Abstract

A scattering theory is presented, which is capable of predicting the light-distribution pattern that arises when a clad optical fiber of arbitrary core refractive-index distribution is illuminated by a laser beam perpendicular to its axis. Theoretical predictions are compared to experimental results in the backscattering direction, with excellent agreement for fibers of practical interest having core and cladding diameters on the order to 50 and 150 μm, respectively. Considerable sensitivity to parameter variations has been observed and implications for diameter and refractive-index distribution determinations are discussed.

© 1975 Optical Society of America

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References

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  1. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, New York, 1969), Ch. 6.
  2. M. Koedam, Phill. Tech. Rev. 27, 208 (1966)
  3. F. P. Gagliano, R. M. Lumley, and L. S. Watkins, Proc. IEEE 57, 114 (1969).
    [CrossRef]
  4. S. R. Seshadri, Can. J. Phys. 42, 860 (1964).
    [CrossRef]
  5. A. R. Jones and E. R. Wooding, Electron Lett. 1, 171 (1965).
    [CrossRef]
  6. H. M. Presby, J. Opt. Soc. Am. 64, 280 (1974).
    [CrossRef]
  7. L. S. Watkins, J. Opt. Soc. Am. 64, 767 (1974).
    [CrossRef]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  9. M. Kerker and E. Matijevic, J. Opt. Soc. Am. 51, 506 (1961).
    [CrossRef]
  10. W. A. Farone and M. Kerker, J. Opt. Soc. Am. 56, 481 (1966).
    [CrossRef]
  11. J. L. Lundberg, J. Colloid Interface Sci. 29, 565 (1969).
    [CrossRef]
  12. W. B. Gandrud and M. J. Saunders, private communication, 29December1972.
  13. H. M. Presby, Appl. Phys. Lett. 24, 422 (1974).
    [CrossRef]
  14. H. M. Presby and D. Marcuse, Appl. Opt. 13, 2882 (1974).
    [CrossRef] [PubMed]
  15. R. Burman, IEEE Trans. AP-13, 646 (1965).
    [CrossRef]
  16. J. Feinstein, J. Geophys. Res. 56, 37 (1951).
    [CrossRef]
  17. S. E. Miller, E. A. J. Marcatili, and Tingye Li, Proc. IEEE 61, 1703 (1973).
    [CrossRef]
  18. D. Gloge and E. A. J. Marcatili, Bell Syst. Tech. J.,  52, 1563 (1973).
    [CrossRef]
  19. J. B. MacChesney, P. B. O’Connor, and H. M. Presby, Proc. IEEE 62, 1278 (1974)
    [CrossRef]
  20. G. W. Tasker and W. G. French, Proc. IEEE 62, 1281 (1974).
    [CrossRef]
  21. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  22. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Std. (U. S.) Appl. Math. Ser. (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965).
  23. D. Marcuse, Appl. Opt. 13, 1903 (1974).
    [CrossRef] [PubMed]
  24. C. A. Burrus and R. D. Standley, Appl. Opt. 13, 2365 (1974).
    [CrossRef] [PubMed]
  25. D. Marcuse and H. M. Presby, Bell Syst. Tech. J. 54, 1 (1975).
    [CrossRef]

1975 (1)

D. Marcuse and H. M. Presby, Bell Syst. Tech. J. 54, 1 (1975).
[CrossRef]

1974 (8)

1973 (2)

S. E. Miller, E. A. J. Marcatili, and Tingye Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

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

1969 (2)

J. L. Lundberg, J. Colloid Interface Sci. 29, 565 (1969).
[CrossRef]

F. P. Gagliano, R. M. Lumley, and L. S. Watkins, Proc. IEEE 57, 114 (1969).
[CrossRef]

1966 (2)

1965 (2)

A. R. Jones and E. R. Wooding, Electron Lett. 1, 171 (1965).
[CrossRef]

R. Burman, IEEE Trans. AP-13, 646 (1965).
[CrossRef]

1964 (1)

S. R. Seshadri, Can. J. Phys. 42, 860 (1964).
[CrossRef]

1961 (1)

1951 (1)

J. Feinstein, J. Geophys. Res. 56, 37 (1951).
[CrossRef]

Burman, R.

R. Burman, IEEE Trans. AP-13, 646 (1965).
[CrossRef]

Burrus, C. A.

Farone, W. A.

Feinstein, J.

J. Feinstein, J. Geophys. Res. 56, 37 (1951).
[CrossRef]

French, W. G.

G. W. Tasker and W. G. French, Proc. IEEE 62, 1281 (1974).
[CrossRef]

Gagliano, F. P.

F. P. Gagliano, R. M. Lumley, and L. S. Watkins, Proc. IEEE 57, 114 (1969).
[CrossRef]

Gandrud, W. B.

W. B. Gandrud and M. J. Saunders, private communication, 29December1972.

Gloge, D.

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

Jones, A. R.

A. R. Jones and E. R. Wooding, Electron Lett. 1, 171 (1965).
[CrossRef]

Kerker, M.

W. A. Farone and M. Kerker, J. Opt. Soc. Am. 56, 481 (1966).
[CrossRef]

M. Kerker and E. Matijevic, J. Opt. Soc. Am. 51, 506 (1961).
[CrossRef]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, New York, 1969), Ch. 6.

Koedam, M.

M. Koedam, Phill. Tech. Rev. 27, 208 (1966)

Li, Tingye

S. E. Miller, E. A. J. Marcatili, and Tingye Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Lumley, R. M.

F. P. Gagliano, R. M. Lumley, and L. S. Watkins, Proc. IEEE 57, 114 (1969).
[CrossRef]

Lundberg, J. L.

J. L. Lundberg, J. Colloid Interface Sci. 29, 565 (1969).
[CrossRef]

MacChesney, J. B.

J. B. MacChesney, P. B. O’Connor, and H. M. Presby, Proc. IEEE 62, 1278 (1974)
[CrossRef]

Marcatili, E. A. J.

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

S. E. Miller, E. A. J. Marcatili, and Tingye Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

Marcuse, D.

Matijevic, E.

Miller, S. E.

S. E. Miller, E. A. J. Marcatili, and Tingye Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

O’Connor, P. B.

J. B. MacChesney, P. B. O’Connor, and H. M. Presby, Proc. IEEE 62, 1278 (1974)
[CrossRef]

Presby, H. M.

D. Marcuse and H. M. Presby, Bell Syst. Tech. J. 54, 1 (1975).
[CrossRef]

H. M. Presby, J. Opt. Soc. Am. 64, 280 (1974).
[CrossRef]

J. B. MacChesney, P. B. O’Connor, and H. M. Presby, Proc. IEEE 62, 1278 (1974)
[CrossRef]

H. M. Presby, Appl. Phys. Lett. 24, 422 (1974).
[CrossRef]

H. M. Presby and D. Marcuse, Appl. Opt. 13, 2882 (1974).
[CrossRef] [PubMed]

Saunders, M. J.

W. B. Gandrud and M. J. Saunders, private communication, 29December1972.

Seshadri, S. R.

S. R. Seshadri, Can. J. Phys. 42, 860 (1964).
[CrossRef]

Standley, R. D.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Tasker, G. W.

G. W. Tasker and W. G. French, Proc. IEEE 62, 1281 (1974).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Watkins, L. S.

L. S. Watkins, J. Opt. Soc. Am. 64, 767 (1974).
[CrossRef]

F. P. Gagliano, R. M. Lumley, and L. S. Watkins, Proc. IEEE 57, 114 (1969).
[CrossRef]

Wooding, E. R.

A. R. Jones and E. R. Wooding, Electron Lett. 1, 171 (1965).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

H. M. Presby, Appl. Phys. Lett. 24, 422 (1974).
[CrossRef]

Bell Syst. Tech. J. (2)

D. Marcuse and H. M. Presby, Bell Syst. Tech. J. 54, 1 (1975).
[CrossRef]

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

Can. J. Phys. (1)

S. R. Seshadri, Can. J. Phys. 42, 860 (1964).
[CrossRef]

Electron Lett. (1)

A. R. Jones and E. R. Wooding, Electron Lett. 1, 171 (1965).
[CrossRef]

IEEE Trans. (1)

R. Burman, IEEE Trans. AP-13, 646 (1965).
[CrossRef]

J. Colloid Interface Sci. (1)

J. L. Lundberg, J. Colloid Interface Sci. 29, 565 (1969).
[CrossRef]

J. Geophys. Res. (1)

J. Feinstein, J. Geophys. Res. 56, 37 (1951).
[CrossRef]

J. Opt. Soc. Am. (4)

Phill. Tech. Rev. (1)

M. Koedam, Phill. Tech. Rev. 27, 208 (1966)

Proc. IEEE (4)

F. P. Gagliano, R. M. Lumley, and L. S. Watkins, Proc. IEEE 57, 114 (1969).
[CrossRef]

S. E. Miller, E. A. J. Marcatili, and Tingye Li, Proc. IEEE 61, 1703 (1973).
[CrossRef]

J. B. MacChesney, P. B. O’Connor, and H. M. Presby, Proc. IEEE 62, 1278 (1974)
[CrossRef]

G. W. Tasker and W. G. French, Proc. IEEE 62, 1281 (1974).
[CrossRef]

Other (5)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Std. (U. S.) Appl. Math. Ser. (U. S. Government Printing Office, Washington, D. C., 1964; Dover, New York, 1965).

W. B. Gandrud and M. J. Saunders, private communication, 29December1972.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation, (Academic, New York, 1969), Ch. 6.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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Figures (23)

FIG. 1
FIG. 1

Geometry for scattering of incident plane wave by an infinite cylinder.

FIG. 2
FIG. 2

Calculated scattered-light pattern for an unclad fiber with kb = 665.26, over the range 180°–148°.

FIG. 3
FIG. 3

Observed scattered-light pattern, for same fiber as Fig. 2.

FIG. 4
FIG. 4

Observed scattered-light pattern, for same fiber as Figs. 2 and 3, for a different rotational orientation than Fig. 3.

FIG. 5
FIG. 5

Calculated scattered-light pattern for step-index fiber, with kb = 546.1.

FIG. 6
FIG. 6

Observed scattered-light pattern for same fiber as Fig. 5.

FIG. 7
FIG. 7

Refractive-index profile of approximate step-index fiber (solid line) and approximation of general theory (dashed line).

FIG. 8
FIG. 8

Calculated scattered-light pattern for step-index fiber, kb = 645.4, assuming exact step index.

FIG. 9
FIG. 9

Observed scattered-light pattern for same fiber as Fig. 8.

FIG. 10
FIG. 10

Pattern observed with same fiber as Fig. 9, but with a different rotational orientation.

FIG. 11
FIG. 11

Scattered-light pattern for same fiber as Fig. 8, but calculated with approximating profile.

FIG. 12
FIG. 12

Refractive-index profile of graded-index fiber kb = 635.47 (solid line) and approximation of general theory (dashed line).

FIG. 13
FIG. 13

Calculated scattered-light pattern for graded-index fiber kb = 635.47.

FIG. 14
FIG. 14

Observed scattered-light pattern for same fiber as Fig. 13.

FIG. 15
FIG. 15

Calculated scattered-light pattern of a fiber with a = 23 μm, Δ = 0.009 for a step-index profile.

FIG. 16
FIG. 16

Scattered-light pattern calculated for a fourth-power-law profile.

FIG. 17
FIG. 17

Scattered-light pattern calculated for a parabolic profile (α = 2).

FIG. 18
FIG. 18

Refractive-index profile of graded-index fiber kb = 645.397 (solid line) and two approximating profiles (dashed and dot–dashed lines). The 3 curves overlap up to approximately 18 μm but are separated for clarity.

FIG. 19
FIG. 19

Scattered-light pattern for graded-index fiber kb = 645.397, calculated from dashed approximation.

FIG. 20
FIG. 20

Scattered-light pattern for same fiber as Fig. 19, but calculated from dot–dashed approximation.

FIG. 21
FIG. 21

Scattered-light pattern observed with same fiber as Figs. 19 and 20.

FIG. 22
FIG. 22

Calculated scattered-light pattern for fiber of Fig. 12 but with 1 μm change of core radius.

FIG. 23
FIG. 23

Scattered-light pattern for fiber of Figs. 12 and 22, but with a 1 μm change of over-all diameter.

Equations (33)

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E = E p + E sc .
H r = i ω μ 0 1 r E z ϕ ,
H ϕ = i ω μ 0 E z r
E p = A i e i k x = 1 2 A i ν = ( i ) ν [ H ν ( 1 ) ( k r ) + H ν ( 2 ) ( k r ) ] e i ν ϕ ,
k = 2 π λ 0 = ω ( 0 μ 0 )
E = ν = [ B ν H ν ( 1 ) ( k r ) + C ν H ν ( 2 ) ( k r ) ] e i ν ϕ .
B ν = ( i ) ν 2 A i
E sc = ν = { [ C ν ( i ) ν 2 A i ] H ν ( 2 ) ( k r ) e i ν ϕ } .
E = ν = D ν F ν ( n 2 k r ) e i ν ϕ .
E sc = A i ( 2 π k r ) 1 / 2 e i ( k r π / 4 ) × ν = 0 ( 2 ν e i α ν cos α ν cos ν ϕ ) ,
ν = { 1 2 for ν = 0 1 for ν 0
tan α ν = F ν ( n 2 k b ) N ν ( k b ) n 2 F ν ( n 2 k b ) N ν ( k b ) F ν ( n 2 k b ) J ν ( k b ) n 2 F ν ( n 2 k b ) J ν ( k b ) .
d 2 F ν d r 2 + 1 r d F ν d r + ( n 2 ( r ) k 2 ν 2 r 2 ) F ν = 0 .
n ( r ) = n 2 .
F ν ( n 2 k r ) = J ν ( n 2 k r ) .
n ( r ) = { n 1 for r < a n 2 for b r a ,
F ν ( n 2 k r ) = J ν ( n 1 k r ) for r < a
F ν ( n 2 k r ) = 1 2 π k a { [ n 1 J ν + 1 ( n 1 k a ) N 2 ( n 2 k a ) n 2 J ν ( n 1 k a ) N ν + 1 ( n 2 k a ) ] J ν ( n 2 k r ) [ n 1 J ν + 1 ( n 1 k a ) J ν ( n 2 k a ) n 2 J ν ( n 1 k a ) J ν + 1 ( n 2 k a ) ] N ν ( n 2 k r ) } , for b r a .
n ( r ) = { μ = 0 5 D 2 μ ( r a ) 2 μ , r < a n 2 , b r a ,
G ν ( z ) = J ν ( z ) for r < a
z = 0 r n ( x ) k d x .
G ν ( z ) = | [ 1 + A ν ( z ) ] J ν ( z ) + B ν ( z ) N ν ( z ) .
A ν ( z ) = π 2 D 0 { μ = 1 5 [ ( 2 μ ) 2 2 μ + 1 D 2 μ 1 Z a 2 μ 0 z x 2 μ J ν ( x ) N ν ( x ) d x ] + 2 ν 2 μ = 1 5 [ 2 μ 2 μ + 1 D 2 μ 1 Z a 2 μ 0 z x 2 μ 1 J ν ( x ) N ν ( x ) d x ] }
B ν ( z ) = π 2 D 0 { μ = 1 5 [ ( 2 μ ) 2 2 μ + 1 D 2 μ 1 Z a 2 μ 0 z x 2 μ J ν ( x ) J ν ( x ) d x ] + 2 ν 2 μ = 1 5 [ 2 μ 2 μ + 1 D 2 μ 1 Z a 2 μ 0 z x 2 μ 1 J ν 2 ( x ) d x ] } .
F ν ( n 2 k r ) = K ν J ν ( n 2 k r ) + M ν N ν ( n 2 k r ) ,
K ν = 1 2 π n 2 k a [ G ν ( Z a ) N ν ( n 2 k a ) G ν ( Z a ) N ν ( n 2 k a ) ]
M ν = 1 2 π n 2 k a [ G ν ( Z a ) J ν ( n 2 k a ) G ν ( Z a ) J ν ( n 2 k a ) ] .
J ν max ( x ) = 0
J ν max ( x ) = 10 30
0 z x 2 μ J ν ( x ) N ν ( x ) d x = 1 2 0 z x 2 μ [ J ν 1 ( x ) J ν + 1 ( x ) ] N ν ( x ) d x
0 z x 2 μ 1 J ν ( x ) N ν ( x ) d x = 1 2 ν 0 z x 2 μ [ J ν 1 ( x ) + J ν ( x ) ] N ν ( x ) d x .
0 z x n + 1 J ν + 1 ( x ) N ν ( x ) d x = n 1 4 n [ n 2 ( 2 ν + 1 ) 2 ] 0 z x n 1 J ν + 1 ( x ) N ν ( x ) d x + 1 n + 1 × { z n + 2 2 ( J ν + 1 N ν + J ν + 1 N ν ) + ν + 1 2 2 n z n + 1 ( J ν + 1 N ν J ν + 1 N ν ) n 4 z n + 1 ( J ν + 1 N ν + J ν + 1 N ν ) + z n 4 ( n 2 2 ν 2 2 ν 1 ) J ν + 1 N ν }
0 z x n + 1 J ν 1 ( x ) N ν ( x ) d x = n 1 4 n [ n 2 ( 2 ν 1 ) 2 ] 0 x x n 1 J ν 1 ( x ) N ν ( x ) d x + 1 n + 1 × { z n + 2 2 ( J ν 1 N ν + J ν 1 N ) ν 1 2 2 n z n + 1 ( J ν 1 N ν J ν 1 N ) n 4 z n + 1 ( J ν 1 N ν + J ν + 1 N ν ) + z n 4 ( n 2 2 ν 2 2 ν 1 ) J ν 1 N ν } .