 Ω sin Ω ζ 1 1 / 2 z n = A z n ,
A = cos 2 Ω sin Ω ζ 1 1 / 2 = cos 2 Ω sin Ω [ ( N 1 x 0 tan Ω - L 1 r a ) 2 + ( N 1 y 0 tan Ω - M 1 r a ) 2 - ( M 1 x 0 - L 1 y 0 ) 2 ] 1 / 2 .
r n cos ϕ n = [ ( N 1 x 0 - L 1 z 0 ) 2 sin 2 Ω + ( N 1 y 0 - M 1 z 0 ) 2 sin 2 Ω - ( M 1 x 0 - L 1 y 0 ) 2 cos 2 Ω ] 1 / 2 = C .
cos ϕ n Z n = A .
Z n = r n tan Ω .
r n cos ϕ n = tan Ω A = C ,
cos ϕ 1 = [ L 0 2 + M 0 2 - ( M 0 x 0 - L 0 y 0 r a ) 2 ] 1 / 2 n 1 ,
S n = ( L n , M n , N n ) ,
L n = sin θ n cos β n , M n = sin θ n sin β n , N n = - cos θ n .
tan β n = M n L n .
γ n = α - β n .
x n = x n cos α - y n sin α , y n = x n sin α - y n cos α .
tan α = y n / x n .
sin γ n = L n y n - M n x n r n ( L n 2 + M n 2 ) 1 / 2 ;
r n sin θ n sin γ n = L n y n - M n x n ,
L n y n - M n x n = L n - 1 y n - 1 - M n - 1 x n - 1 = L 1 y 1 - M 1 x 1 .
r n sin θ n sin γ n = C ,
S 1 = S 1 = S 1 ( L 1 , M 1 , N 1 ) , cos ϕ 0 = sin θ 0 cos γ 0 .
L 1 = - sin θ cos γ ,
M 1 = - sin θ sin γ ,
N 1 = - cos θ .
cos ϕ n = cos Ω r b [ N 1 2 ( x 0 2 + y 0 2 ) tan 2 Ω - 2 N 1 r a ( L 1 x 0 + M 1 y 0 ) tan Ω + L 1 2 r a 2 ] 1 / 2 ,
z n L - l = r b cotan Ω ,
tan θ cos γ t g Ω ,
- L 1 - N 1 tan Ω .
cos ϕ n = r a cos Ω r b cos ϕ 0 .
δ max = r a cos Ω r b N 1 tan Ω ,
δ max = | N 1 tan Ω L 1 | = tan Ω tan θ cos γ 1.
cos Ω 1 ,
r b cos ϕ n = r a cos ϕ 0 .
r n cos ϕ n = constant .
r 1 r a , ϕ 1 ϕ 0 .
r b cos ϕ n r n cos ϕ n = r 1 cos ϕ 1 r a cos ϕ 0 .
cos ϕ n = sin θ n cos γ n , cos ϕ 0 = sin θ 0 cos γ 0 , γ 0 = γ n = 0
r a sin θ 0 = r b sin θ n .
NA m u = ( n 1 2 - n 2 2 ) 1 / 2 ,
F m u = π 2 a 2 I 0 n 1 2 NA m u 2 ,
I ( θ ) = I 0 cos θ ,
cos ϕ = sin θ cos γ ( n 1 2 - n 2 2 ) 1 / 2 n 1 ,
sin θ M = ( n 1 2 - n 2 2 ) 1 / 2 n 1 ,
sin θ c = n n 1 ,
cos γ ¯ = sin θ M sin θ .
0 γ π / 2             for             0 θ θ M ,
γ ¯ γ π / 2             for θ M θ θ c .
F s u = 8 π a 2 θ = 0 θ M γ = 0 π / 2 I ( θ ) t t α η ( θ ) e - β P × cos 2 γ sin θ d γ d θ + 8 π a 2 θ = θ M θ N γ = γ ¯ ( θ ) π / 2 I ( θ ) t t α η ( θ ) e - β P × cos 2 γ sin θ d γ d θ ,
n 1 sin θ N = n sin θ N ,
F s u ( θ N ) = π 2 a 2 I 0 sin 2 θ N - 2 π a 2 I 0 [ sin θ M ( sin 2 θ N - sin 2 θ M ) 1 / 2 + ( sin 2 θ N - 2 sin 2 θ M ) cos - 1 ( sin θ M sin θ N ) ] .
F s u ( θ N ) = π 2 a 2 I 0 sin 2 θ N .
F s u ( θ c ) = π 2 a 2 I 0 n 1 2 NA s u 2 ,
NA s u 2 = n 2 - 2 / π { [ ( n 1 2 - n 2 2 ) ( n 2 - n 1 2 + n 2 2 ) ] 1 / 2 + [ n 2 - 2 ( n 1 2 - n 2 2 ) ] cos - 1 [ ( n 1 2 - n 2 2 ) 1 / 2 n ] }
NA m t = r b r a ( n 1 2 - n 2 2 ) 1 / 2 = r b r a NA m u .
F m t = π 2 r a 2 I 0 n 1 2 NA m t 2 .
cos ϕ n = sin θ n cos γ n ( n 1 2 - n 2 2 ) 1 / 2 n 1 .
sin θ 0 cos γ 0 r b r a cos Ω ( n 1 2 - n 2 2 ) 1 / 2 n 1 .
sin θ M t = r b r a cos Ω ( n 1 2 - n 2 2 ) 1 / 2 n 1 ,
cos γ ¯ t = sin θ M t sin θ .
0 γ 0 π 2             for             0 θ 0 θ M t ,
γ ¯ t γ 0 π 2             for             0 θ 0 θ c .
F s t ( θ N ) = π 2 r a 2 I 0 sin 2 θ N - 2 π r a 2 I 0 [ sin θ M t ( sin 2 θ N - sin 2 θ M t ) 1 / 2 + ( sin 2 θ N - 2 sin 2 θ M t ) cos - 1 ( sin θ M t sin θ N ) ]             for             θ N > θ M t
F s t ( θ N ) = π 2 r a 2 I 0 sin 2 θ N             for             θ N θ M t .
NA s t 2 = n 2 - 2 / π { R ( n 1 2 - n 2 2 ) 1 / 2 [ n 2 - R 2 ( n 1 2 - n 2 2 ) ] 1 / 2 + [ n 2 - 2 R 2 ( n 1 2 - n 2 2 ) ] cos - 1 [ R ( n 1 2 - n 2 2 ) 1 / 2 n ] } ,
R = r b / r a cos Ω .
F s t = π 2 r a 2 I 0 n 1 2 NA s t 2 .
Δ = R ( n 1 2 - n 2 2 ) 1 / 2 n .
NA m = n Δ ,
NA s = n { 1 - 2 π [ Δ ( 1 - Δ 2 ) 1 / 2 + ( 1 - 2 Δ 2 ) cos - 1 Δ ] } 1 / 2 .

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