Abstract

GRIN-rod lenses, ball lenses, and plano-convex rod lenses are currently proposed for fiber-optic devices. From analyses of the aberrations of these lens types it turned out that equivalent lenses in a well-designed coupling have nearly the same aberration. We derive valuable relations between the coupling efficiency and the optimal lens parameters. For each above-mentioned lens type, coupling efficiencies of 95% (0.2-dB insertion loss) can be obtained if typical communication fibers are incorporated. We also present a lens connector that has been fabricated in a factory-type production line. Its performance reaches the theoretical limits. Finally the influence of misalignments in the lens coupling system is investigated.

© 1981 Optical Society of America

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References

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  1. K. Thyagarajan, A. Rohra, A. K. Ghatak, Appl. Opt. 19, 266 (1980).
    [CrossRef] [PubMed]
  2. W. J. Tomlinson, Appl. Opt. 19, 1117 (1980).
    [CrossRef] [PubMed]
  3. W. J. Tomlinson, Appl. Opt. 16, 2180 (1977).
    [CrossRef] [PubMed]
  4. K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimuto, Fiber Integrated Opt. 2, 1 (1979).
    [CrossRef]
  5. A. Nicia, Electron. Lett. 14, 511 (1978).
    [CrossRef]
  6. J. C. North, J. H. Stewart, in Digest, Fifth ECOC, Amsterdam, paper 9-4.
  7. K. Iga, Appl. Opt. 19, 1039 (1980).
    [CrossRef] [PubMed]
  8. M. Kawazu, Y. Ogura, Appl. Opt. 19, 1105 (1980).
    [CrossRef] [PubMed]
  9. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), pp. 84–87.
  10. K. Aoyama, J. Minowa, Appl. Opt. 18, 2834 (1979).
    [CrossRef] [PubMed]
  11. S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
    [CrossRef]
  12. D. T. Moore, Appl. Opt. 19, 1035 (1980).
    [CrossRef] [PubMed]
  13. A. R. Cooper, M. A. Él-Leil, Appl. Opt. 19, 1087 (1980).
    [CrossRef] [PubMed]
  14. SLS Selfoc Micro Lens, Nippon Sheet Glass Co. Ltd.
  15. Optical Glass Catalog 3050/66, Schott.
  16. C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).
  17. D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).
  18. Y. Daido, T. Iwama, E. Miyauchi, Trans. Inst. Electron Commun. Eng. Jpn. 61-E, 816 (1978).
  19. P. di Vita, U. Rossi, Opt. Quantum Electron. 10, 107 (1978).
    [CrossRef]
  20. K. Nawata, IEEE J. Quantum Electron. QE-16, 618 (1980).
    [CrossRef]

1980 (7)

1979 (2)

K. Aoyama, J. Minowa, Appl. Opt. 18, 2834 (1979).
[CrossRef] [PubMed]

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimuto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

1978 (4)

A. Nicia, Electron. Lett. 14, 511 (1978).
[CrossRef]

C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).

Y. Daido, T. Iwama, E. Miyauchi, Trans. Inst. Electron Commun. Eng. Jpn. 61-E, 816 (1978).

P. di Vita, U. Rossi, Opt. Quantum Electron. 10, 107 (1978).
[CrossRef]

1977 (1)

1976 (1)

D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).

1968 (1)

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Aoyama, K.

Cooper, A. R.

Daido, Y.

Y. Daido, T. Iwama, E. Miyauchi, Trans. Inst. Electron Commun. Eng. Jpn. 61-E, 816 (1978).

di Vita, P.

P. di Vita, U. Rossi, Opt. Quantum Electron. 10, 107 (1978).
[CrossRef]

Él-Leil, M. A.

Ghatak, A. K.

Gloge, D.

D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).

Iga, K.

Ishikawa, R.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimuto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

Iwama, T.

Y. Daido, T. Iwama, E. Miyauchi, Trans. Inst. Electron Commun. Eng. Jpn. 61-E, 816 (1978).

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), pp. 84–87.

Kawakami, S.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Kawazu, M.

Kobayashi, K.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimuto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

Mettler, S. C.

C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).

Miller, C. M.

C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).

Minemura, K.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimuto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

Minowa, J.

Miyauchi, E.

Y. Daido, T. Iwama, E. Miyauchi, Trans. Inst. Electron Commun. Eng. Jpn. 61-E, 816 (1978).

Moore, D. T.

Nawata, K.

K. Nawata, IEEE J. Quantum Electron. QE-16, 618 (1980).
[CrossRef]

Nicia, A.

A. Nicia, Electron. Lett. 14, 511 (1978).
[CrossRef]

Nishizawa, J.

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

North, J. C.

J. C. North, J. H. Stewart, in Digest, Fifth ECOC, Amsterdam, paper 9-4.

Ogura, Y.

Rohra, A.

Rossi, U.

P. di Vita, U. Rossi, Opt. Quantum Electron. 10, 107 (1978).
[CrossRef]

Stewart, J. H.

J. C. North, J. H. Stewart, in Digest, Fifth ECOC, Amsterdam, paper 9-4.

Sugimuto, S.

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimuto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

Thyagarajan, K.

Tomlinson, W. J.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), pp. 84–87.

Appl. Opt. (8)

Bell Syst. Tech. J. (2)

C. M. Miller, S. C. Mettler, Bell Syst. Tech. J. 57, 3167 (1978).

D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).

Electron. Lett. (1)

A. Nicia, Electron. Lett. 14, 511 (1978).
[CrossRef]

Fiber Integrated Opt. (1)

K. Kobayashi, R. Ishikawa, K. Minemura, S. Sugimuto, Fiber Integrated Opt. 2, 1 (1979).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. Nawata, IEEE J. Quantum Electron. QE-16, 618 (1980).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
[CrossRef]

Opt. Quantum Electron. (1)

P. di Vita, U. Rossi, Opt. Quantum Electron. 10, 107 (1978).
[CrossRef]

Trans. Inst. Electron Commun. Eng. Jpn. (1)

Y. Daido, T. Iwama, E. Miyauchi, Trans. Inst. Electron Commun. Eng. Jpn. 61-E, 816 (1978).

Other (4)

J. C. North, J. H. Stewart, in Digest, Fifth ECOC, Amsterdam, paper 9-4.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1976), pp. 84–87.

SLS Selfoc Micro Lens, Nippon Sheet Glass Co. Ltd.

Optical Glass Catalog 3050/66, Schott.

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Figures (19)

Fig. 1
Fig. 1

Basic structure of lens coupling.

Fig. 2
Fig. 2

Fundamental imaging parameters: (a) GRIN-rod lens; (b) ball lens; and (c) rod lens.

Fig. 3
Fig. 3

Universal arrangement for optimal lens coupling.

Fig. 4
Fig. 4

Effect of spherical aberration.

Fig. 5
Fig. 5

Influence of material refractive index on the characteristic transverse spherical aberration /fN.A.3 of the lens coupling system.

Fig. 6
Fig. 6

Relationship between transverse spherical aberration and launching angle θ showing the effect of defocused fiber ends. Solid curves were obtained by exact ray-tracing; dashed curves by third-order aberration theory: (a) using high refractive-index ball lenses; (b) using low refractive-index ball lenses.

Fig. 7
Fig. 7

Correspondence between separation mismatch in butt joint connections and aberration in a lens coupling.

Fig. 8
Fig. 8

Distribution of light of a fiber end.

Fig. 9
Fig. 9

Definition of elemental area element dA.

Fig. 10
Fig. 10

Relationship between coupling efficiency η and defocusing of the fiber ends for a step-index fiber with 100-μm core diam and N.A. = 0.3. Solid curves were obtained by exact ray-tracing techniques, dashed curves by using the approximations derived in Sec. IV: (a) using high refractive-index ball lenses; (b) using low refractive-index ball lenses.

Fig. 11
Fig. 11

Relationship between coupling efficiency η and defocusing of the fiber ends for a graded-index fiber with 50 μm core diam and N.A. = 0.21. Solid curves were obtained by exact ray-tracing techniques, dashed curves by using the approximations derived in Sec. IV: (a) using high refractive-index ball lenses; (b) using low refractive-index ball lenses.

Fig. 12
Fig. 12

Visualization of the model used for deriving approximated relations between coupling efficiency and mismatch in lens connections.

Fig. 13
Fig. 13

Relationship between coupling efficiency and basic mismatches if ball lenses with r = 2476 μm, n = 1.83, and graded-index fibers with 50-μm core diam, N.A. = 0.21 are incorporated. Solid curves were obtained by exact ray-tracing techniques, dashed curves by using the approximations derived in Sec. V.

Fig. 14
Fig. 14

Lens connector: (a) separated; (b) connected.

Fig. 15
Fig. 15

(a) Extended view of the lens connector; (b) exploded view of the lens connector.

Fig. 16
Fig. 16

Coupling efficiency of 70 lens connectors, using ball lenses with r = 2476 μm and n = 1.83 and incorporating step-index fibers with 100-μm core diam, N.A. = 0.3.

Fig. 17
Fig. 17

Coupling efficiency of 68 lens connectors, using ball lenses with r = 2476 μm and n = 1.83 and incorporating graded-index fibers with 50-μm core diam, N.A. = 0.26.

Fig. 18
Fig. 18

Ray path in a GRIN-rod lens with a half-pitch length.

Fig. 19
Fig. 19

Ray path in the ball lens system.

Equations (70)

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n 2 ( r ) = n 0 2 [ 1 - ( g r ) 2 + h 4 ( g r ) 4 + h 6 ( g r ) 6 ] ,
f = 1 / n 0 g ,
H = 1 / n 0 g .
f = n 2 ( n - 1 ) r ,
H = r .
f = 1 n - 1 r ,
H = 1 n - 1 r .
n 2 ( r ) = n 0 2 [ 1 - ( g r ) 2 ] .
E = π 2 1 n 0 2 f N . A . 3 ,
E = ¼ [ n ( n - 1 ) 2 - 1 ] f N . A . 3 ,
E = 1 n 2 ( n - 1 ) 2 f N . A . 3 ,
E = - 2 z θ + ¼ [ n ( n - 1 ) 2 - 1 ] f θ 3 ,
n ( ρ ) = n a [ 1 - 2 Δ ( ρ / a ) α ] 1 / 2 , ρ < a , = n a ( 1 - 2 Δ ) 1 / 2 , ρ a ,
N . A . ( ρ ) = N . A . [ 1 - ( ρ / a ) α ] 1 / 2 , ρ < a , = 0 , ρ a ,
P = L cos θ d A d Ω ,
d Ω = sin θ d θ d φ .
d A = 2 p a cos γ d γ ,
p 2 / a 2 = ρ 1 / a 2 - sin 2 γ ,
( ρ 1 / a ) 2 = ( 1 - sin 2 θ N . A . 2 ) 2 / α .
d A = 2 a 2 cos γ [ ( 1 - sin 2 θ N . A . 2 ) 2 / α - sin 2 γ ] 1 / 2 d γ .
P = 8 π a 2 L θ = 0 θ c γ = 0 γ 1 ( θ ) [ ( 1 - sin 2 θ N . A . 2 ) 2 / α - sin 2 γ ] 1 / 2 × cos γ sin θ cos θ d γ d θ ,
θ c = arcsin ( N . A . ) ,
sin γ 1 = ( 1 - sin 2 θ N . A . 2 ) 1 / α .
P 0 = α α + 2 π 2 a 2 L N . A . 2 .
P = 8 π a 2 L θ = 0 θ 1 γ = 0 γ 2 ( θ ) { [ ( 1 - sin 2 θ N . A . 2 ) 2 / α - sin 2 γ ] 1 / 2 - C ( θ ) } × cos γ sin θ cos θ d γ d θ ,
C ( θ ) = E ( θ ) / 2 a .
sin 2 γ 2 = ( 1 - sin 2 θ N . A . 2 ) 2 / α - C 2 .
C ( θ 1 ) = ( 1 - sin 2 θ 1 N . A . 2 ) 1 / α .
η = P / P 0 ,
sin x = sin θ N . A . ,
η = 4 π α + 2 α 0 x 1 sin x cos x × [ cos 4 / α x arccos ( C cos - 2 / α x ) - ( cos 4 / α x - C 2 ) 1 / 2 C ] d x ,
C ( x 1 ) = cos 2 / α x 1 .
η = 1 - 8 π α + 2 α 0 π / 2 sin x cos 1 + 2 / α x C d x .
C ( x ) = - Δ a sin x + Δ a sin 3 x .
η = 1 - 8 π α + 2 α 0 π / 2 sin x cos x 1 + 2 / α × - Δ s sin x + Δ a sin 3 x d x .
η s . i . = 1 - 8 π 0 π / 2 sin x cos x - Δ s sin x + Δ a sin 3 x d x .
η s . i . = 1 - 8 π [ 1 5 - 1 3 Δ s Δ a + 4 15 ( Δ s Δ a ) 5 / 2 ] Δ a .
η s . i . = 1 - 0.19 Δ a
Δ s = 0.63 Δ a
η g . i . = 1 - 16 π 0 π / 2 sin x cos 2 x - Δ s sin x + Δ a sin 3 x d x .
η g . i . = 1 - 16 π [ π 32 - x 0 8 - ( π 16 - x 0 4 ) sin 2 x 0 + ( 1 8 sin x 0 - 1 6 sin 3 x 0 + 1 6 sin 5 x 0 ) cos x 0 ] Δ a ,
η g . i . = 1 - 0.21 Δ a
Δ s = 0.5 Δ a
η s . i . = 1 - 8 π | 1 3 Δ s + 1 5 Δ a | ,             Δ s / Δ a > 1 ,             Δ s / Δ a < 0 ,
η g . i . = 1 - | - Δ s + 1 2 Δ a | ,             Δ s / Δ a > 1 , Δ s / Δ a < 0 ,
a ˜ = f N . A . ,
N . A . ˜ = a / f .
η s . i . = η opt ( 1 - 0.64 d / f N . A . ) ,
η g . i . = η opt ( 1 - 0.85 d / f N . A . ) ;
η s . i . = η opt ( 1 - 0.64 α f / a ) ,
η g . i . = η opt ( 1 - 0.85 α f / a ) ;
η s . i . = η opt ( 1 - 0.43 s a / f 2 N . A . ) ;
η g . i . = η opt ( 1 - 0.5 s a / f 2 N . A . ) .
η g . i . = 1 - ½ Δ a .
C 0 2 d 2 r d z 2 + n 0 2 g r = 0 ,
C 0 = n ( r ) cos θ .
d 2 r d z 2 + g 2 cos 2 θ 0 r = 0 ,
r = sin θ 0 g sin ( g z cos θ 0 ) .
E = sin θ 0 g sin ( π cos θ 0 ) .
E - ½ π g θ 0 3 .
sin θ 1 = ( f / r ) sin θ ,
δ = 2 ( θ 1 - θ 2 ) - θ .
δ = 2 [ arcsin ( f r sin θ ) - arcsin ( f n r sin θ ) ] - θ .
δ 1 8 [ n ( n - 1 ) 2 - 1 ] θ 3 .
φ = 2 ( φ 1 - φ 2 ) + δ .
φ θ + 2 δ .
b = r sin φ 1 sin φ .
E = ( f - r sin φ 1 sin φ ) tan φ .
E = [ 1 - sin θ sin ( θ + 2 δ ) ] f tan ( θ + 2 δ ) ,
E 2 δ f .

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