Abstract

Formulas have been derived allowing the determination of the parameter ka (k = 2π/λ, a is the fiber radius) based on the approximate geometric theory for dielectric nonabsorbing unclad fibers of circular cross section from the course backscattered light intensity. The application of the method is restricted to fibers with the refractive indices within the interval (√2,2) and for ka > 100. In this region the method yields results with an accuracy of about 5%.

© 1976 Optical Society of America

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References

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  1. H. M. Presby, J. Opt. Soc. Am. 64, 280 (1974).
    [CrossRef]
  2. D. Marcuse, Appl. Opt. 14, 1528 (1975).
    [CrossRef] [PubMed]
  3. D. Marcuse, H. M. Presby, J. Opt. Soc. Am. 65, 367 (1975).
    [CrossRef]
  4. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 200, 207, 214.
  5. L. S. Watkins, J. Opt. Soc. Am. 64, 767 (1974).
    [CrossRef]
  6. T. Yamaguchi, Appl. Opt. 14, 1111 (1975).
    [CrossRef] [PubMed]
  7. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  8. E. J. Meehan, A. E. Gyberg, Appl. Opt. 12, 551 (1973).
    [CrossRef] [PubMed]
  9. P. J. Livesey, F. W. Billmeyer, J. Colloid Interface Sci. 30, 447 (1969).
    [CrossRef]
  10. H. M. Presby, D. Marcuse, Appl. Opt. 13, 2882 (1974).
    [CrossRef] [PubMed]

1975 (3)

1974 (3)

1973 (1)

1969 (1)

P. J. Livesey, F. W. Billmeyer, J. Colloid Interface Sci. 30, 447 (1969).
[CrossRef]

Billmeyer, F. W.

P. J. Livesey, F. W. Billmeyer, J. Colloid Interface Sci. 30, 447 (1969).
[CrossRef]

Gyberg, A. E.

Kerker, M.

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

Livesey, P. J.

P. J. Livesey, F. W. Billmeyer, J. Colloid Interface Sci. 30, 447 (1969).
[CrossRef]

Marcuse, D.

Meehan, E. J.

Presby, H. M.

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 200, 207, 214.

Watkins, L. S.

Yamaguchi, T.

Appl. Opt. (4)

J. Colloid Interface Sci. (1)

P. J. Livesey, F. W. Billmeyer, J. Colloid Interface Sci. 30, 447 (1969).
[CrossRef]

J. Opt. Soc. Am. (3)

Other (2)

H. C. Van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), pp. 200, 207, 214.

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

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Figures (3)

Fig. 1
Fig. 1

Schematic view of scattering geometry, definitions of angles used in analysis. 1,2,3,4 incident, 1′,2′,3′,4′ scattered rays outgoing the same angle ψ. ij angles of incidence, rj angles of refraction, θ scattering angle, ψ = πθ angle of backscattered light.

Fig. 2
Fig. 2

Backscattered light intensity in relative units for ka = 600 and n = 1.457: (a) a result of interference of rays 1 and 2; (b) the same for rays 2 and 3; (c) a result of approximate geometric theory involving rays 1,2,3, and 4 according to Eq. (1). Lower part of (d) is a result of an exact wave analysis after Marcuse.2

Fig. 3
Fig. 3

Role of ray 4 for higher refractive indices (n = 1.575, polyamide): (a) intensity corresponding to contributions of rays 1,2,3; (b) intensity corresponding to contributions of 1,2,3, and 4.

Tables (2)

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Table I Changes in Ray Phases due to Passage Through Phase Lines

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Table II List of ka Values Obtained by Using Approximate Formulas

Equations (17)

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I = A | p p D p 1 / 2 exp ( i σ p ) | 2 ,
r 1 ( i j ) = [ cos i j - ( n 2 - sin 2 i j ) 1 / 2 ] / [ cos i j + ( n 2 - sin 2 i j ) 1 / 2 ] , j = 1 , 2 , 3 , 4.
D ( i 1 ) = ( cos i 1 ) / 2 , D ( i j ) = [ ( n 2 - sin 2 i j ) 1 / 2 cos i j ] / [ 2 ( n 2 - sin 2 i j ) 1 / 2 - 4 cos i j ] j = 2 , 3 , 4.
σ ( i j ) = δ ( i j ) + ( π / 2 ) ( p + 1 - q / 2 + s / 2 - 2 k ) j , j = 1 , 2 , 3 , 4 ,
δ ( i 1 ) = k a [ 1 - cos ( ψ / 2 ) ] δ ( i j ) = k a [ 4 n cos r j + 1 - 2 cos i j + cos ( ψ / 2 ) ] , j = 2 , 3 , 4.
cos 2 i = ( n 2 - 1 ) / ( p 2 - 1 ) .
[ σ ( i 2 ) - σ ( i 1 ) ] ψ 2 - [ σ ( i 2 ) - σ ( i 1 ) ] ψ 1 = 2 π ,
ka = π [ 2 n ( cos r 21 - cos r 22 ) + cos ( ψ 1 / 2 ) - cos ( ψ 2 / 2 ) + cos i 22 - cos i 21 ] - 1 ,
sin r j = ( sin i j ) / n .
[ σ ( i 3 ) - σ ( i 2 ) ] ψ 3 - [ σ ( i 3 ) - σ ( i 2 ) ] ψ 2 = 2 π ,
ka = π [ 2 n ( cos r 33 + cos r 22 - cos r 32 - cos r 23 ) + cos i 32 + cos i 23 - cos i 33 - cos i 22 ] - 1 .
ψ = 4 r - 2 i ,
sin ( ψ / 2 ) = sin ( 2 r - i ) .
sin ( ψ / 2 ) = ( 2 sin i / n ) ( 1 - sin 2 i / n 2 ) 1 / 2 cos i - ( 1 - 2 sin 2 i / n 2 ) sin i .
sin ( ψ / 2 ) = 2 y [ ( 1 - n 2 y 2 ) ( 1 - y 2 ) ] 1 / 2 - ( 1 - 2 y 2 ) n y ,
y 4 - a 1 y 3 + a 2 y 2 + ( 1 / 2 ) a 1 y + ( 1 / 4 ) a 1 2 / n 2 = 0 ,
ψ 4 , max = 4 arcsin ( 1 / n ) - π ,

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