Abstract

The coupling efficiency between two multimode fibers with an angular misalignment is calculated. For the practically interesting cases of parabolic and step-index profiles, closed-form expressions and simple approximations are derived. Furthermore a general loss formula for small tilt angles and arbitrary profile exponents is presented.

© 1983 Optical Society of America

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References

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  1. D. Marcuse, Appl. Opt. 14, 3016 (1975).
    [CrossRef] [PubMed]
  2. R. E. Wagner, C. R. Sandahl, Appl. Opt. 21, 1381 (1982).
    [CrossRef] [PubMed]
  3. S. C. Mettler, Bell Syst. Tech. J. 58, 2163 (1979).
  4. E.-G. Neumann, NTZ Arch. 2, 159 (1980).
  5. E.-G. Neumann, W. Weidhaas, AEU Arch. fuer Elektron. und Uebertragungstech. Electron. and Commun. 30, 448 (1976).
  6. D. Gloge, Bell Syst. Tech. J. 55, 905 (1976).
  7. P. Di Vita, U. Rossi, Opt. Quantum Electron. 10, 107 (1978).
    [CrossRef]
  8. G. Grau, Optische Nachrichtentechnik (Springer, Berlin, 1981), pp.287–293.

1982 (1)

1980 (1)

E.-G. Neumann, NTZ Arch. 2, 159 (1980).

1979 (1)

S. C. Mettler, Bell Syst. Tech. J. 58, 2163 (1979).

1978 (1)

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

1976 (2)

E.-G. Neumann, W. Weidhaas, AEU Arch. fuer Elektron. und Uebertragungstech. Electron. and Commun. 30, 448 (1976).

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

1975 (1)

Di Vita, P.

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

Gloge, D.

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

Grau, G.

G. Grau, Optische Nachrichtentechnik (Springer, Berlin, 1981), pp.287–293.

Marcuse, D.

Mettler, S. C.

S. C. Mettler, Bell Syst. Tech. J. 58, 2163 (1979).

Neumann, E.-G.

E.-G. Neumann, NTZ Arch. 2, 159 (1980).

E.-G. Neumann, W. Weidhaas, AEU Arch. fuer Elektron. und Uebertragungstech. Electron. and Commun. 30, 448 (1976).

Rossi, U.

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

Sandahl, C. R.

Wagner, R. E.

Weidhaas, W.

E.-G. Neumann, W. Weidhaas, AEU Arch. fuer Elektron. und Uebertragungstech. Electron. and Commun. 30, 448 (1976).

AEU Arch. fuer Elektron. und Uebertragungstech. Electron. and Commun. (1)

E.-G. Neumann, W. Weidhaas, AEU Arch. fuer Elektron. und Uebertragungstech. Electron. and Commun. 30, 448 (1976).

Appl. Opt. (2)

Bell Syst. Tech. J. (2)

S. C. Mettler, Bell Syst. Tech. J. 58, 2163 (1979).

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

NTZ Arch. (1)

E.-G. Neumann, NTZ Arch. 2, 159 (1980).

Opt. Quantum Electron. (1)

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

Other (1)

G. Grau, Optische Nachrichtentechnik (Springer, Berlin, 1981), pp.287–293.

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Figures (4)

Fig. 1
Fig. 1

Graphic definitions for an angular misalignment between two fibers.

Fig. 2
Fig. 2

(a) Overlap of the numerical apertures. (b) Plan view of overlap area of the cones.

Fig. 3
Fig. 3

Projection of fiber endfaces within the S tilt plane.

Fig. 4
Fig. 4

Power coupling efficiency of step-index and graded-index fibers as a function of the normalized misalignment sin ( α / 2 ) / sin θ c 0. Solid curves represent Eqs. (24) and (28); dotted curves are the approximations (25) and (29).

Equations (38)

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n 2 ( r ) = { n 0 2 [ 1 2 Δ ( r / a ) g ] for 0 r 1 , n 1 2 for r 1 ,
P 2 = S Ω 2 L cos ϑ d Ω 2 d S 2
P 2 = Ω 2 L cos ϑ d Ω 2 .
Ω 0 = A 0 R 2 ,
Ω 0 = 4 R 2 β = 0 β max ρ min ρ max d A 0 ,
d A 0 = R 2 sin ϑ d ϑ d β ,
β max = arccos ( sin α / 2 sin θ c ) ,
ρ min = R sin α / 2 cos β ,
ρ max = R sin θ c .
P 2 = 4 0 β max ϑ min ϑ max L cos ϑ sin ϑ d ϑ d β .
sin ϑ min = sin α / 2 cos β ,
sin ϑ max = sin θ c .
P 2 ( r ) = 2 L [ sin 2 θ c arctan ( sin 2 θ c sin 2 α / 2 1 ) sin 2 α / 2 ( sin 2 θ c sin 2 α / 2 1 ) 1 / 2 ] .
x 2 + y 2 cos 2 α / 2 = r 2 .
P 2 = 4 0 r max 0 y max ( r ) P 2 ( r ) r d y d r ( r 2 y 2 cos 2 α / 2 ) 1 / 2 .
P 2 = 2 π cos α / 2 0 r max P 2 ( r ) r d r .
r max = a ( 1 sin 2 α / 2 sin 2 θ c 0 ) 1 / g ,
sin 2 θ c 2 Δ [ 1 ( r / a ) g ] .
υ 2 = sin 2 θ c sin 2 α / 2 1 ,
P 2 g = 4 π L a 2 cos α / 2 sin 4 α / 2 sin 2 θ c B [ 0 υ max υ 2 d υ 0 υ max ( υ + υ 3 ) arctan ( υ ) d υ ] .
υ max = ( sin 2 θ c 0 sin 2 α / 2 1 ) 1 / 2 .
P 2 g = 1 4 B [ ( υ max 4 + 2 υ max 2 + 1 ) arctan ( υ max ) 5 3 υ max 3 υ max ] .
P 2 g = 1 4 B [ sin 4 θ c 0 sin 4 α / 2 arctan ( sin 2 θ c 0 sin 2 α / 2 1 ) 1 / 2 5 3 ( sin 2 θ c 0 sin 2 α / 2 1 ) 3 / 2 ( sin 2 θ c 0 sin 2 α / 2 1 ) ] .
P 0 g = 2 π 2 L cos α / 2 0 a 2 Δ [ 1 ( r / a ) 2 ] r d r .
P 0 g = L π 2 a 2 sin 2 θ c 0 2 cos α / 2 ,
η g = 2 π { arccos ( sin α / 2 sin θ c 0 ) sin 4 α / 2 sin 4 θ c 0 [ 5 3 ( sin 2 θ c 0 sin 2 α / 2 1 ) 3 / 2 + ( sin 2 θ c 0 sin 2 α / 2 1 ) 1 / 2 ] } .
η g 1 16 3 π sin α / 2 sin θ c 0 .
P 2 s = 2 L [ 2 π sin 2 θ c 0 sin α / 2 arctan ( sin 2 θ c 0 sin 2 α / 2 1 ) 1 / 2 sin 2 α / 2 ( sin 2 θ c 0 sin 2 α / 2 1 ) 1 / 2 0 a r d r ] ;
P 0 s = 2 P 0 g ,
η s = 2 π [ arctan ( sin 2 θ c 0 sin 2 α / 2 1 ) 1 / 2 sin 2 α / 2 sin 2 θ c 0 ( sin 2 θ c 0 sin 2 α / 2 1 ) 1 / 2 ] .
η s 1 4 π sin α / 2 sin θ c 0 .
P 2 a = 4 π L cos α / 2 ( π 2 0 r max sin 2 θ c r d r 2 sin α / 2 0 r max sin θ c r d r ) .
w = r / a
z 2 = ( r / a ) g
P 2 a = 4 π a 2 L cos α / 2 [ π 2 0 1 2 Δ ( 1 w g ) w d w 4 sin α / 2 2 Δ g 0 1 z 4 / g 1 ( 1 z 2 ) 1 / 2 d z ] .
P 2 a = 4 π a 2 L cos α / 2 { π Δ ( 1 2 1 g + 2 ) sin α / 2 [ Γ ( 2 / g + 1 ) Γ ( 3 / 2 ) Γ ( 2 / g + 3 / 2 ) ] } .
P 0 a = 2 a 2 π 2 L sin 2 θ c 0 cos α / 2 ( 1 2 1 g + 2 ) ,
η a = 1 4 π ( g + 2 ) g sin α / 2 sin θ c 0 [ Γ ( 2 / g + 1 ) Γ ( 3 / 2 ) Γ ( 2 / g + 3 / 2 ) ] .

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