Abstract

The upside-down taper lens optical fiber end is a new type of fiber end that may be useful for micro-optic image systems. We give the index distribution of this kind of fiber end, and the ray refraction formula at a spherical surface and the ray trace equation are used to analyze the ray transformation properties. Under the paraxial approximation, the optical transformation matrix of the upside-down taper lens fiber end is given.

© 1992 Optical Society of America

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References

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  1. Yuan Libo, Shou 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. Yuan Libo, Shou Reilan, “The light gathering properties and the effect on pulse spreading of an upside down taper lens fiber end,” Opt. Fibers Electr. Cables 1, 33–37 (1990). (In Chinese.)
  3. A. W. Snyder, “Understanding monomode optical fibers,” Proc. IEEE 69, 6–13 (1981).
    [CrossRef]
  4. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 3, p. 97.
  5. 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]
  6. C. Gómez-Reino, J. Linares, “Paraxial Fourier transforming and imaging properties of a GRIN lens with revolution symmetry: GRIN lens law,” Appl. Opt. 25, 3418–3424 (1986).
    [CrossRef] [PubMed]

1990

Yuan Libo, Shou 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]

Yuan Libo, Shou Reilan, “The light gathering properties and the effect on pulse spreading of an upside down taper lens fiber end,” Opt. Fibers Electr. Cables 1, 33–37 (1990). (In Chinese.)

1986

1984

1981

A. W. Snyder, “Understanding monomode optical fibers,” Proc. IEEE 69, 6–13 (1981).
[CrossRef]

Cuadrado, J. M.

Gomez-Reino, C.

Gómez-Reino, C.

Larrea, E.

Libo, Yuan

Yuan Libo, Shou Reilan, “The light gathering properties and the effect on pulse spreading of an upside down taper lens fiber end,” Opt. Fibers Electr. Cables 1, 33–37 (1990). (In Chinese.)

Yuan Libo, Shou 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]

Linares, J.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 3, p. 97.

Perez, M. V.

Reilan, Shou

Yuan Libo, Shou Reilan, “The light gathering properties and the effect on pulse spreading of an upside down taper lens fiber end,” Opt. Fibers Electr. Cables 1, 33–37 (1990). (In Chinese.)

Yuan Libo, Shou 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]

Snyder, A. W.

A. W. Snyder, “Understanding monomode optical fibers,” Proc. IEEE 69, 6–13 (1981).
[CrossRef]

Appl. Opt.

Opt. Fibers Electr. Cables

Yuan Libo, Shou Reilan, “The light gathering properties and the effect on pulse spreading of an upside down taper lens fiber end,” Opt. Fibers Electr. Cables 1, 33–37 (1990). (In Chinese.)

Proc. IEEE

A. W. Snyder, “Understanding monomode optical fibers,” Proc. IEEE 69, 6–13 (1981).
[CrossRef]

Sensors Actuators A

Yuan Libo, Shou 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

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 3, p. 97.

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Figures (3)

Fig. 1
Fig. 1

Cross section of the UDTL fiber end.

Fig. 2
Fig. 2

UDTL fiber end core coordinate system.

Fig. 3
Fig. 3

Refractive-index distribution of the taper core part.

Equations (22)

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R 0 = D 2 + h 2 2 h ,
Ω = arctan ( D L ) .
n 2 ( r ) = n 0 2 ( 1 - A 0 2 r 2 ) ,
n 2 ( r , z ) = n 0 2 [ 1 - A 2 ( z ) r 2 ] ,
r = D - ztg Ω = D ( 1 - z L ) .
n 2 ( r , z ) r = D [ 1 - ( z / L ) ] = n 2 ( r ) r = a 0 ,
n 0 2 [ 1 - A 2 ( z ) D 2 ( 1 - z L ) 2 ] = n 0 2 ( 1 - A 0 2 a 0 2 ) .
A ( z ) = A 0 a 0 D [ 1 - ( z / L ) ] ,             0 z L ( D - a 0 ) D .
n 2 ( r , z ) = n 0 2 { 1 - A 0 2 a 0 2 D 2 [ 1 - ( z / L ) ] 2 r 2 } .
d d s [ n ( r ) d R d s ] = n ( r ) ,
d 2 R ( z ) d z 2 + A 2 ( z ) R ( z ) = 0.
R ( 0 ) = r 0 , d R d z | z = - h = ( n 0 - 1 n 0 R 0 ) r 0 + 1 n 0 θ 0 ,
R ( 0 ) = r 0 , d R d z | z = 0 = ( n 0 - 1 n 0 R 0 ) r 0 + 1 n 0 θ 0 .
R ( z ) = C 1 [ g ( z ) ] 1 / 2 cos [ η ln g ( z ) ] + C 2 [ g ( z ) ] 1 / 2 sin [ η ln g ( z ) ] ,
R ( z ) = r 0 [ g ( z ) ] 1 / 2 { cos [ η ln g ( z ) ] - 1 2 η sin [ η ln g ( z ) ] } + r 0 ( n 0 - 1 ) n 0 R 0 [ g ( z ) ] 1 / 2 sin [ η ln g ( z ) ] - θ 0 L n 0 η [ g ( z ) ] 1 / 2 sin [ η ln g ( z ) ] ,
d R ( z ) d z = r 0 A 0 2 a 0 2 L D 2 [ g ( z ) ] 1 / 2 sin [ η ln g ( z ) ] - r 0 ( n 0 - 1 ) n 0 R 0 [ g ( z ) ] 1 / 2 { cos [ η ln g ( z ) ] + 1 2 η sin [ η ln g ( z ) } + θ 0 1 n 0 [ g ( z ) ] 1 / 2 { cos [ η ln g ( z ) ] + 1 2 η sin [ η ln g ( z ) ] } .
[ r θ ] = [ R 2 ( z ) R 1 ( z ) / n 0 n 0 [ d R 2 ( z ) / d z ] d R 1 ( z ) / d z ] × [ 1 0 ( 1 - n 0 ) / R 0 1 ] [ r 0 θ 0 ] = T 1 · T 2 [ r 0 θ 0 ] ,
T 1 = [ R 2 ( z ) R 1 ( z ) / n 0 n 0 [ d R 2 ( z ) / d z ] d R 1 ( z ) / d z ] , T 2 = [ 1 0 ( 1 - n 0 ) / R 0 1 ] ,
R ( z ) = - L η [ g ( z ) ] 1 / 2 sin [ η ln g ( z ) ] , d R ( z ) d z = 1 [ g ( z ) ] 1 / 2 { cos [ η ln g ( z ) ] + 1 2 η sin [ η ln g ( z ) ] } , R ( z ) = [ g ( z ) ] 1 / 2 { cos [ η ln g ( z ) ] - 1 2 η sin [ η ln g ( z ) ] } , d R ( z ) d z = A 0 2 a 0 2 L D 2 [ g ( z ) ] 1 / 2 sin [ η ln g ( z ) ] .
T = T 1 · T 2 = [ A B C D ] ,
A = R 2 ( z ) - ( n 0 - 1 ) n 0 R 0 R 1 ( z ) , B = 1 n 0 R 1 ( z ) , C = n 0 d R 2 ( z ) d z - ( n 0 - 1 ) R 0 d R 1 ( z ) d z , D = d R 1 ( z ) d z ,
det T = A D - B C = 1.

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