Abstract

Analytical and graphical results are presented that describe the changes of a mode as it propagates through a multimode step-index linear optical-fiber taper.

© 1986 Optical Society of America

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References

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  1. 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]
  2. 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]
  3. S. E. Miller, E. A. J. Marcatili, T. Li, “Research towards optical fiber transmission systems. Pt. II,” Proc. IEEE 61, 1726–1751 (1973).
    [CrossRef]
  4. T. Ozeki, B. S. Kawasaki, “Optical directional coupler using tapered sections in multimode fibers,” Appl. Phys. Lett. 28, 528–529 (1976).
    [CrossRef]
  5. F. Szarka, A. Lightstone, J. W. Y. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integr. Opt. 3, 285–298 (1980).
    [CrossRef]
  6. Y. F. Li, J. W. Y. Lit, “Coupling efficiency of a multimode biconical taper coupler,” J. Opt. Soc. Am. A 2, 1301–1306 (1985).
    [CrossRef]
  7. Y. F. Li, J. W. Y. Lit, “Transmission properties of a multimode optical fiber taper,” J. Opt. Soc. Am. A 2, 462–468 (1985).
    [CrossRef]
  8. T. Ozeki, B. S. Kawasaki, “Mode behavior in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (1976).
    [CrossRef]
  9. See Eq. (17) in Ref. 7 and the equations that show L= L″ in Ref. 8.
  10. See Eqs. (9)–(11) in Ref. 8.
  11. In Ref. 8, the refractive index n1of the core and the wavelength λ have not been given. These, we believe, are necessary for the calculation of mb and ub. As a result, we could not compare our results with similar ones in Ref. 8.

1985 (2)

1980 (1)

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

1976 (2)

T. Ozeki, B. S. Kawasaki, “Mode behavior 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]

Hughes, R.

F. Szarka, A. Lightstone, J. W. Y. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integr. 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]

Kawasaki, B. S.

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

T. Ozeki, B. S. Kawasaki, “Mode behavior in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (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]

Li, Y. F.

Lightstone, A.

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

Lit, J. W. Y.

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]

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]

Ozeki, T.

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

T. Ozeki, B. S. Kawasaki, “Mode behavior in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (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]

Szarka, F.

F. Szarka, A. Lightstone, J. W. Y. Lit, R. Hughes, “A review of biconical taper couplers,” Fiber Integr. 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]

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 behavior in a tapered multimode fiber,” Electron. Lett. 12, 407–408 (1976).
[CrossRef]

Fiber Integr. Opt. (1)

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

J. Opt. Soc. Am. A (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 (3)

See Eq. (17) in Ref. 7 and the equations that show L= L″ in Ref. 8.

See Eqs. (9)–(11) in Ref. 8.

In Ref. 8, the refractive index n1of the core and the wavelength λ have not been given. These, we believe, are necessary for the calculation of mb and ub. As a result, we could not compare our results with similar ones in Ref. 8.

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Figures (7)

Fig. 1
Fig. 1

Geometry of a ray in a taper section. The ellipse is a cross section normal to the axis of the taper.

Fig. 2
Fig. 2

Geometry for tracing a ray in a multimode linear fiber taper.

Fig. 3
Fig. 3

Change in radial mode order m of a ray between the entrance and the exit planes of a taper as a function of the input parameter La/ua. Δ = 0.1, va = 33.0, λ = 0.6328 μm, n1 = 1.49, Ω = 0.001 rad, and R = 5.0. The solid lines are obtained from Eq. (25). ▷ and ◇ are from ray tracing. A, ua/va = 0.3; B, ua/va = 1.0.

Fig. 4
Fig. 4

Change in m of a ray between the entrance and the exit planes of a taper as a function of La/ua. Δ = 0.1 va = 33.0, λ = 0.6328 μm, n1 = 1.49, Ω = 0.01 rad, and R = 10.0. A, ua/va = 0.8; B, ua/va = 1.0.

Fig. 5
Fig. 5

Function y = (x−2 − 1)1/2 − cos−1x against x = L/u [see expression (27)].

Fig. 6
Fig. 6

ua and ub, Lateral component of a wave vector multiplied by the core radius in the entrance and the exit planes, respectively. ub/ua − 1 is the fractional change of u between the two planes. Solid lines are obtained from Eq. (22), dashed lines from Eq. (29), and ◇ and ⬠ from ray tracing. A, ua/va = 0.3; B, ua/va = 1.0. Values of the other parameters are the same as those in Fig. 3.

Fig. 7
Fig. 7

Same as Fig. 6. Δ = 0.1, va = 33.0, λ = 0.6328 μm, n1 = 1.49, Ω = 0.01 rad, and R = 10.0. A, ua/va = 0.3; B, ua/va = 1.0.

Equations (29)

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r cos ϕ = C 1
r sin θ sin γ = C 2 .
cos ϕ = sin θ cos γ cos Ω - cos θ sin Ω .
r n sin θ n sin γ n = r 1 sin θ 1 sin γ 1 ,
r n ( sin θ n cos γ n cos Ω - cos θ n sin Ω ) = r 1 ( sin θ 1 cos γ 1 cos Ω - cos θ 1 sin Ω ) .
A = r 1 r n ( sin θ 1 cos γ 1 - cos θ 1 tan Ω )
B = r 1 r n sin θ 1 sin γ 1 ,
sin θ n = [ A 2 + B 2 + 2 A tan Ω ( 1 - A 2 - B 2 ) 1 / 2 ] 1 / 2
sin γ n = B / sin θ n ,
1 + tan 2 Ω 1 ,
tan 2 Ω 1.
r 1 r a ,
r n r b ,
r 1 r n r a r b = 1 F ;
A = 1 R ( sin θ 1 cos γ 1 - cos θ 1 tan Ω ) ,
B = 1 R sin θ 1 sin γ 1 .
A = 1 R { 2 Δ u a v a [ 1 - ( L a u a ) 2 ] 1 / 2 - tan Ω [ 1 - 2 Δ ( u a v a ) 2 ] 1 / 2 } ,
B = 1 R 2 Δ × u a v a × L a u a = 2 Δ R L a v a ,
v a = k n 1 r a ( 2 Δ ) 1 / 2
Δ = ( n 1 2 - n 2 2 ) / 2 n 1 2 ,
( n 1 - n 2 ) / n 1 .
u b = r b n 1 k sin θ n = r b n 1 k [ A 2 + B 2 + 2 A tan Ω ( 1 - A 2 - B 2 ) 1 / 2 ] 1 / 2
m b = u b π { [ 1 - ( L b u b ) 2 ] 1 / 2 - L b u b cos - 1 ( L b u b ) } ,
L b / u b = B / [ A 2 + B 2 + 2 A tan Ω ( 1 - A 2 - B 2 ) 1 / 2 ] 1 / 2 .
Δ m = m b - m a = u b π { [ 1 - ( L b u b ) 2 ] 1 / 2 - L b u b cos - 1 ( L b u b ) } - u a π { [ 1 - ( L a u a ) 2 ] 1 / 2 - L a u a cos - 1 ( L a u a ) } = L π ( { [ ( L b u b ) 2 - 1 ] 1 / 2 - cos - 1 ( L b u b ) } - { [ ( L a u a ) 2 - 1 ] 1 / 2 - cos - 1 ( L a u a ) } ) .
Δ m < 1 ,
- π L < { [ ( L b u b ) - 2 - 1 ] 1 / 2 - cos - 1 ( L b u b ) } - { [ ( L a u a ) - 2 - 1 ] 1 / 2 - cos - 1 ( L a u a ) } < π L .
u b / u a - 1 = Ω ( R - 1 ) cos γ a / sin θ a ,
= Ω ( R - 1 ) [ 1 - ( L a / u a ) 2 ] 1 / 2 / [ ( 2 Δ ) 1 / 2 u a / v a ] .

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