Abstract

This paper describes a new method (the scattering-pattern method) for determining the refractive-index profile in an optical fiber from its scattering pattern for a normally incident laser beam. The proposed method is applicable to an arbitrary profile and is nondestructive. The spatial resolution is high, and the accuracy is good when the fiber diameter and the refractive-index variation are relatively small. The drawback is that a large number of data are required; however, this difficulty has been overcome by using the automated measuring system described in this paper. The profile obtained shows good agreement with design data.

© 1976 Optical Society of America

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References

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  1. The outline of this method was once described in a short communication: T. Okoshi, K. Hotate, Opt. Quantum Electron. 8, 78 (1976).
    [CrossRef]
  2. H. M. Presby, J. Opt. Soc. Am. 64, 280 (1974).
    [CrossRef]
  3. L. S. Watkins, J. Opt. Soc. Am. 64, 767 (1974).
    [CrossRef]
  4. W. E. Martin, Appl. Opt. 13, 2112 (1974).
    [CrossRef] [PubMed]
  5. T. Kitano et al., IEEE J. Quantum Electron. QE-9, 967 (1973).
    [CrossRef]
  6. W. Eickhoff et al., Opt. Quantum Electron. 7, 109 (1975).
    [CrossRef]
  7. Y. S. Liu, Appl. Opt. 13, 1255 (1974).
    [CrossRef] [PubMed]
  8. A. M. Hunter et al., Appl. Opt. 14, 634 (1975).
    [CrossRef]
  9. E. G. Rawson, Appl. Opt. 13, 2370 (1974).
    [CrossRef] [PubMed]
  10. J. L. Lundberg, J. Colloid Interface Sci. 29, 565 (1969).
    [CrossRef]
  11. A. Papoulis, in Generalized Networks (Polytechnic Press, Brooklyn, 1966) 753.

1976

The outline of this method was once described in a short communication: T. Okoshi, K. Hotate, Opt. Quantum Electron. 8, 78 (1976).
[CrossRef]

1975

W. Eickhoff et al., Opt. Quantum Electron. 7, 109 (1975).
[CrossRef]

A. M. Hunter et al., Appl. Opt. 14, 634 (1975).
[CrossRef]

1974

1973

T. Kitano et al., IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

1969

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

Eickhoff, W.

W. Eickhoff et al., Opt. Quantum Electron. 7, 109 (1975).
[CrossRef]

Hotate, K.

The outline of this method was once described in a short communication: T. Okoshi, K. Hotate, Opt. Quantum Electron. 8, 78 (1976).
[CrossRef]

Hunter, A. M.

Kitano, T.

T. Kitano et al., IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

Liu, Y. S.

Lundberg, J. L.

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

Martin, W. E.

Okoshi, T.

The outline of this method was once described in a short communication: T. Okoshi, K. Hotate, Opt. Quantum Electron. 8, 78 (1976).
[CrossRef]

Papoulis, A.

A. Papoulis, in Generalized Networks (Polytechnic Press, Brooklyn, 1966) 753.

Presby, H. M.

Rawson, E. G.

Watkins, L. S.

Appl. Opt.

IEEE J. Quantum Electron.

T. Kitano et al., IEEE J. Quantum Electron. QE-9, 967 (1973).
[CrossRef]

J. Colloid Interface Sci.

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

J. Opt. Soc. Am.

Opt. Quantum Electron.

W. Eickhoff et al., Opt. Quantum Electron. 7, 109 (1975).
[CrossRef]

The outline of this method was once described in a short communication: T. Okoshi, K. Hotate, Opt. Quantum Electron. 8, 78 (1976).
[CrossRef]

Other

A. Papoulis, in Generalized Networks (Polytechnic Press, Brooklyn, 1966) 753.

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Figures (16)

Fig. 1
Fig. 1

Basic part of the optical setup for measuring the scattering pattern.

Fig. 2
Fig. 2

Coordinates and symbols used in the analysis: (a) the coordinate system; (b) symbols.

Fig. 3
Fig. 3

Relation between the wavenumber vectors.

Fig. 4
Fig. 4

Permittivity profile suitable for determining the polarity sign [Es(θ)].

Fig. 5
Fig. 5

Schematic illustration of the scattering pattern given by Eq. (16): (a) scattered field generated in the core; (b) small ripple generated at the fiber surface; (c) total scattered field; (d) total scattered power.

Fig. 6
Fig. 6

A drawing for determining the polarity.

Fig. 7
Fig. 7

Another drawing for determining the polarity.

Fig. 8
Fig. 8

An example of the normalized scattering power computed by using Eq. (32).

Fig. 9
Fig. 9

Normalized refractive-index profiles computed from rigorous scattering patterns by using Eq. (14). Dots and small circles correspond to 0.1% and 0.5% refractive-index diffgrences, respectively. In these cases the sampling interval of the scattering pattern and the maximum sampling angle are 0.001 rad and 1.0 rad, respectively.

Fig. 10
Fig. 10

Error E obtained with various simulations as a function of the product rcore × Δn′.

Fig. 11
Fig. 11

Error E as a function of the sampling interval Δθs in case of a uniform-core fiber with rcore = 10 μm and Δn′ = 0.1%. In this case the maximum sampling angle is 1.0 rad.

Fig. 12
Fig. 12

Refractive-index profile obtained with a simulation in which the range of the scattering pattern is not satisfactorily wide; in this case 0–0.1 rad.

Fig. 13
Fig. 13

Automated system for measuring the refractive-index profile.

Fig. 14
Fig. 14

Optical setup for measuring the scattering pattern.

Fig. 15
Fig. 15

A part of the measured scattering pattern σ(θ) for a uniform-core fiber with rcore = 10 μm and Δn = 0.3%. The ordinate shows log[σ(θ)], and the sampling interval in the abscissa is 0.586 × 10−4 rad. The maximum sampling angle was 0.698 rad; only one-third of the pattern (0–0.22 rad) is shown in this figure (a graphic line-printer output.)

Fig. 16
Fig. 16

Refractive-index profile computed from the scattering pattern shown in Fig. 15 (a graphic line-printer output).

Tables (1)

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Table I Compensation Coefficient α as a Function of the Value 0 r core Δ n ( r ) d r ( = β )

Equations (44)

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d E s = - k 0 2 4 π R 1 ( r ) 0 A 0 exp ( j ω t ) exp ( - j k 0 R ) d V · u ,
R = r cos ϕ + R ,
R = R - r cos ( ϕ - θ ) - z p R z q ( R r ) ,
R = R - 2 r sin θ 2 [ sin ( ϕ - θ 2 ) ] - z p R z q .
1 ( r ) d r π ,
E s k 0 2 A 0 4 π R - l / 2 l / 2 0 2 π 0 a 1 ( r ) 0 r exp ( - j k 0 R ) d r d ϕ d z = k 0 2 2 R A 0 l sinc k z l 2 × 0 a 1 ( r ) 0 r J 0 ( r k t ) d r ,
k z = [ ( k 0 ) / R ] z p ,
k t = 2 k 0 sin ( θ / 2 ) .
k t = k s - k i
P s t = W 2 Z 0 - + E s 2 d z p = W k 0 3 A 0 2 l 4 Z 0 R [ 0 1 ( r ) 0 r J 0 ( r k t ) d r ] 2 ,
σ = 2 π R P s t / P i t ,
P i t = A 0 2 l W / 2 Z 0 ,
sign [ E s ( θ ) ] 1 π k 0 3 / 2 [ σ ( k t ) ] 1 / 2 = 0 1 ( r ) 0 r J 0 ( r k t ) d r .
1 ( r ) 0 = 1 π k 0 3 / 2 0 sign [ E s ( θ ) ] [ σ ( k t ) ] 1 / 2 k t J 0 ( k t r ) d k t .
1 ( r ) 0 = k 0 1 / 2 π 0 sign [ E s ( θ ) ] [ σ ( θ ) ] 1 / 2 J 0 ( 2 k 0 sin θ 2 · r ) sin θ d θ .
n ( r ) = n 0 [ 1 + 1 ( r ) / 0 ] 1 / 2 ,
sign [ E s ( θ ) ] [ σ ( θ ) ] 1 / 2 = π k 0 3 / 2 × 0 r c o r e 1 ( r ) - clad 0 r J 0 ( 2 k 0 sin θ 2 · r ) d r + π k 0 3 / 2 clad 0 r clad J 1 [ 2 k 0 r clad sin ( θ / 2 ) ] 2 k 0 r clad sin ( θ / 2 ) ,
F ( r ) = 0 k f ( k ) J 0 ( r k ) d k ,
f ( k ) = 0 r F ( r ) J 0 ( k r ) d r .
F ( r ) = 0 for r r a ,
F ( r ) = k = 1 2 [ J 1 ( α 0 k ) ] 2 r a 2 f ( α 0 k r a ) J 0 ( α 0 k r a r ) ,
1 ( r ) 0 = 1 π k 0 3 / 2 k = 1 { 2 [ J 1 ( α 0 k ) ] 2 r core 2 × sign [ E s ( k k ) ] [ σ ( k k ) ] 1 / 2 J 0 ( k k · r ) } ,
k k = α 0 k / r core ,
θ k = 2 sin - 1 ( α 0 k 2 k 0 r core ) .
α 0 k 2 k 0 r core             ( θ π ) .
Δ θ s = γ [ 3 / ( k 0 r core ) ] ,
[ σ ( k t ) ] 1 / 2 = 0             for k t > k m .
r k = α 0 k / k m             ( k = 1 , 2 , 3 ) .
Δ r = r k + 1 - r k π / k m ,
Δ r min = π / 2 k 0 = λ 0 / 4.
E s = A 0 λ 0 π R exp [ - j ( 2 π λ 0 R - π 4 ) ] ( b 0 + 2 n = 1 b n cos n θ ) ,
b n = m J n ( α ) J n ( m α ) - J n ( α ) J n ( m α ) m H n ( α ) J n ( m α ) - H n ( α ) J n ( m α ) ,
[ σ ( θ ) ] 1 / 2 = sign [ σ ( θ ) ] 1 / 2 2 λ 0 π | b 0 + 2 n = 1 b n cos n θ | .
E = n - n / Δ n ,
Δ θ s < 0.9 / k 0 r core ( rad ) .
n ˜ ( r 1 ) = 1 2 Δ r r 1 - Δ r r 1 + Δ r n ( r ) d r ,
F ( r ) = 0 for r r a ,
f ( k ) = r a 2 0 1 r ˜ F ˜ ( r ˜ ) J 0 ( k r a r ˜ ) d r ˜ ,
F ˜ ( r ˜ ) = k = 1 a k J 0 ( α 0 k r ˜ ) ,
a k = 2 [ J 1 ( α 0 k ) ] 2 0 1 t F ˜ ( t ) J 0 ( α 0 k t ) d t .
a k = 2 [ J 1 ( α 0 k ) ] 2 r a 2 f ( α 0 k r a ) .
Δ n c ( r ) = r core ( 1 - α ) r + α r core Δ n ( r ) ,
0 r core Δ n ( r ) d r .
0 r core Δ n ( r ) d r .

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