Abstract

A mathematical formula is derived from the multiple-beam Fizeau fringers crossing an elliptical multilayer fiber. A formula for the two-beam interference fringes crossing this fiber is also given. A model that divides the elliptical fiber cross-section into elliptical zones of constant refractive index is assumed. The index profile of the poly(aryl-ether-ether-ketone) fiber is calculated by using the mathematical formulas. Microinterferograms of both the multiple-beam Fizeau fringes and the totally duplicated image of the fiber from a two-beam intereference microscope are used for the determination of the refractive-index profile of the fiber. The cross-sectional shape is determined from the diffraction pattern of a He–Ne laser beam. Microinterferograms are given for illustration.

© 1992 Optical Society of America

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References

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  1. W. E. Morton, J. W. S. Hearle, Physical Properties of Textile Fibers, (Textile Institute, London, 1975), pp. 573–578.
  2. R. C. Faust, “An interferometric method of studying local variation in the refractive indices of solids,” Proc. Phys. Soc. London Ser. B 65, 48–61 (1952).
    [CrossRef]
  3. R. C. Faust, “The use of the Baker interference microscope for the study of optically heterogeneous specimens,” Quart. J. Microsc. Sci. 97, 569–591 (1956).
  4. N. Barakat, “Interferometric studies on fibers, part I: theory of interferometric determination of indices of fibers,” Text. Res. J. 14, 167–170 (1971).
    [CrossRef]
  5. A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Multiple-beam interferometric studies on fibers with irregular transverse sections,” J. Phys. D 18, 1773–1780 (1985).
    [CrossRef]
  6. A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Interferometric determination of optical properties of fibers with irregular transverse sections and having a skin-core structure,” J. Phys. D 18, 2321–2328 (1985).
    [CrossRef]
  7. N. Barakat, A. A. Hamza, Interferometry of Fibrous Materials (Hilger, London, 1990).
  8. P. L. Chu, “Nondestructive refractive-index profile measurement of elliptical optical fiber or preform,” Electron. Lett. 15, 357–361 (1979).
    [CrossRef]
  9. A. A. Hamza, M. A. Kabeel, “Multiple-beam Fizeau fringes crossing a cylindrical multilayer fiber,” J. Phys. D 19, 1175–1182 (1986).
    [CrossRef]
  10. H. Hanns, “Interferometric measurement of the microstructure of synthetic fibers,” Kolloid Z. Z. Polym. 250, 765–774 (1972).
  11. S. C. Simmens, “Birefringence determination in objects of irregular cross-sectional shape and constant weight per unit length,” Nature (London) 181, 1260–1261 (1958).
    [CrossRef]
  12. S. M. Curry, A. L. Schawlow, “Measuring the diameter of a hair by diffraction,” Am. J. Phys. 42, 12–15 (1974).
    [CrossRef]
  13. L. Lynch, “The use of light-scattering profiles of single fibers to measure and compare the within-fiber diameter variability of wool tops,” Text. Res. J. 44, 203–205 (1974).
    [CrossRef]
  14. N. Barakat, H. A. El-Hennawi, “Interferometric studies on fibers, part II: interferometric determination of refractive indices and birefringence of acrylic fibers,” Text. Res. J. 41, 391–396 (1971).
    [CrossRef]
  15. M. Pluta, “Interference microscopy of polymer fibers,” J. Microsc. 96, 309–332 (1972).
    [CrossRef]
  16. M. Pluta, “A double refracting interference microscope with a continuously variable amount and direction of wavefront shear,” Opt. Acta 18, 661–675 (1971).
    [CrossRef]

1986 (1)

A. A. Hamza, M. A. Kabeel, “Multiple-beam Fizeau fringes crossing a cylindrical multilayer fiber,” J. Phys. D 19, 1175–1182 (1986).
[CrossRef]

1985 (2)

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Multiple-beam interferometric studies on fibers with irregular transverse sections,” J. Phys. D 18, 1773–1780 (1985).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Interferometric determination of optical properties of fibers with irregular transverse sections and having a skin-core structure,” J. Phys. D 18, 2321–2328 (1985).
[CrossRef]

1979 (1)

P. L. Chu, “Nondestructive refractive-index profile measurement of elliptical optical fiber or preform,” Electron. Lett. 15, 357–361 (1979).
[CrossRef]

1974 (2)

S. M. Curry, A. L. Schawlow, “Measuring the diameter of a hair by diffraction,” Am. J. Phys. 42, 12–15 (1974).
[CrossRef]

L. Lynch, “The use of light-scattering profiles of single fibers to measure and compare the within-fiber diameter variability of wool tops,” Text. Res. J. 44, 203–205 (1974).
[CrossRef]

1972 (2)

H. Hanns, “Interferometric measurement of the microstructure of synthetic fibers,” Kolloid Z. Z. Polym. 250, 765–774 (1972).

M. Pluta, “Interference microscopy of polymer fibers,” J. Microsc. 96, 309–332 (1972).
[CrossRef]

1971 (3)

M. Pluta, “A double refracting interference microscope with a continuously variable amount and direction of wavefront shear,” Opt. Acta 18, 661–675 (1971).
[CrossRef]

N. Barakat, “Interferometric studies on fibers, part I: theory of interferometric determination of indices of fibers,” Text. Res. J. 14, 167–170 (1971).
[CrossRef]

N. Barakat, H. A. El-Hennawi, “Interferometric studies on fibers, part II: interferometric determination of refractive indices and birefringence of acrylic fibers,” Text. Res. J. 41, 391–396 (1971).
[CrossRef]

1958 (1)

S. C. Simmens, “Birefringence determination in objects of irregular cross-sectional shape and constant weight per unit length,” Nature (London) 181, 1260–1261 (1958).
[CrossRef]

1956 (1)

R. C. Faust, “The use of the Baker interference microscope for the study of optically heterogeneous specimens,” Quart. J. Microsc. Sci. 97, 569–591 (1956).

1952 (1)

R. C. Faust, “An interferometric method of studying local variation in the refractive indices of solids,” Proc. Phys. Soc. London Ser. B 65, 48–61 (1952).
[CrossRef]

Barakat, N.

N. Barakat, “Interferometric studies on fibers, part I: theory of interferometric determination of indices of fibers,” Text. Res. J. 14, 167–170 (1971).
[CrossRef]

N. Barakat, H. A. El-Hennawi, “Interferometric studies on fibers, part II: interferometric determination of refractive indices and birefringence of acrylic fibers,” Text. Res. J. 41, 391–396 (1971).
[CrossRef]

N. Barakat, A. A. Hamza, Interferometry of Fibrous Materials (Hilger, London, 1990).

Chu, P. L.

P. L. Chu, “Nondestructive refractive-index profile measurement of elliptical optical fiber or preform,” Electron. Lett. 15, 357–361 (1979).
[CrossRef]

Curry, S. M.

S. M. Curry, A. L. Schawlow, “Measuring the diameter of a hair by diffraction,” Am. J. Phys. 42, 12–15 (1974).
[CrossRef]

El-Hennawi, H. A.

N. Barakat, H. A. El-Hennawi, “Interferometric studies on fibers, part II: interferometric determination of refractive indices and birefringence of acrylic fibers,” Text. Res. J. 41, 391–396 (1971).
[CrossRef]

Faust, R. C.

R. C. Faust, “The use of the Baker interference microscope for the study of optically heterogeneous specimens,” Quart. J. Microsc. Sci. 97, 569–591 (1956).

R. C. Faust, “An interferometric method of studying local variation in the refractive indices of solids,” Proc. Phys. Soc. London Ser. B 65, 48–61 (1952).
[CrossRef]

Hamza, A. A.

A. A. Hamza, M. A. Kabeel, “Multiple-beam Fizeau fringes crossing a cylindrical multilayer fiber,” J. Phys. D 19, 1175–1182 (1986).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Multiple-beam interferometric studies on fibers with irregular transverse sections,” J. Phys. D 18, 1773–1780 (1985).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Interferometric determination of optical properties of fibers with irregular transverse sections and having a skin-core structure,” J. Phys. D 18, 2321–2328 (1985).
[CrossRef]

N. Barakat, A. A. Hamza, Interferometry of Fibrous Materials (Hilger, London, 1990).

Hanns, H.

H. Hanns, “Interferometric measurement of the microstructure of synthetic fibers,” Kolloid Z. Z. Polym. 250, 765–774 (1972).

Hearle, J. W. S.

W. E. Morton, J. W. S. Hearle, Physical Properties of Textile Fibers, (Textile Institute, London, 1975), pp. 573–578.

Kabeel, M. A.

A. A. Hamza, M. A. Kabeel, “Multiple-beam Fizeau fringes crossing a cylindrical multilayer fiber,” J. Phys. D 19, 1175–1182 (1986).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Multiple-beam interferometric studies on fibers with irregular transverse sections,” J. Phys. D 18, 1773–1780 (1985).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Interferometric determination of optical properties of fibers with irregular transverse sections and having a skin-core structure,” J. Phys. D 18, 2321–2328 (1985).
[CrossRef]

Lynch, L.

L. Lynch, “The use of light-scattering profiles of single fibers to measure and compare the within-fiber diameter variability of wool tops,” Text. Res. J. 44, 203–205 (1974).
[CrossRef]

Morton, W. E.

W. E. Morton, J. W. S. Hearle, Physical Properties of Textile Fibers, (Textile Institute, London, 1975), pp. 573–578.

Pluta, M.

M. Pluta, “Interference microscopy of polymer fibers,” J. Microsc. 96, 309–332 (1972).
[CrossRef]

M. Pluta, “A double refracting interference microscope with a continuously variable amount and direction of wavefront shear,” Opt. Acta 18, 661–675 (1971).
[CrossRef]

Schawlow, A. L.

S. M. Curry, A. L. Schawlow, “Measuring the diameter of a hair by diffraction,” Am. J. Phys. 42, 12–15 (1974).
[CrossRef]

Simmens, S. C.

S. C. Simmens, “Birefringence determination in objects of irregular cross-sectional shape and constant weight per unit length,” Nature (London) 181, 1260–1261 (1958).
[CrossRef]

Sokkar, T. Z. N.

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Interferometric determination of optical properties of fibers with irregular transverse sections and having a skin-core structure,” J. Phys. D 18, 2321–2328 (1985).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Multiple-beam interferometric studies on fibers with irregular transverse sections,” J. Phys. D 18, 1773–1780 (1985).
[CrossRef]

Am. J. Phys. (1)

S. M. Curry, A. L. Schawlow, “Measuring the diameter of a hair by diffraction,” Am. J. Phys. 42, 12–15 (1974).
[CrossRef]

Electron. Lett. (1)

P. L. Chu, “Nondestructive refractive-index profile measurement of elliptical optical fiber or preform,” Electron. Lett. 15, 357–361 (1979).
[CrossRef]

J. Microsc. (1)

M. Pluta, “Interference microscopy of polymer fibers,” J. Microsc. 96, 309–332 (1972).
[CrossRef]

J. Phys. D (3)

A. A. Hamza, M. A. Kabeel, “Multiple-beam Fizeau fringes crossing a cylindrical multilayer fiber,” J. Phys. D 19, 1175–1182 (1986).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Multiple-beam interferometric studies on fibers with irregular transverse sections,” J. Phys. D 18, 1773–1780 (1985).
[CrossRef]

A. A. Hamza, T. Z. N. Sokkar, M. A. Kabeel, “Interferometric determination of optical properties of fibers with irregular transverse sections and having a skin-core structure,” J. Phys. D 18, 2321–2328 (1985).
[CrossRef]

Kolloid Z. Z. Polym. (1)

H. Hanns, “Interferometric measurement of the microstructure of synthetic fibers,” Kolloid Z. Z. Polym. 250, 765–774 (1972).

Nature (London) (1)

S. C. Simmens, “Birefringence determination in objects of irregular cross-sectional shape and constant weight per unit length,” Nature (London) 181, 1260–1261 (1958).
[CrossRef]

Opt. Acta (1)

M. Pluta, “A double refracting interference microscope with a continuously variable amount and direction of wavefront shear,” Opt. Acta 18, 661–675 (1971).
[CrossRef]

Proc. Phys. Soc. London Ser. B (1)

R. C. Faust, “An interferometric method of studying local variation in the refractive indices of solids,” Proc. Phys. Soc. London Ser. B 65, 48–61 (1952).
[CrossRef]

Quart. J. Microsc. Sci. (1)

R. C. Faust, “The use of the Baker interference microscope for the study of optically heterogeneous specimens,” Quart. J. Microsc. Sci. 97, 569–591 (1956).

Text. Res. J. (3)

N. Barakat, “Interferometric studies on fibers, part I: theory of interferometric determination of indices of fibers,” Text. Res. J. 14, 167–170 (1971).
[CrossRef]

L. Lynch, “The use of light-scattering profiles of single fibers to measure and compare the within-fiber diameter variability of wool tops,” Text. Res. J. 44, 203–205 (1974).
[CrossRef]

N. Barakat, H. A. El-Hennawi, “Interferometric studies on fibers, part II: interferometric determination of refractive indices and birefringence of acrylic fibers,” Text. Res. J. 41, 391–396 (1971).
[CrossRef]

Other (2)

W. E. Morton, J. W. S. Hearle, Physical Properties of Textile Fibers, (Textile Institute, London, 1975), pp. 573–578.

N. Barakat, A. A. Hamza, Interferometry of Fibrous Materials (Hilger, London, 1990).

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Figures (7)

Fig. 1
Fig. 1

(a) Cross section of an elliptical multilayer fiber (elliptical-zone model) in a silvered-liquid-wedge interferometer. The minor axis is parallel to the direction of the incident beam of the light source. aQ and bQ are the major and minor axes of the elliptical cross section and nL is the refractive index of the liquid. (b) Schematic representation of the resulting fringe; F’s, areas enclosed under the fringe deviation.

Fig. 2
Fig. 2

The diffraction patterns of a PEEK fiber.

Fig. 3
Fig. 3

The cross-sectional shape of a PEEK fiber obtained by the diffraction technique.

Fig. 4
Fig. 4

Microinterferogram showing the multiple-beam Fizeau fringes in transmission crossing a PEEK fiber; λ, 546 nm; nL, 1.6295 at 20°C.

Fig. 5
Fig. 5

Microinterferogram showing differentially sheared images for a sample of the PEEK fiber by using the Pluta microscope; λ, 546 nm; nL = 1.6295 at 20°C.

Fig. 6
Fig. 6

Refractive-index profile n of a PEEK fiber taken by using the microinterferograms of the multiple-beam Fizeau fringes in transmission by (a) the line method and (b) the area method.

Fig. 7
Fig. 7

Refractive-index profile n of PEEK fibers by using the microinterferograms of a two-beam interference polarizing microscope: (a) the line method and (b) the area method.

Equations (17)

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OPL = ( t 2 y 1 ) n L + 2 ( y 1 y 2 ) n 1 + 2 ( y 1 y 3 ) n 2 + + ( y m 1 y m ) n m 1 + 2 y m n m ,
N λ = 2 n L t + 4 y 1 ( n 1 n L ) + 4 y 2 ( n 2 n 1 ) + 4 y 3 ( n 3 n 2 ) + 4 y m ( n m n m 1 ) ,
2 n L tan ( N λ 2 n L tan z ) = 4 Q = 1 m ( n Q n Q 1 ) y Q ,
λ 2 h z = 2 Q = 1 m ( n Q n Q 1 ) y Q ,
y Q 2 = ( a Q 2 x 2 ) e Q 2 , Q = 1 , 2 , 3 , , m ,
λ 4 h z m = Q = 1 m ( n Q n Q 1 ) ( a Q 2 x 2 ) 1 / 2 e Q .
λ 4 h F m = Q = 1 m ( n Q n Q 1 ) A Q , m ,
F m = a m + 1 a m z d x , A Q , m = e Q a m + 1 a m ( a Q 2 x 2 ) 1 / 2 d x ,
I = e Q α β ( D 2 x 2 ) 1 / 2 d x ,
I = e Q θ 2 θ 1 D 2 cos 2 θ d θ .
I = e Q D 2 2 ( sin 1 β D sin 1 α D ) + e Q 2 [ β ( D 2 β 2 ) 1 / 2 α ( D 2 α 2 ) 1 / 2 ] .
A Q , m = e Q a Q 2 2 ( sin 1 a m a Q sin 1 a m + 1 a Q ) + e Q 2 [ a m ( a Q 2 a m 2 ) 1 / 2 a m + 1 ( a Q 2 a m + 1 2 ) 1 / 2 ] .
n m = n m 1 + λ F m 4 h A m , m ( Q = 1 m 1 ( n Q n Q 1 ) { a Q e Q 2 ( sin 1 a m a Q sin 1 a m + 1 a Q ) + e Q 2 [ a m 2 ( a Q 2 a m 2 ) 1 / 2 ] [ a m + 1 ( a Q 2 a m + 1 2 ) 1 / 2 ] } A m , m ) ,
A m , m = a m 2 e m 2 [ sin 1 a m a m sin 1 ( a m + 1 a m ) ] [ a m + 1 e m 2 ( a m 2 a m + 1 2 ) 1 / 2 ] .
λ 2 h z m = Q = 1 m ( n Q n Q 1 ) ( a Q 2 x 2 ) e Q , Q = 1 , 2 , 3 , , m ,
n m = n m 1 + λ F m 2 h A m , m ( Q = 1 m 1 ( n Q n Q 1 ) { a Q e Q 2 ( sin 1 a m a Q sin 1 a m + 1 a Q ) + e Q 2 [ a m ( a Q 2 a m 2 ) 1 / 2 ] [ a m + 1 ( a Q 2 a m + 1 2 ) 1 / 2 ] } A m , m ) ,
x m = ± m λ L / b ,

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