Abstract

Using mode theory, we calculate the crosstalk between overmoded optical fibers or light pipes when one fiber is illuminated by a focused collimated beam. The presence of many modes allows the summation of crosstalk power between individual modes to be converted to an integral expression. The result is applied to two parallel, identical fibers, a hexagonal array of fibers, lossy fibers, and fibers of unequal diameter. In practice, the light pipes must be nearly identical and touching for significant crosstalk.

© 1976 Optical Society of America

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References

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  1. N. S. Kapany, Fiber Optics (Academic, New York, 1967), Chap. 3.
  2. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 10.
  3. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), Chap. 3.
  4. A. W. Snyder, J. Opt. Soc. Am. 62, 1267 (1972).
    [Crossref]
  5. P. D. McIntyre and A. W. Snyder, J. Opt. Soc. Am. 63, 1518 (1973).
    [Crossref]
  6. A. H. Cherin and E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).
    [Crossref]
  7. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1926), Chap. 9.
  8. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 2.
  9. A. W. Snyder, Appl. Phys. 4, 273 (1974).
    [Crossref]
  10. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
    [Crossref]
  11. A. W. Snyder and C. Pask, Opt. Commun. 15, 314 (1975).
    [Crossref]
  12. J. A. Arnuad, Bell Syst. Tech. J. 54, 1431 (1975).
    [Crossref]
  13. P. D. McIntyre and A. W. Snyder, J. Opt. Soc. Am. 64, 285 (1974).
    [Crossref] [PubMed]
  14. A. W. Snyder and D. J. Mitchell, J. Opt. Soc. Am. 64, 599 (1974).
    [Crossref]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).
  16. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970).
  17. A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
    [Crossref]
  18. A. W. Snyder, C. Pask, and D. J. Mitchell, J. Opt. Soc. Am. 63, 59 (1973).
    [Crossref] [PubMed]

1975 (3)

A. H. Cherin and E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).
[Crossref]

A. W. Snyder and C. Pask, Opt. Commun. 15, 314 (1975).
[Crossref]

J. A. Arnuad, Bell Syst. Tech. J. 54, 1431 (1975).
[Crossref]

1974 (3)

1973 (2)

1972 (1)

1969 (2)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
[Crossref]

Arnuad, J. A.

J. A. Arnuad, Bell Syst. Tech. J. 54, 1431 (1975).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), Chap. 3.

Cherin, A. H.

A. H. Cherin and E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1926), Chap. 9.

Kapany, N. S.

N. S. Kapany, Fiber Optics (Academic, New York, 1967), Chap. 3.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), Chap. 3.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 10.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 2.

McIntyre, P. D.

Mitchell, D. J.

Murphy, E. J.

A. H. Cherin and E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).
[Crossref]

Pask, C.

Snyder, A. W.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970).

Appl. Phys. (1)

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

Bell Syst. Tech. J. (2)

A. H. Cherin and E. J. Murphy, Bell Syst. Tech. J. 54, 17 (1975).
[Crossref]

J. A. Arnuad, Bell Syst. Tech. J. 54, 1431 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1138 (1969).
[Crossref]

A. W. Snyder, IEEE Trans. Microwave Theory Tech. 17, 1130 (1969).
[Crossref]

J. Opt. Soc. Am. (5)

Opt. Commun. (1)

A. W. Snyder and C. Pask, Opt. Commun. 15, 314 (1975).
[Crossref]

Other (7)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1926), Chap. 9.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 2.

N. S. Kapany, Fiber Optics (Academic, New York, 1967), Chap. 3.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972), Chap. 10.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), Chap. 3.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970).

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Figures (7)

FIG. 1
FIG. 1

Illumination of optical waveguide. The core has a circular cross section of diameter dco and refractive index nco. ncl and no are the refractive indexes of the cladding and the medium external to the fiber. The lens, of diameter dL and focal length f, is illuminated by a highly collimated on-axis beam with azimuthal symmetry. FP is the focal plane of the lens.

FIG. 2
FIG. 2

(a) Two Identical light pipes embedded in an infinite medium of refractive index ncl, with a center-to-center separation of ds. (b) Two cladded optical fibers in an external medium no. Because only coupling between meridional rays is important (see text), the situation in Fig. 2(a) represents a good approximation of the situation in Fig. 2(b), especially when the diameter of the cladding ≫ dco.

FIG. 3
FIG. 3

Percent of crosstalk power (power in the unilluminated fiber) between two identical parallel fibers when one is illuminated as in Fig. 1, with unity total power (Pin = 1), calculated using Eqs. (5) and (6). The dimensionless length parameter L is given by Eq. (7). The ideal optics case (—), and the imperfect optics with K2 = 2(– – –) and K2 = 5(· – · –) are shown.

FIG. 4
FIG. 4

Percent crosstalk power at position Z along the axis of the unilluminated fiber. V = 150, θc = 0.2, dco = 100 μ, Um = 50. Excitation by a perfect optical system Eq. (5). The curves for ds/dco = 1, 1.01 have been truncated to avoid confusion with the other curves.

FIG. 5
FIG. 5

Hexagonal array of identical elements. (a) Light pipes of refractive index nco embedded in infinite medium of refractive index ncl. (b) Closely packed optical fibers. As in Fig. 2(b), Fig. 5(b) represents a good approximation to Fig. 5(a).

FIG. 6
FIG. 6

The maximum crosstalk-power transfer F as a function of the fractional difference, Δdco/dco, in core diameter, for V = 200 (——), and as a comparison, for 11E11 modes with V = 2.4(– – –).

FIG. 7
FIG. 7

Comparison of the exact mode-sum (– · – · –) of Pct/Pin for Um = 50, with the integral approximation (——) for a perfect-optics system. The integral is a function of L only whereas the sum is a function of L and Um, which is a measure of the number of modes contributing to the sum. When Um ≃ 500, the sum curve is too close to the integral curve to be shown here.

Equations (41)

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θ m n o d L / 2 n c o f ,
D ( θ ) = e - ( k θ / θ m ) 2 ,
V = π d c o n c o θ c / λ ,
θ c = { 1 - n c l 2 / n c o 2 } 1 / 2 .
P c t P i n 1 2 ( 1 - sin L L )
P c t P i n L 2 L 2 + K 4
L = 4 θ c Z d c o ( θ m θ c ) 2 e - 2 v ( D - 1 ) ( π V D ) 1 / 2 ,
α = α c o + α c l - α c o v ( θ θ c ) 2 .
P c t P i n = 1 2 L 2 + γ 2 ( K 2 + γ 2 ) L 2 + ( K 2 + γ 2 ) 2 e - α c o z ,
γ 2 = ( α c l - α c o ) Z v ( θ m θ c ) 2 .
F = 1 / ( 1 + X 2 ) ,
X = 1 2 ( π D V 3 ) 1 / 2 ( Δ d c o / d c o ) e 2 V ( D - 1 ) .
p c t q = p i n q sin 2 C q Z ,
C q = 2 θ c d c o U q 2 V 5 / 2 e - 2 v ( D - 1 ) ( π D ) 1 / 2 ,
P c t = q = 1 m P i n q sin 2 C q Z ,
P i n = q = 1 m P i n q .
P c t 1 π 0 U m P i n ( U ) sin 2 [ C ( U ) Z ] d U ,
P i n 1 π 0 U m P i n ( U ) d U .
U = V sin θ / sin θ c
V ( θ / θ c ) ,
E = E 0 J 1 ( B R ) / B R ,
B = V sin θ m / sin θ c ,
P i n q = | A c o E × h q * · Z ˆ d A | 2 / | A e q × h q * · Z ˆ d A | 2 ,
P i n q 2 π 2 ( 0 μ ) 1 / 2 ( E 0 ρ c o B ) 2 U q I q ;
I q = B 0 1 J 0 ( U q R ) J 1 ( B R ) d R ,
P i n ( U ) = U D ( U ) .
D ( U ) = D 0 ,
D ( U ) = D 0 exp [ - ( K U / U m ) 2 ] ,
P c t P i n = 1 2 ( 1 - sin L L )
P c t P i n = 1 2 L 2 L 2 + K 4 .
L = 4 θ c Z d c o ( θ m θ c ) 2 e - 2 V ( D - 1 ) ( π V D ) 1 / 2 .
θ m n o θ 0 / n c o ,
P c t = q = 1 m 6 7 P i n q sin 2 ( 7 C q Z ) ,
P c t q = P i n q sin 2 ( C q Z ) e - α q Z ,
α q α c o + ( α c l - α c o ) ( U q 2 / V 3 ) α c o + α c l - α c o V ( θ θ c ) 2 .
P c t P i n = 1 2 L 2 + γ 2 ( K 2 + γ 2 ) L 2 + ( K 2 + γ 2 ) 2 e - α c o Z ,
γ 2 = ( α c l - α c o ) Z ( U m 2 / V 3 ) .
P c t = F q = 1 m P i n q sin 2 ( C q Z / F ) ,
F = [ 1 + ( Δ β q / 2 C q ) 2 ] - 1 ,
Δ β q 2 θ c ( Δ d c o / d c o 2 ) ( U q 2 / V ) ,
Δ β q / 2 C q 1 2 ( π D V 3 ) 1 / 2 ( Δ d c o / d c o ) e 2 v ( D - 1 ) ,