Abstract

A new nondestructive technique is presented for determining the refractive index profile of an optical fiber from its backscattered pattern arising from a normally incident laser beam to the fiber axis. The proposed method requires no sample preparation or index matching liquid. The principle of the method is to construct a deflection function from the measured pattern. The index profile can then be determined by the inversion of an Abel integral equation. Good agreement is obtained between the index profile determined by this technique and that measured by the near-field scanning technique.

© 1979 Optical Society of America

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References

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  1. M. E. Marhic, P. S. Ho, M. Epstein, Appl. Phys. Lett. 26, 574 (1975).
    [CrossRef]
  2. Y. Ohtsuka, Y. Shimizu, Appl. Opt. 16, 1050 (1977).
    [PubMed]
  3. K. Iga, Y. Kokubun, in Technical Digest of International Conference Integrated Optics and Optical Fiber Communication, Japan (1977), p. 403.
  4. M. J. Saunders, W. B. Gardner, Appl. Opt. 16, 2369 (1977).
    [CrossRef]
  5. P. L. Chu, T. W. Whitbread, Appl. Opt.18, 000 (1979), same issue to be published.
  6. T. Okoshi, K. Hotate, Appl. Opt. 15, 2756 (1976).
    [CrossRef] [PubMed]
  7. L. S. Watkins, J. Opt. Soc. Am. 64, 767 (1974).
    [CrossRef]
  8. D. H. Smithgall, L. S. Watkins, R. E. Frazee, Appl. Opt. 16, 2395 (1977).
    [CrossRef] [PubMed]
  9. H. M. Presby, J. Opt. Soc. Am. 64, 280 (1974).
    [CrossRef]
  10. H. M. Presby, D. Marcuse, Appl. Opt. 13, 2882 (1974).
    [CrossRef] [PubMed]
  11. P. L. Chu, Electron. Lett. 12, 14 (1976).
    [CrossRef]
  12. P. L. Chu, Electron. Lett. 12, 155 (1976).
    [CrossRef]
  13. P. L. Chu, C. Saekeang, Y. M. Chow, Electron. Lett. 13, 41 (1977).
    [CrossRef]
  14. H. M. Presby, Appl. Opt. 15, 492 (1976).
    [CrossRef] [PubMed]
  15. C. Saekeang, P. L. Chu, J. Opt. Soc. Am. 68, 1298 (1978).
    [CrossRef]
  16. P. L. Chu, Electron. Lett. 13, 763 (1977).
    [CrossRef]
  17. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  18. U. Buck, J. Chem. Phys. 54, 1923 (1971).
    [CrossRef]
  19. F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
    [CrossRef]

1978 (1)

1977 (5)

P. L. Chu, Electron. Lett. 13, 763 (1977).
[CrossRef]

P. L. Chu, C. Saekeang, Y. M. Chow, Electron. Lett. 13, 41 (1977).
[CrossRef]

Y. Ohtsuka, Y. Shimizu, Appl. Opt. 16, 1050 (1977).
[PubMed]

M. J. Saunders, W. B. Gardner, Appl. Opt. 16, 2369 (1977).
[CrossRef]

D. H. Smithgall, L. S. Watkins, R. E. Frazee, Appl. Opt. 16, 2395 (1977).
[CrossRef] [PubMed]

1976 (5)

P. L. Chu, Electron. Lett. 12, 14 (1976).
[CrossRef]

P. L. Chu, Electron. Lett. 12, 155 (1976).
[CrossRef]

T. Okoshi, K. Hotate, Appl. Opt. 15, 2756 (1976).
[CrossRef] [PubMed]

H. M. Presby, Appl. Opt. 15, 492 (1976).
[CrossRef] [PubMed]

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

1975 (1)

M. E. Marhic, P. S. Ho, M. Epstein, Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

1974 (3)

1971 (1)

U. Buck, J. Chem. Phys. 54, 1923 (1971).
[CrossRef]

Adams, M. J.

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

Buck, U.

U. Buck, J. Chem. Phys. 54, 1923 (1971).
[CrossRef]

Chow, Y. M.

P. L. Chu, C. Saekeang, Y. M. Chow, Electron. Lett. 13, 41 (1977).
[CrossRef]

Chu, P. L.

C. Saekeang, P. L. Chu, J. Opt. Soc. Am. 68, 1298 (1978).
[CrossRef]

P. L. Chu, Electron. Lett. 13, 763 (1977).
[CrossRef]

P. L. Chu, C. Saekeang, Y. M. Chow, Electron. Lett. 13, 41 (1977).
[CrossRef]

P. L. Chu, Electron. Lett. 12, 14 (1976).
[CrossRef]

P. L. Chu, Electron. Lett. 12, 155 (1976).
[CrossRef]

P. L. Chu, T. W. Whitbread, Appl. Opt.18, 000 (1979), same issue to be published.

Epstein, M.

M. E. Marhic, P. S. Ho, M. Epstein, Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

Frazee, R. E.

Gardner, W. B.

M. J. Saunders, W. B. Gardner, Appl. Opt. 16, 2369 (1977).
[CrossRef]

Ho, P. S.

M. E. Marhic, P. S. Ho, M. Epstein, Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

Hotate, K.

Iga, K.

K. Iga, Y. Kokubun, in Technical Digest of International Conference Integrated Optics and Optical Fiber Communication, Japan (1977), p. 403.

Kokubun, Y.

K. Iga, Y. Kokubun, in Technical Digest of International Conference Integrated Optics and Optical Fiber Communication, Japan (1977), p. 403.

Marcuse, D.

Marhic, M. E.

M. E. Marhic, P. S. Ho, M. Epstein, Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

Ohtsuka, Y.

Okoshi, T.

Payne, D. N.

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

Presby, H. M.

Saekeang, C.

C. Saekeang, P. L. Chu, J. Opt. Soc. Am. 68, 1298 (1978).
[CrossRef]

P. L. Chu, C. Saekeang, Y. M. Chow, Electron. Lett. 13, 41 (1977).
[CrossRef]

Saunders, M. J.

M. J. Saunders, W. B. Gardner, Appl. Opt. 16, 2369 (1977).
[CrossRef]

Shimizu, Y.

Sladen, F. M. E.

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

Smithgall, D. H.

Watkins, L. S.

Whitbread, T. W.

P. L. Chu, T. W. Whitbread, Appl. Opt.18, 000 (1979), same issue to be published.

Appl. Opt. (6)

Appl. Phys. Lett. (2)

M. E. Marhic, P. S. Ho, M. Epstein, Appl. Phys. Lett. 26, 574 (1975).
[CrossRef]

F. M. E. Sladen, D. N. Payne, M. J. Adams, Appl. Phys. Lett. 28, 255 (1976).
[CrossRef]

Electron. Lett. (4)

P. L. Chu, Electron. Lett. 13, 763 (1977).
[CrossRef]

P. L. Chu, Electron. Lett. 12, 14 (1976).
[CrossRef]

P. L. Chu, Electron. Lett. 12, 155 (1976).
[CrossRef]

P. L. Chu, C. Saekeang, Y. M. Chow, Electron. Lett. 13, 41 (1977).
[CrossRef]

J. Chem. Phys. (1)

U. Buck, J. Chem. Phys. 54, 1923 (1971).
[CrossRef]

J. Opt. Soc. Am. (3)

Other (3)

P. L. Chu, T. W. Whitbread, Appl. Opt.18, 000 (1979), same issue to be published.

K. Iga, Y. Kokubun, in Technical Digest of International Conference Integrated Optics and Optical Fiber Communication, Japan (1977), p. 403.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (11)

Fig. 1
Fig. 1

Geometry of incident and scattered rays used in the uniform approximation analysis.

Fig. 2
Fig. 2

Calculated backscattered light patterns for a parabolic-index fiber with kb = 600, core-to-cladding ratio a/b = 0.425 and Δ = 0.01288: (a) due to UR(ϕ); (b) due to U0(ϕ) and UR(ϕ).

Fig. 3
Fig. 3

The deflection function for the parabolic-index fiber. The heavily drawn curve shows the part of Φ(y) which is known.

Fig. 4
Fig. 4

Ray scattering by unclad fiber in the forward direction.

Fig. 5
Fig. 5

Backward ray scattering by clad fiber.

Fig. 6
Fig. 6

Given refractive index profile (solid line) and the calculated profile from the proposed method (denoted by stars).

Fig. 7
Fig. 7

Experimental setup.

Fig. 8
Fig. 8

Backscattered patterns from Southampton University fiber: (a) the measured pattern; (b) the calculated pattern from the uniform approximation with index profile obtained from the proposed method.

Fig. 9
Fig. 9

Calculated deflection function for Southampton University fiber.

Fig. 10
Fig. 10

Refractive index profile of Southampton University fiber. The solid line is determined by a near-field measurement, and the stars are the profile obtained from the reported method.

Fig. 11
Fig. 11

Optical pathlength of a ray.

Equations (35)

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I ( ϕ ) = U 0 ( ϕ ) + U R ( ϕ ) 2 ,
U 0 ( ϕ ) = ( 2 π k Φ ( y 3 ) ) 1 / 2 A ( y 3 ) exp [ i k ψ ( y 3 , ϕ ) + i π / 4 ] ,
U R ( ϕ ) = ( 2 π k ) 1 / 2 [ P · A i ( - w ) + i Q · A i ( - w ) ] exp [ i k B ( ϕ ) ] ,
P = π 1 / 2 w 1 / 4 [ A ( y 2 ) Φ ( y 2 ) 1 / 2 + A ( y 1 ) Φ ( y 1 ) 1 / 2 ] ,
Q = π 1 / 2 w - 1 / 4 [ A ( y 2 ) Φ ( y 2 ) 1 / 2 - A ( y 1 ) Φ ( y 1 ) 1 / 2 ] ,
w = { 3 4 k [ ψ ( y 2 ) - ψ ( y 1 ) ] } 2 / 3 .
ψ ( y ) = 0 y Φ ( ξ ) d ξ + y ϕ + ψ o ,
Φ ( y ) = - ϕ ;             ψ o = 4 0 b n ( r ) d r ,
Φ 2 ( y ) = 4 sin - 1 ( y / n 1 b ) - 2 sin - 1 ( y / b ) - π .
4 3 w N 3 / 2 = ( N - 3 4 ) 2 π .
4 3 w N 3 / 2 = k [ ψ ( y 2 , N ) - ψ ( y 1 , N ) ] .
4 3 w N 3 / 2 = k [ y 1 , N y 2 , N Φ ( y ) d y + ϕ N ( y 2 , N - y 1 , N ) ] .
Δ B = 4 3 k ( w i + 1 3 / 2 - w i 3 / 2 ) ,
Δ B = y 2 , i ( ϕ i + 1 - ϕ i ) + y 2 , i y 2 , i + 1 Φ 2 ( y ) d y + ( y 2 , i + 1 - y 2 , i ) ϕ i + 1 - Δ C ,
Δ C = 1 2 ( y 1 , i + y 1 , i + 1 ) ( ϕ i + 1 - ϕ i ) .
y 1 , i + y 1 , i + 1 = 2 y 2 , i + 2 ( K 2 - K 1 ) / ( ϕ i + 1 - ϕ i ) ,
K 1 = 4 3 ( w i + 1 3 / 2 - w i 3 / 2 ) / k ,
K 2 = y 2 , i y 2 , i + 1 Φ 2 ( y ) d y + ( y 2 , i + 1 - y 2 , i ) ϕ i + 1 ,
Θ ( y ) = π - 2 n ( a ) y r o d r r [ r 2 n 2 ( r ) - n 2 ( a ) y 2 ] 1 / 2 ,
t ( r ) = r n ( r ) / n ( a ) ,
Θ ( y ) = - 2 y y h ( t ) d t ( t 2 - y 2 ) 1 / 2 ,
h ( t ) = log [ r ( t ) / t ] .
h ( t ) = 1 π t Θ ( y ) d y ( y 2 - t 2 ) 1 / 2 = 1 π 0 cosh - 1 ( a / t ) Θ ( t cosh x ) d x ,
r ( t ) = t · exp [ h ( t ) ] ,
n [ r ( t ) ] = n ( a ) exp [ - h ( t ) ] .
y = y / n 1 ,
Θ ( y ) = 1 2 [ Φ ( y ) - 4 sin - 1 ( y / n 1 b ) + 2 sin - 1 ( y / b ) + π ] .
Ψ ( y ) = 4 r 0 b n ( r ) d s = 4 r 0 b n ( r ) d s d r d r = 4 r 0 b r n 2 ( r ) d r [ r 2 n 2 ( r ) - y 2 ] 1 / 2 = 4 r 0 b [ r 2 n 2 ( r ) - y 2 ] 1 / 2 ( 1 / r ) d r + ( y 2 / r ) d r [ r 2 n 2 ( r ) - y 2 ] 1 / 2 = 4 r 0 b [ r 2 n 2 ( r ) - y 2 ] 1 / 2 r d r + 4 y 2 r 0 b d r r [ r 2 n 2 ( r ) - y 2 ] 1 / 2 ,
θ ( y ) = y r 0 b d r r [ r 2 n 2 ( r ) - y 2 ] 1 / 2 ,
I ( y ) = r 0 b [ r 2 n 2 ( r ) - y 2 ] 1 / 2 r d r ,
d I d y = - y r 0 b d r r [ r 2 n 2 ( r ) - y 2 ] 1 / 2 = - θ ( y ) ,
I ( y ) = - 0 y θ ( ξ ) d ξ + I ( 0 ) = - 0 y θ ( ξ ) d ξ + 0 b n ( r ) d r .
Φ ( y ) = - ϕ = π - 2 α - 4 θ ( y )
4 θ = π - 2 α - Φ ( y ) .
Ψ ( y ) = - 4 0 y θ ( ξ ) d ξ + 4 0 b n ( r ) d r + 4 y θ ( y ) = - 0 y [ π - 2 α - Φ ( ξ ) ] d ξ + y [ π - 2 α - Φ ( y ) ] + 4 0 b n ( r ) d r = 0 y Φ ( ξ ) d ξ + y ϕ + 4 0 b n ( r ) d r .

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