Abstract

We introduce and analyze the upside-down taper lens end drawn from step-index fibers. Also, we model the refractive-index distribution and present the ABCD transformation matrix of this fiber end under paraxial approximation. The analysis can be useful for designing micro-optic image systems and laser diodes to single-mode fiber coupling optics.

© 1998 Optical Society of America

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References

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  1. Y. Libo, S. Reilan, “Formation and power distribution properties of an upside down taper lens at the end of an optical fiber,” Sensors Actuators A 23, 1158–1161 (1990).
    [CrossRef]
  2. L. Yuan, A. Qui, “Analysis of a single-mode fiber with taper lens end,” J. Opt. Soc. Am. A 9, 950–952 (1992).
    [CrossRef]
  3. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 8, p. 286; Chap. 11, p. 443.
  4. C. Gomez-Reino, E. Larrea, M. V. Perez, J. M. Cuadrado, “Transmittance function and modal propagation in a conical gradient-index rod,” Appl. Opt. 23, 1107–1109 (1984).
    [CrossRef] [PubMed]
  5. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 2, p. 37.

1992 (1)

1990 (1)

Y. Libo, S. Reilan, “Formation and power distribution properties of an upside down taper lens at the end of an optical fiber,” Sensors Actuators A 23, 1158–1161 (1990).
[CrossRef]

1984 (1)

Cuadrado, J. M.

Gomez-Reino, C.

Larrea, E.

Libo, Y.

Y. Libo, S. Reilan, “Formation and power distribution properties of an upside down taper lens at the end of an optical fiber,” Sensors Actuators A 23, 1158–1161 (1990).
[CrossRef]

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 8, p. 286; Chap. 11, p. 443.

Perez, M. V.

Qui, A.

Reilan, S.

Y. Libo, S. Reilan, “Formation and power distribution properties of an upside down taper lens at the end of an optical fiber,” Sensors Actuators A 23, 1158–1161 (1990).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 2, p. 37.

Yuan, L.

Appl. Opt. (1)

J. Opt. Soc. Am. A (1)

Sensors Actuators A (1)

Y. Libo, S. Reilan, “Formation and power distribution properties of an upside down taper lens at the end of an optical fiber,” Sensors Actuators A 23, 1158–1161 (1990).
[CrossRef]

Other (2)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chap. 8, p. 286; Chap. 11, p. 443.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 2, p. 37.

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Figures (2)

Fig. 1
Fig. 1

Coordinate system that describes the UDTL step-index fiber end core.

Fig. 2
Fig. 2

Refractive-index distribution of the step-index fiber end core before and after tapering.

Equations (20)

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R 0 = h 2 + D 2 2 h .
r = D 1 - z L ,
n 2 = n 1 2 , n 2 2 , r r a , > a ,
n 2 r ,   z = n 1 2 exp - r 2 g 2 z ,
n 2 | r = D 1 - z L = n 2 2 .
g z = A 1 - z L , 0 z L D - a D ,
A = 1 D 2   ln n 1 n 2 1 / 2 .
n 2 r ,   z = n 1 2 exp - r 2 A 2 1 - z L 2 .
d d s n r ,   z d R d s = n r ,   z ,
δ n r ,   z δ z d r d z + n r ,   z d 2 r d z 2 = δ n r ,   z δ r .
d 2 r d z 2 + A 2 1 - z L 2   r = 0 .
r z = B 1 1 - z L 1 / 2 cos   k z + C 1 1 - z L 1 / 2 sin   k z ,
r 0 = r 0 , d r d z | z = - h d r d z | z = 0 = n 1 - 1 n 1 R 0   r 0 + θ 0 n 1 ,
r z = r 0 1 - z L 1 / 2 cos   k z - 1 2 α sin   k z - r 0 L α n 1 - 1 n 1 R 0 1 - z L 1 / 2 sin   k z - θ 0 n 1 1 - z L 1 / 2 L α sin   k z , d r z d z = r 0 A 2 L α 1 - z L 1 / 2 sin   k z + r 0 n 1 - 1 n 1 R 0 1 - z L 1 / 2 × cos   k z + 1 2 α sin   k z + θ 0 1 n 1 1 - z L 1 / 2 × cos   k z + 1 2 α sin   k z .
r θ = T r 0 θ 0 = T 1 T 2 r 0 θ 0 ,
T 1 = r 2 z d r 2 z d z r 1 z d r 1 z d z ,
r 1 z = - L α 1 - z L 1 / 2 sin   k z , d r 1 z d z = 1 1 - z L 1 / 2 cos   k z + 1 2 α sin   k z , r 2 z = 1 - z L 1 / 2 cos   k z - 1 2 α sin   k z , d r 2 z d z = A 2 L α 1 - z L 1 / 2 sin   k z .
T 2 = 1 n 1 - 1 n 1 R 0 0 1 n 1 .
T = A C B D ,
A = r 2 z + n 1 - 1 n 1 R 0   r 1 z , B = r 1 z n 1 , C = d r 2 z d z + n 1 - 1 n 1 R 0 d r 1 z d z , D = 1 n 1 d r 1 z d z ,

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