Abstract

The power launching into multimode optical fibers with any profile of refractive index is analyzed by means of the geometrical optics. First two formulas of the numerical aperture are obtained, one including and the other excluding the contribution of leaky skew rays. Each one of these two numerical apertures has a certain domain of application, and the launching efficiencies from incoherent sources into optical fibers are calculated in both cases. Useful formulas and plots of launching efficiency for a few refractive index profiles are given. Finally the power losses due to source-fiber coupling errors (separation, displacement, or misalignment) are evaluated, and an interesting effect of virtual lengthening of the fiber is found.

© 1976 Optical Society of America

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References

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  1. S. Kawakami, J. Nishizawa, IEEE Trans. Microwave Theory Tech. MTT-16, 814 (1968).
    [CrossRef]
  2. D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).
  3. R. Bouillie, A. Cozannet, K. H. Steiner, M. Treheux, Appl. Opt. 13, 1045 (1974).
    [CrossRef] [PubMed]
  4. R. Olshansky, D. B. Keck, Appl. Opt. 15, 483 (1976).
    [CrossRef] [PubMed]
  5. L. B. Felsen, Electron. Lett. 10, 95 (1974).
    [CrossRef]
  6. A. W. Snyder, Appl. Phys. 4, 273 (1974).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).
  8. B. G. Levich, Theoretical Physics (Quantum Mechanics) (North Holland, Amsterdam, 1973), Vol. 3.
  9. P. Di Vita, R. Vannucci, “Loss Mechanisms of Leaky Skew Rays in Optical Fibres,” to be published.
  10. M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 238 (1975).
    [CrossRef]
  11. P. Di Vita, R. Vannucci, Appl. Phys. 7, 249 (1975).
    [CrossRef]
  12. F. Albertin, P. Di Vita, R. Vannucci, Opto-electron. 6, 369 (1974).
    [CrossRef]
  13. P. Di Vita, R. Vannucci, Opt. Commun. 14, 139 (1975).
    [CrossRef]
  14. H. Matsumura, Opt. Quantum Electron. 7, 81 (1975).
    [CrossRef]

1976 (1)

1975 (4)

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 238 (1975).
[CrossRef]

P. Di Vita, R. Vannucci, Appl. Phys. 7, 249 (1975).
[CrossRef]

P. Di Vita, R. Vannucci, Opt. Commun. 14, 139 (1975).
[CrossRef]

H. Matsumura, Opt. Quantum Electron. 7, 81 (1975).
[CrossRef]

1974 (4)

F. Albertin, P. Di Vita, R. Vannucci, Opto-electron. 6, 369 (1974).
[CrossRef]

L. B. Felsen, Electron. Lett. 10, 95 (1974).
[CrossRef]

A. W. Snyder, Appl. Phys. 4, 273 (1974).
[CrossRef]

R. Bouillie, A. Cozannet, K. H. Steiner, M. Treheux, Appl. Opt. 13, 1045 (1974).
[CrossRef] [PubMed]

1973 (1)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

1968 (1)

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

Adams, M. J.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 238 (1975).
[CrossRef]

Albertin, F.

F. Albertin, P. Di Vita, R. Vannucci, Opto-electron. 6, 369 (1974).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

Bouillie, R.

Cozannet, A.

Di Vita, P.

P. Di Vita, R. Vannucci, Appl. Phys. 7, 249 (1975).
[CrossRef]

P. Di Vita, R. Vannucci, Opt. Commun. 14, 139 (1975).
[CrossRef]

F. Albertin, P. Di Vita, R. Vannucci, Opto-electron. 6, 369 (1974).
[CrossRef]

P. Di Vita, R. Vannucci, “Loss Mechanisms of Leaky Skew Rays in Optical Fibres,” to be published.

Felsen, L. B.

L. B. Felsen, Electron. Lett. 10, 95 (1974).
[CrossRef]

Gloge, D.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Kawakami, S.

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

Keck, D. B.

Levich, B. G.

B. G. Levich, Theoretical Physics (Quantum Mechanics) (North Holland, Amsterdam, 1973), Vol. 3.

Marcatili, E. A. J.

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Matsumura, H.

H. Matsumura, Opt. Quantum Electron. 7, 81 (1975).
[CrossRef]

Nishizawa, J.

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

Olshansky, R.

Payne, D. N.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 238 (1975).
[CrossRef]

Sladen, F. M. E.

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 238 (1975).
[CrossRef]

Snyder, A. W.

A. W. Snyder, Appl. Phys. 4, 273 (1974).
[CrossRef]

Steiner, K. H.

Treheux, M.

Vannucci, R.

P. Di Vita, R. Vannucci, Opt. Commun. 14, 139 (1975).
[CrossRef]

P. Di Vita, R. Vannucci, Appl. Phys. 7, 249 (1975).
[CrossRef]

F. Albertin, P. Di Vita, R. Vannucci, Opto-electron. 6, 369 (1974).
[CrossRef]

P. Di Vita, R. Vannucci, “Loss Mechanisms of Leaky Skew Rays in Optical Fibres,” to be published.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

Appl. Opt. (2)

Appl. Phys. (2)

A. W. Snyder, Appl. Phys. 4, 273 (1974).
[CrossRef]

P. Di Vita, R. Vannucci, Appl. Phys. 7, 249 (1975).
[CrossRef]

Bell Syst. Tech. J. (1)

D. Gloge, E. A. J. Marcatili, Bell Syst. Tech. J. 52, 1563 (1973).

Electron. Lett. (2)

L. B. Felsen, Electron. Lett. 10, 95 (1974).
[CrossRef]

M. J. Adams, D. N. Payne, F. M. E. Sladen, Electron. Lett. 11, 238 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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

Opt. Commun. (1)

P. Di Vita, R. Vannucci, Opt. Commun. 14, 139 (1975).
[CrossRef]

Opt. Quantum Electron. (1)

H. Matsumura, Opt. Quantum Electron. 7, 81 (1975).
[CrossRef]

Opto-electron. (1)

F. Albertin, P. Di Vita, R. Vannucci, Opto-electron. 6, 369 (1974).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1964).

B. G. Levich, Theoretical Physics (Quantum Mechanics) (North Holland, Amsterdam, 1973), Vol. 3.

P. Di Vita, R. Vannucci, “Loss Mechanisms of Leaky Skew Rays in Optical Fibres,” to be published.

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Figures (9)

Fig. 1
Fig. 1

Angular and position coordinate of a light ray in an optical fiber.

Fig. 2
Fig. 2

Launching efficiencies Λ* (full lines) and Λ (chain lines) against the exponent x of the refractive index profile (38) in fibers with n1 = 1.456, (n02n12)1/2 = 0.140, n′ = 1, a = b, T0 = 1. The different curves are calculated for increasing values of the source parameter l.

Fig. 3
Fig. 3

Ratio of excited leaky skew rays: η, against the exponent x of the profile (38) in fibers with the same data of Fig. 2 and for different values of the source parameter l.

Fig. 4
Fig. 4

Launching efficiencies Λ* (full lines) and Λ (chain lines) against the source radius b (normalized to the fiber radius a), for different values of the exponent x of the profile (38), in fibers with the same data of Fig. 2 and for a Lambertian source.

Fig. 5
Fig. 5

Geometry of coupling errors between source and fiber.

Fig. 6
Fig. 6

Launching efficiencies Λ* (full lines) and Λ (chain lines) against the separation s (normalized to a), for different values of the exponent x of the profile (38), in fibers with the same data of Fig. 2 and for a Lambertian source.

Fig. 7
Fig. 7

Profiles of the acceptance cone for meridional rays for different values of the exponent x of the profile (38) (for x → +∞ the step index fiber is obtained) in fibers with the same data of Fig. 2.

Fig. 8
Fig. 8

Launching efficiencies Λ* (full lines) and Λ (chain lines) against the displacement u (normalized to a) for different values of the exponent x of the profile (38), in fibers with the same data of Fig. 2, for a Lambertian source, and for s = a/2.

Fig. 9
Fig. 9

Launching efficiencies Λ* (full lines) and Λ (chain lines) against the misalignment angle α, for different values of the exponent x of the profile (38), in fibers with the same data of Fig. 2, for a Lambertian source, and for s = a.

Equations (49)

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A ( r ) · exp [ 2 π i S ( r ) / λ 0 ] ,
( S ) 2 = n 2 ( r ) ,
S ( r , ψ , z ) = ± r [ n 2 ( r ) - k 2 - h 2 r 2 ] 1 / 2 d r + h ψ + k z ,
r = r ( z ) , ψ = ψ ( z ) , z = z ,
k r ˙ = ± [ n 2 ( r ) - k 2 - ( h 2 / r 2 ) ] 1 / 2 ,
k r 2 ψ ˙ = h
h = n r 2 ψ ˙ ( r ˙ 2 + r 2 ψ ˙ 2 + 1 ) 1 / 2 ;
k = n ( r ˙ 2 + r 2 ψ ˙ 2 + 1 ) 1 / 2 .
h = n · r · sin φ · sin θ ;
k = n · cos θ .
h = n ( r 0 ) · r 0 · sin φ 0 · sin θ 0 ;
k = n ( r 0 ) · cos θ 0 .
n 2 ( r c ) - k 2 - ( h 2 / r c 2 ) = 0.
n 2 ( r ) - k 2 - ( h 2 / r 2 ) = ( k 2 r ˙ 2 )
n 2 ( a ) - k 2 - ( h 2 / a 2 ) = 0 ,
A n ( r 0 ) · sin θ 0 M ,
A ( r 0 , φ 0 ) = { [ n 2 ( r 0 ) - n 1 2 1 - r 0 2 a 2 sin 2 φ 0 ] 1 / 2 for n ( r 0 ) · r 0 · sin φ 0 a n 1 , n ( r 0 ) , for n ( r 0 ) · r 0 sin φ 0 a n 1 .
k = n 1 .
A * ( r 0 ) = [ n 2 ( r 0 ) - n 1 2 ] 1 / 2 .
Λ W 0 / W .
r = r 0 , ψ = ψ 0 , θ = θ 0 , φ = φ 0 ,
n sin θ 0 = n ( r 0 ) sin θ 0 .
R ( r 0 , ψ 0 , θ 0 , φ 0 ) = R ( r , ψ , θ , φ ) .
W = 0 b r d r 0 2 π d ψ 0 2 π d φ × 0 π / 2 R ( r , ψ , θ , φ ) sin θ · cos θ d θ ,
W 0 = 0 a r 0 d r 0 0 2 π d ψ 0 0 2 π d φ 0 0 θ M R ( r 0 , ψ 0 , θ 0 , φ 0 ) × T ( r 0 , θ 0 ) · sin θ 0 · cos θ 0 d θ 0 ,
n sin θ M ( r 0 , φ 0 ) = A ( r 0 , φ 0 ) ,
n sin θ M ( r 0 ) = A * ( r 0 ) .
T ( r 0 , θ 0 ) = cos 2 ( θ 0 - θ 0 ) - cos 2 ( θ 0 + θ 0 ) 2 sin 2 ( θ 0 + θ 0 ) · [ 1 + 1 cos 2 ( θ 0 - θ 0 ) ] ,
T ( r 0 ) = T ( r 0 , 0 ) = 4 n · n ( r 0 ) [ n + n ( r 0 ) ] 2 .
T 0 M = n 0 ( n + n ¯ ) 2 ( n 0 n ¯ + n 2 ) · ( n 0 + n ¯ ) ,
Λ = 2 T 0 b 2 n 2 0 a n 2 ( r 0 ) - n 1 2 ( 1 - r 0 2 / a 2 ) 1 / 2 r 0 d r 0 , Λ * = 2 T 0 b 2 n 2 0 a [ n 2 ( r 0 ) - n 1 2 ] r 0 d r 0 .
W ( r 0 ) = n 2 ( r 0 ) - n 1 2 n 0 2 - n 1 2 · W ( 0 ) ( 1 - r 0 2 a 2 ) 1 / 2 ;             W * ( r 0 ) = n 2 ( r 0 ) - n 1 2 n 0 2 - n 1 2 · W * ( 0 )
n 2 ( r ) = 0 β j j r 2 j ;
Λ = 2 T 0 n 2 · a 2 b 2 1 β j j a 2 j [ ( 2 j ) ! ! ( 2 j + 1 ) ! ! - 1 ]
Λ * = T 0 n 2 · a 2 b 2 1 β j j a 2 j · ( - j j + 1 ) .
Λ = 2 3 T 0 a 2 b 2 · n 0 2 - n 1 2 n 2 ;
Λ * = 1 2 T 0 a 2 b 2 · n 0 2 - n 1 2 n 2 .
n 2 ( r ) = n 0 2 ( 1 - β r x ) ; x > 0 ,
Λ = 2 T 0 · a 2 b 2 · n 0 2 - n 1 2 n 2 · [ 1 - π 2 · Γ ( x + 2 2 ) / Γ ( x + 3 2 ) ]
Λ * = T 0 · a 2 b 2 · n 0 2 - n 1 2 n 2 · ( x x + 2 ) .
R ( θ ) = R 0 · cos l θ ; l 0 ,
Λ = T 0 π b 2 0 a r 0 d r 0 · 0 2 π ( 1 - { 1 - n 2 ( r 0 ) - n 1 2 n 2 [ 1 - ( r 0 2 / a 2 ) sin 2 φ 0 ] } ( l + 2 ) / 2 ) d φ 0 ,
Λ * = 2 T 0 b 2 0 a { 1 - [ 1 - n 2 ( r 0 ) - n 1 2 n 2 ] ( l + 2 ) / 2 } r 0 d r 0 .
η [ ( Λ - Λ * ) / Λ ] = 1 - ( Λ * / Λ ) ,
Λ * ( b ) = T 0 n 2 · 2 [ n 2 ( b ) - n 1 2 ] + x [ n 0 2 - n 1 2 ] 2 + x .
{ r = r ( r 0 , ψ 0 , θ 0 , φ 0 ) ; ψ = ψ ( r 0 , ψ 0 , θ 0 , φ 0 ) ; θ = θ ( r 0 , ψ 0 , θ 0 , φ 0 ) ; φ = φ ( r 0 , ψ 0 , θ 0 , φ 0 ) ,
R ( r 0 , ψ 0 , θ 0 , φ 0 ) = R [ r ( r 0 , ψ 0 , θ 0 , φ 0 ) , ψ ( r 0 , ψ 0 , θ 0 , φ 0 ) , θ ( r 0 , ψ 0 , θ 0 , φ 0 ) , φ ( r 0 , ψ 0 , θ 0 , φ 0 ) ] .
p = n 2 · cos 2 φ 0 - ( d n 2 / d r ) r = a .
p = n 2 n 0 2 - n 1 2 · a · cos 2 φ 0 x .

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