Abstract

A ray theory of light backscattering from an optical fiber of arbitrary refractive-index profile in the rainbow region is presented. The backscattered light field is expressed in terms of Airy function and its first derivative, and is uniformly valid throughout the rainbow region. Thus this theory is called the uniform approximation approach. The theory gives results in excellent agreement with those from the wave theory but it has the advantage of providing us with a better understanding of the scattering process and is also less time consuming in computation. Numerical results of backscattered light patterns from five different fibers are given.

© 1978 Optical Society of America

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References

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  1. L. S. Watkins, “Scattering from side-illuminated clad glass fibers for determination of fiber parameters,” J. Opt. Soc. Am. 64, 767–772 (1974).
    [Crossref]
  2. H. M. Presby, “Refractive index and diameter measurements of unclad optical fibers,” J. Opt. Soc. Am. 64, 280–284 (1974).
    [Crossref]
  3. D. Marcuse, “Light scattering from unclad fibers: ray theory,” Appl. Opt. 14, 1528–1532 (1975).
    [Crossref] [PubMed]
  4. P. L. Chu, “Determination of diameter of unclad optical fibre,” Electron. Lett. 12, 14–16 (1976).
    [Crossref]
  5. P. L. Chu, “Determination of diameters and refractive indices of step-index optical fibres,” ibid., 155–157 (1976).
  6. J. W. Y. Lit, “Radius of unclad fibre from backscattered radiation pattern,” J. Opt. Soc. Am. 65, 1311–1315 (1975).
    [Crossref]
  7. H. M. Presby and D. Marcuse, “Refractive index and diameter determinations of step index optical fibers and preforms.” Appl. Opt. 13, 2882–2885 (1974).
    [Crossref] [PubMed]
  8. D. Marcuse, “Light scattering from elliptical fibers,” Appl. Opt. 13, 1903–1905 (1974).
    [Crossref] [PubMed]
  9. P. L. Chu, C. Saekeang, and Y. M. Chow, “Determinations of refractive index and ellipticity of unclad elliptical optical fibre,” Electrn. Lett. 13, 41–42 (1977).
    [Crossref]
  10. H. M. Presby, “Ellipticity measurement of optical fibers,” Appl. Opt. 15, 492–494 (1976).
    [Crossref] [PubMed]
  11. T. Okoshi and K. Hotate, “Refractive index profile of an optical fiber: its measurement by the scattering-pattern method,” Appl. Opt. 15, 2756–2764 (1976).
    [Crossref] [PubMed]
  12. K. Iga and Y. Kokubun, “Precise measurement of the refractive index profile of optical fibres by nondestructive interference method,” IOOC Technical Digest, 403–406 (1977).
  13. M. E. Marhic, R. S. Ho, and M. Epstein, “Nondestructive refractive-index profile measurements of clad optical fibers,” Appl. Phys. Lett. 26, 574–575 (1975).
    [Crossref]
  14. Y. Ohtsuka and Y. Shimizu, “Radial distribution of the refractive index in light focusing rods: determination using interphako microscopy,” Appl. Opt. 16, 1050–1053 (1977).
    [PubMed]
  15. M. J. Saunders and W. B. Gardner, “Nondstructive interferometric measurement of the delta and alpha of clad optical fibers,” Appl. Opt. 16, 2369–2371 (1977).
    [Crossref]
  16. P. L. Chu, “Nondestructive measurement of index profile of an optical fibre preform,” Electron. Lett. 13, 736–738 (1977).
    [Crossref]
  17. D. Marcuse and H. M. Presby, “Light scattering from optical fibers with arbitrary refractive-index distribution,” J. Opt. Soc. Am. 65, 367–375 (1975).
    [Crossref]
  18. P. L. Chu, “Application of geometrical optics to scattering of plane electromagnetic waves by graded index optical fibre,” Proc. URSI Symposium on Electromagnetic Wave Theory, 144–146 (1977).
  19. M. V. Berry, “Uniform approximation for potential scattering involving a rainbow,” Proc. Phys. Soc. 89, 479–490 (1966).
    [Crossref]
  20. U. Buck, “Inversion of molecular scattering data,” Rev. Mod. Phys. 46, 369–389 (1974).
    [Crossref]
  21. H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957).
  22. J. Holoubek, “Light scattering from unclad fibers: approximate ray theory of backscattered light,” Appl. Opt. 15, 2751–2755 (1976).
    [Crossref] [PubMed]
  23. M. J. Lighthill, Fourier Analysis and Generalised Functions, (Cambridge University, London, 1970).
  24. C. Chester, B. Friedmann, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
    [Crossref]
  25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, (Dover, New York, 1972).
  26. Z. Kopal, Numerical Analysis, (Chapman and Hall, London, 1957).

1977 (6)

P. L. Chu, C. Saekeang, and Y. M. Chow, “Determinations of refractive index and ellipticity of unclad elliptical optical fibre,” Electrn. Lett. 13, 41–42 (1977).
[Crossref]

Y. Ohtsuka and Y. Shimizu, “Radial distribution of the refractive index in light focusing rods: determination using interphako microscopy,” Appl. Opt. 16, 1050–1053 (1977).
[PubMed]

M. J. Saunders and W. B. Gardner, “Nondstructive interferometric measurement of the delta and alpha of clad optical fibers,” Appl. Opt. 16, 2369–2371 (1977).
[Crossref]

P. L. Chu, “Nondestructive measurement of index profile of an optical fibre preform,” Electron. Lett. 13, 736–738 (1977).
[Crossref]

K. Iga and Y. Kokubun, “Precise measurement of the refractive index profile of optical fibres by nondestructive interference method,” IOOC Technical Digest, 403–406 (1977).

P. L. Chu, “Application of geometrical optics to scattering of plane electromagnetic waves by graded index optical fibre,” Proc. URSI Symposium on Electromagnetic Wave Theory, 144–146 (1977).

1976 (4)

1975 (4)

1974 (5)

1966 (1)

M. V. Berry, “Uniform approximation for potential scattering involving a rainbow,” Proc. Phys. Soc. 89, 479–490 (1966).
[Crossref]

1957 (1)

C. Chester, B. Friedmann, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

Abramowitz, M.

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

Berry, M. V.

M. V. Berry, “Uniform approximation for potential scattering involving a rainbow,” Proc. Phys. Soc. 89, 479–490 (1966).
[Crossref]

Buck, U.

U. Buck, “Inversion of molecular scattering data,” Rev. Mod. Phys. 46, 369–389 (1974).
[Crossref]

Chester, C.

C. Chester, B. Friedmann, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

Chow, Y. M.

P. L. Chu, C. Saekeang, and Y. M. Chow, “Determinations of refractive index and ellipticity of unclad elliptical optical fibre,” Electrn. Lett. 13, 41–42 (1977).
[Crossref]

Chu, P. L.

P. L. Chu, C. Saekeang, and Y. M. Chow, “Determinations of refractive index and ellipticity of unclad elliptical optical fibre,” Electrn. Lett. 13, 41–42 (1977).
[Crossref]

P. L. Chu, “Nondestructive measurement of index profile of an optical fibre preform,” Electron. Lett. 13, 736–738 (1977).
[Crossref]

P. L. Chu, “Application of geometrical optics to scattering of plane electromagnetic waves by graded index optical fibre,” Proc. URSI Symposium on Electromagnetic Wave Theory, 144–146 (1977).

P. L. Chu, “Determination of diameter of unclad optical fibre,” Electron. Lett. 12, 14–16 (1976).
[Crossref]

P. L. Chu, “Determination of diameters and refractive indices of step-index optical fibres,” ibid., 155–157 (1976).

Epstein, M.

M. E. Marhic, R. S. Ho, and M. Epstein, “Nondestructive refractive-index profile measurements of clad optical fibers,” Appl. Phys. Lett. 26, 574–575 (1975).
[Crossref]

Friedmann, B.

C. Chester, B. Friedmann, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

Gardner, W. B.

M. J. Saunders and W. B. Gardner, “Nondstructive interferometric measurement of the delta and alpha of clad optical fibers,” Appl. Opt. 16, 2369–2371 (1977).
[Crossref]

Ho, R. S.

M. E. Marhic, R. S. Ho, and M. Epstein, “Nondestructive refractive-index profile measurements of clad optical fibers,” Appl. Phys. Lett. 26, 574–575 (1975).
[Crossref]

Holoubek, J.

Hotate, K.

Iga, K.

K. Iga and Y. Kokubun, “Precise measurement of the refractive index profile of optical fibres by nondestructive interference method,” IOOC Technical Digest, 403–406 (1977).

Kokubun, Y.

K. Iga and Y. Kokubun, “Precise measurement of the refractive index profile of optical fibres by nondestructive interference method,” IOOC Technical Digest, 403–406 (1977).

Kopal, Z.

Z. Kopal, Numerical Analysis, (Chapman and Hall, London, 1957).

Lighthill, M. J.

M. J. Lighthill, Fourier Analysis and Generalised Functions, (Cambridge University, London, 1970).

Lit, J. W. Y.

Marcuse, D.

Marhic, M. E.

M. E. Marhic, R. S. Ho, and M. Epstein, “Nondestructive refractive-index profile measurements of clad optical fibers,” Appl. Phys. Lett. 26, 574–575 (1975).
[Crossref]

Ohtsuka, Y.

Okoshi, T.

Presby, H. M.

Saekeang, C.

P. L. Chu, C. Saekeang, and Y. M. Chow, “Determinations of refractive index and ellipticity of unclad elliptical optical fibre,” Electrn. Lett. 13, 41–42 (1977).
[Crossref]

Saunders, M. J.

M. J. Saunders and W. B. Gardner, “Nondstructive interferometric measurement of the delta and alpha of clad optical fibers,” Appl. Opt. 16, 2369–2371 (1977).
[Crossref]

Shimizu, Y.

Stegun, I. A.

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

Ursell, F.

C. Chester, B. Friedmann, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

van de Hulst, H. C.

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

Watkins, L. S.

Appl. Opt. (8)

Appl. Phys. Lett. (1)

M. E. Marhic, R. S. Ho, and M. Epstein, “Nondestructive refractive-index profile measurements of clad optical fibers,” Appl. Phys. Lett. 26, 574–575 (1975).
[Crossref]

Electrn. Lett. (1)

P. L. Chu, C. Saekeang, and Y. M. Chow, “Determinations of refractive index and ellipticity of unclad elliptical optical fibre,” Electrn. Lett. 13, 41–42 (1977).
[Crossref]

Electron. Lett. (2)

P. L. Chu, “Determination of diameter of unclad optical fibre,” Electron. Lett. 12, 14–16 (1976).
[Crossref]

P. L. Chu, “Nondestructive measurement of index profile of an optical fibre preform,” Electron. Lett. 13, 736–738 (1977).
[Crossref]

IOOC Technical Digest (1)

K. Iga and Y. Kokubun, “Precise measurement of the refractive index profile of optical fibres by nondestructive interference method,” IOOC Technical Digest, 403–406 (1977).

J. Opt. Soc. Am. (4)

Proc. Camb. Phil. Soc. (1)

C. Chester, B. Friedmann, and F. Ursell, “An extension of the method of steepest descents,” Proc. Camb. Phil. Soc. 53, 599–611 (1957).
[Crossref]

Proc. Phys. Soc. (1)

M. V. Berry, “Uniform approximation for potential scattering involving a rainbow,” Proc. Phys. Soc. 89, 479–490 (1966).
[Crossref]

Proc. URSI Symposium on Electromagnetic Wave Theory (1)

P. L. Chu, “Application of geometrical optics to scattering of plane electromagnetic waves by graded index optical fibre,” Proc. URSI Symposium on Electromagnetic Wave Theory, 144–146 (1977).

Rev. Mod. Phys. (1)

U. Buck, “Inversion of molecular scattering data,” Rev. Mod. Phys. 46, 369–389 (1974).
[Crossref]

Other (5)

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

P. L. Chu, “Determination of diameters and refractive indices of step-index optical fibres,” ibid., 155–157 (1976).

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

Z. Kopal, Numerical Analysis, (Chapman and Hall, London, 1957).

M. J. Lighthill, Fourier Analysis and Generalised Functions, (Cambridge University, London, 1970).

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Figures (11)

FIG. 1
FIG. 1

Geometry of incident and scattered rays used in the analysis. This diagram defines the deflection angle and the reference planes for the phase function ψ(y).

FIG. 2
FIG. 2

A typical deflection function found in the case of rainbow scattering.

FIG. 3
FIG. 3

Calculated backscattered light patterns for an unclad fiber with kb = 665.25 and n = 1.457: (a) a result of the uniform approximation; (b) a result of the exact wave analysis after Marcuse and Presby.17

FIG. 4
FIG. 4

Refractive-index profile of an approximate step-index fiber (solid line: Fig. 7 in Marcuse and Presby’s paper17) and approximate profile (dashed line) used in our analysis.

FIG. 5
FIG. 5

Calculated backscattered light patterns for fiber with approximate profile shown in Fig. 4, kb = 645.4 and a/b =0.6923: (a) a result of the uniform approximation; (b) a result of the perturbational wave analysis.

FIG. 6
FIG. 6

Calculated backscattered light patterns for parabolic-profile fiber with kb = 600, core-to-cladding ratio a/b = 0.425 and Δ = 0.012 88: (a) a result of the uniform approximation; (b) a result of the perturbational wave analysis.

FIG. 7
FIG. 7

Refractive-index distribution of a parabolic-profile fiber with a valley having depth equal to half the peak, kb = 600 and a/b = 0.425.

FIG. 8
FIG. 8

Deflection functions of parabolic-profile fibers: fiber without a dip at core and cladding boundary (dashed line) and fiber with a dip (solid line).

FIG. 9
FIG. 9

Calculated backscattered light patterns for fiber shown in Fig. 7: (a) a result of the uniform approximation; (b) a result of the perturbational wave analysis.

FIG. 10
FIG. 10

Refractive-index profile of graded-index fiber (solid line: Fig. 12 in Marcuse and Presby’s paper17) and the approximate profile (dashed line) used in our analysis.

FIG. 11
FIG. 11

Calculated backscattered light patterns for fiber given in Fig. 10 with kb = 635.47 and a/b = 0.5: (a) a result of the uniform approximation; (b) a result of the perturbational wave analysis.

Equations (36)

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I ( ϕ ) = | ν A ( y ν , ϕ ) e i k ψ ( y ν , ϕ ) | 2 = U ( ϕ ) 2 ,
U ( ϕ ) = m = - - e - i k 2 m π y A ( y , ϕ ) e i k ψ ( y , ϕ ) d y .
ψ ( y , ϕ ) = η ( y ) + y ϕ ,
Φ ( y ) = π - 2 y r 0 d r r [ r 2 n 2 ( r ) - y 2 ] 1 / 2 ,
r 0 2 n 2 ( r 0 ) - y 2 = 0.
d η d y = Φ ( y ) .
d η d y = Φ ( y ) = 2 m π - ϕ .
U ( ϕ ) = - A ( y , ϕ ) e i k [ η ( y ) + y ϕ ] d y .
I ( ϕ ) = U 0 ( ϕ ) + U R ( ϕ ) 2 ,
U 0 ( ϕ ) = ( 2 π k Φ ( y 3 ) ) 1 / 2 A ( y 3 , ϕ ) e i k ψ ( y 3 , ϕ ) + i π / 4 ,
k ψ ( y 3 , ϕ ) = 3 π / 2 + k L 3 ,
U R ( ϕ ) = ( 2 π k Φ ( y 1 ) ) 1 / 2 A ( y 1 , ϕ ) e i k ψ ( y 1 , ϕ ) + i π / 4 + ( 2 π k Φ ( y 2 ) ) 1 / 2 A ( y 2 , ϕ ) e i k ψ ( y 2 , ϕ ) - i π / 4 .
I R ( ϕ ) = 2 π k ( A 2 ( y 1 ) Φ ( y 1 ) + A 2 ( y 2 ) Φ ( y 2 ) + 2 A ( y 1 ) · A ( y 2 ) Φ ( y 1 ) · Φ ( y 2 ) 1 / 2 cos { k [ ψ ( y 2 ) - ψ ( y 1 ) ] - π / 2 } )
y = y r + v
ψ ( y , ϕ ) = η ( y r + v , ϕ ) + ( y r + v ) ϕ = η ( y r ) + η ( y r ) v + η ( y r ) v 2 2 + η ( y r ) v 3 6 + ( y r + v ) ϕ + O ( v 4 ) = η ( y r ) + y r ϕ + ( ϕ - ϕ r ) v - Φ ( y r ) v 3 6 + O ( v 4 ) ,
U R ( ϕ ) = - A ( y , ϕ ) exp { i k ψ ( y r , ϕ ) + i k [ ( ϕ - ϕ r ) v - Φ ( y r ) v 3 / 6 ] } d v .
U R ( ϕ ) = 2 1 / 3 π A ( y r , ϕ ) k Φ ( y r ) 1 / 3 × A i ( - | 2 k 2 Φ ( y r ) | 1 / 3 ( ϕ - ϕ r ) ) e i k ψ ( y r , ϕ )
I R ( ϕ ) = 2 2 / 3 π 2 A 2 ( y r , ϕ ) k Φ ( y r ) 2 / 3 A i 2 ( - | 2 k 2 Φ ( y r ) | 1 / 3 ( ϕ - ϕ r ) ) .
ψ ( y , ϕ ) = - ζ ( ϕ ) z + 1 3 z 3 + B ( ϕ ) .
4 3 ζ 3 / 2 ( ϕ ) = ψ ( y 2 ) - ψ ( y 1 ) ,
B ( ϕ ) = 1 2 [ ψ ( y 2 ) + ψ ( y 1 ) ] ,
U R ( ϕ ) = - A ( y , ϕ ) d y d z exp { i k [ - ζ ( ϕ ) z + 1 / 3 z 3 + B ( ϕ ) ] } d z .
A ( y , ϕ ) d y d z = m = 0 [ z 2 - ζ ( ϕ ) ] m · ( p m + q m z ) .
A ( y , ϕ ) d y d z = p 0 + q 0 z .
U R ( ϕ ) = ( 2 π k ) 1 / 2 ( P · A i ( - w ) + i Q · A i ( - w ) ) e i k B ( ϕ ) ,
P = w 1 / 4 ( A ( y 2 ) Φ ( y 2 ) 1 / 2 + A ( y 1 ) Φ ( y 1 ) 1 / 2 ) ,
Q = 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 .
Φ ( y ) = Φ ( y r ) ( y - y r ) + O [ ( y - y r ) 2 ] ,
I R ( ϕ ) = 2 π k [ P 2 · A i 2 ( - w ) + Q 2 · A i 2 ( - w ) ] .
I ( ϕ ) = U 0 ( ϕ ) + U R ( ϕ ) 2 = 2 π k | A ( y 3 ) Φ ( y 3 ) 1 / 2 e i k ψ ( y 3 , ϕ ) + i π / 4 + [ e i k B ( ϕ ) · { P · A i ( - w ) + i Q · A i ( - w ) } ] | 2 .
Φ ( y ) = π - 2 sin - 1 ( y / b ) - 4 y r 0 b d r r [ r 2 n 2 ( r ) - y 2 ] 1 / 2
ψ ( y , ϕ ) = 2 [ b - ( b 2 - y 2 ) 1 / 2 ] + 4 r 0 b r n 2 ( r ) d r [ r 2 n 2 ( r ) - y 2 ] 1 / 2
L 3 = 2 b ( 1 - cos ϕ / 2 ) .
Φ ( y ) = 4 sin - 1 ( y / n b ) - 2 sin - 1 ( y / b ) - π ,
ψ ( y ) = 2 [ b - ( b 2 - y 2 ) 1 / 2 ] + 4 n b cos [ sin - 1 ( y / n b ) ] ,