Abstract

In this paper, we present a modification of one of the common systems for producing multiple-beam fringes of equal chromatic order (FECO). This modification allows us to capture two FECO patterns of different polarizations simultaneously during dynamic investigations of fibers. Using this modified system, the dispersion curves in cases of parallel and perpendicular polarizations can be measured simultaneously. Also, dispersion of birefringence values along the visible spectrum can be calculated using a single FECO pattern.

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References

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  1. F. A. Jenkins and E. H. White, Fundamentals of Optics (McGraw-Hill, 1976).
  2. R. S. Sirohi, Introduction to Optical Metrology (CRC Press, 2015).
  3. A. Hamza, I. Fouda, T. Sokkar, and M. El-Bakary, “Determination of spectral dispersion curves of polypropylene fibres,” J. Opt. A 1, 359 (1999).
    [Crossref]
  4. A. Hamza, T. Sokkar, and W. Ramadan, “On the microinterferometric determination of refractive indices and birefringence of fibres,” Pure Appl. Opt. 1, 321 (1992).
    [Crossref]
  5. A. Hamza, T. Sokkar, and M. Shahin, “Interferometric determination of optical anisotropy in fibers III: multilayer fibers,” J. Appl. Phys. 70, 4480–4484 (1991).
    [Crossref]
  6. F. El-Diasty, “Characterization of optical fibers by two- and multiple-beam interferometry,” Opt. Laser Eng. 46, 291–305 (2008).
    [Crossref]
  7. K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).
  8. S. Tolansky, “XXXI: New contributions to interferometry. Part V—new multiple beam white light interference fringes and their applications,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 36, 225–236 (1945).
    [Crossref]
  9. S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Oxford University, 1948).
  10. P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photon. 7, 1–65 (2015).
    [Crossref]
  11. W. Koehler, “Multiple-beam fringes of equal chromatic order. Part VII. Mechanism of polishing glass,” J. Opt. Soc. Am. 45, 1015–1020 (1955).
    [Crossref]
  12. R. Gupta and J. Fréchette, “Interferometry of surfaces with well-defined topography in the surface force apparatus,” J. Colloid Interface Sci. 412, 82–88 (2013).
    [Crossref]
  13. S. Diddams and J.-C. Diels, “Dispersion measurements with white-light interferometry,” J. Opt. Soc. Am. B 13, 1120–1129 (1996).
    [Crossref]
  14. C. Peucheret, F. Liu, and R. J. S. Pedersen, “Measurement of small dispersion values in optical components [WDM networks],” Electron. Lett. 35, 409–411 (1999).
    [Crossref]
  15. N. Barakat and A. Hindeleh, “Determination of the refractive indices, birefringence, and tensile properties of normal viscose rayon fibers,” Textile Res. J. 34, 581–584 (1964).
    [Crossref]
  16. N. Barakat and A. Hindeleh, “Interferometric determination of the refractive indices and birefringence of mohair wool fibers,” Textile Res. J. 34, 357–362 (1964).
    [Crossref]
  17. A. Hamza, T. Sokkar, K. El-Farahaty, and M. Raslan, “A novel double-image Fizeau system for accurate investigation of anisotropic polymer fibres,” J. Microsc. 254, 84–94 (2014).
    [Crossref]
  18. T. Sokkar, K. El-Farahaty, and M. Raslan, “Online double-arm of a multiple-beam Fizeau system: I. Optical setup for simultaneous recording of two interferometric patterns in the same frame,” Polym. Test. 29, 1065–1074 (2010).
    [Crossref]
  19. E. Riande, J. Guzmán, M. P. Tarazona, and E. Saiz, “Random-coil configurations of aromatic polyesters: stress-optical behavior of poly (diethylene glycol terephthalate),” J. Polym. Sci., Polym. Phys. Ed. 22, 917–929 (1984).
    [Crossref]
  20. E. Cohen, Quantities, Units and Symbols in Physical Chemistry (RSC Publishing, 2007), p. 28.
  21. F. Hernández-Sánchez and P. Herrera-Franco, “Electrical and thermal properties of recycled polypropylene-carbon black composites,” Polym. Bull. 45, 509–516 (2001).
    [Crossref]
  22. H. de Vries, “A new approach to the continuum theory of birefringence of oriented polymers,” Colloid Polym. Sci. 257, 226–238 (1979).
    [Crossref]
  23. M. DiDomenico, “Material dispersion in optical fiber waveguides,” Appl. Opt. 11, 652–654 (1972).
    [Crossref]

2015 (1)

2014 (1)

A. Hamza, T. Sokkar, K. El-Farahaty, and M. Raslan, “A novel double-image Fizeau system for accurate investigation of anisotropic polymer fibres,” J. Microsc. 254, 84–94 (2014).
[Crossref]

2013 (1)

R. Gupta and J. Fréchette, “Interferometry of surfaces with well-defined topography in the surface force apparatus,” J. Colloid Interface Sci. 412, 82–88 (2013).
[Crossref]

2010 (1)

T. Sokkar, K. El-Farahaty, and M. Raslan, “Online double-arm of a multiple-beam Fizeau system: I. Optical setup for simultaneous recording of two interferometric patterns in the same frame,” Polym. Test. 29, 1065–1074 (2010).
[Crossref]

2008 (1)

F. El-Diasty, “Characterization of optical fibers by two- and multiple-beam interferometry,” Opt. Laser Eng. 46, 291–305 (2008).
[Crossref]

2001 (1)

F. Hernández-Sánchez and P. Herrera-Franco, “Electrical and thermal properties of recycled polypropylene-carbon black composites,” Polym. Bull. 45, 509–516 (2001).
[Crossref]

1999 (2)

C. Peucheret, F. Liu, and R. J. S. Pedersen, “Measurement of small dispersion values in optical components [WDM networks],” Electron. Lett. 35, 409–411 (1999).
[Crossref]

A. Hamza, I. Fouda, T. Sokkar, and M. El-Bakary, “Determination of spectral dispersion curves of polypropylene fibres,” J. Opt. A 1, 359 (1999).
[Crossref]

1996 (1)

1992 (1)

A. Hamza, T. Sokkar, and W. Ramadan, “On the microinterferometric determination of refractive indices and birefringence of fibres,” Pure Appl. Opt. 1, 321 (1992).
[Crossref]

1991 (1)

A. Hamza, T. Sokkar, and M. Shahin, “Interferometric determination of optical anisotropy in fibers III: multilayer fibers,” J. Appl. Phys. 70, 4480–4484 (1991).
[Crossref]

1984 (1)

E. Riande, J. Guzmán, M. P. Tarazona, and E. Saiz, “Random-coil configurations of aromatic polyesters: stress-optical behavior of poly (diethylene glycol terephthalate),” J. Polym. Sci., Polym. Phys. Ed. 22, 917–929 (1984).
[Crossref]

1979 (1)

H. de Vries, “A new approach to the continuum theory of birefringence of oriented polymers,” Colloid Polym. Sci. 257, 226–238 (1979).
[Crossref]

1972 (1)

1964 (2)

N. Barakat and A. Hindeleh, “Determination of the refractive indices, birefringence, and tensile properties of normal viscose rayon fibers,” Textile Res. J. 34, 581–584 (1964).
[Crossref]

N. Barakat and A. Hindeleh, “Interferometric determination of the refractive indices and birefringence of mohair wool fibers,” Textile Res. J. 34, 357–362 (1964).
[Crossref]

1955 (1)

1945 (1)

S. Tolansky, “XXXI: New contributions to interferometry. Part V—new multiple beam white light interference fringes and their applications,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 36, 225–236 (1945).
[Crossref]

Barakat, N.

N. Barakat and A. Hindeleh, “Determination of the refractive indices, birefringence, and tensile properties of normal viscose rayon fibers,” Textile Res. J. 34, 581–584 (1964).
[Crossref]

N. Barakat and A. Hindeleh, “Interferometric determination of the refractive indices and birefringence of mohair wool fibers,” Textile Res. J. 34, 357–362 (1964).
[Crossref]

Cohen, E.

E. Cohen, Quantities, Units and Symbols in Physical Chemistry (RSC Publishing, 2007), p. 28.

de Groot, P.

de Vries, H.

H. de Vries, “A new approach to the continuum theory of birefringence of oriented polymers,” Colloid Polym. Sci. 257, 226–238 (1979).
[Crossref]

Diddams, S.

DiDomenico, M.

Diels, J.-C.

El-Bakary, M.

A. Hamza, I. Fouda, T. Sokkar, and M. El-Bakary, “Determination of spectral dispersion curves of polypropylene fibres,” J. Opt. A 1, 359 (1999).
[Crossref]

El-Diasty, F.

F. El-Diasty, “Characterization of optical fibers by two- and multiple-beam interferometry,” Opt. Laser Eng. 46, 291–305 (2008).
[Crossref]

El-Farahaty, K.

A. Hamza, T. Sokkar, K. El-Farahaty, and M. Raslan, “A novel double-image Fizeau system for accurate investigation of anisotropic polymer fibres,” J. Microsc. 254, 84–94 (2014).
[Crossref]

T. Sokkar, K. El-Farahaty, and M. Raslan, “Online double-arm of a multiple-beam Fizeau system: I. Optical setup for simultaneous recording of two interferometric patterns in the same frame,” Polym. Test. 29, 1065–1074 (2010).
[Crossref]

Fouda, I.

A. Hamza, I. Fouda, T. Sokkar, and M. El-Bakary, “Determination of spectral dispersion curves of polypropylene fibres,” J. Opt. A 1, 359 (1999).
[Crossref]

Fréchette, J.

R. Gupta and J. Fréchette, “Interferometry of surfaces with well-defined topography in the surface force apparatus,” J. Colloid Interface Sci. 412, 82–88 (2013).
[Crossref]

Gupta, R.

R. Gupta and J. Fréchette, “Interferometry of surfaces with well-defined topography in the surface force apparatus,” J. Colloid Interface Sci. 412, 82–88 (2013).
[Crossref]

Guzmán, J.

E. Riande, J. Guzmán, M. P. Tarazona, and E. Saiz, “Random-coil configurations of aromatic polyesters: stress-optical behavior of poly (diethylene glycol terephthalate),” J. Polym. Sci., Polym. Phys. Ed. 22, 917–929 (1984).
[Crossref]

Hamza, A.

A. Hamza, T. Sokkar, K. El-Farahaty, and M. Raslan, “A novel double-image Fizeau system for accurate investigation of anisotropic polymer fibres,” J. Microsc. 254, 84–94 (2014).
[Crossref]

A. Hamza, I. Fouda, T. Sokkar, and M. El-Bakary, “Determination of spectral dispersion curves of polypropylene fibres,” J. Opt. A 1, 359 (1999).
[Crossref]

A. Hamza, T. Sokkar, and W. Ramadan, “On the microinterferometric determination of refractive indices and birefringence of fibres,” Pure Appl. Opt. 1, 321 (1992).
[Crossref]

A. Hamza, T. Sokkar, and M. Shahin, “Interferometric determination of optical anisotropy in fibers III: multilayer fibers,” J. Appl. Phys. 70, 4480–4484 (1991).
[Crossref]

Hernández-Sánchez, F.

F. Hernández-Sánchez and P. Herrera-Franco, “Electrical and thermal properties of recycled polypropylene-carbon black composites,” Polym. Bull. 45, 509–516 (2001).
[Crossref]

Herrera-Franco, P.

F. Hernández-Sánchez and P. Herrera-Franco, “Electrical and thermal properties of recycled polypropylene-carbon black composites,” Polym. Bull. 45, 509–516 (2001).
[Crossref]

Hindeleh, A.

N. Barakat and A. Hindeleh, “Interferometric determination of the refractive indices and birefringence of mohair wool fibers,” Textile Res. J. 34, 357–362 (1964).
[Crossref]

N. Barakat and A. Hindeleh, “Determination of the refractive indices, birefringence, and tensile properties of normal viscose rayon fibers,” Textile Res. J. 34, 581–584 (1964).
[Crossref]

Jenkins, F. A.

F. A. Jenkins and E. H. White, Fundamentals of Optics (McGraw-Hill, 1976).

Koehler, W.

Liu, F.

C. Peucheret, F. Liu, and R. J. S. Pedersen, “Measurement of small dispersion values in optical components [WDM networks],” Electron. Lett. 35, 409–411 (1999).
[Crossref]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).

Pedersen, R. J. S.

C. Peucheret, F. Liu, and R. J. S. Pedersen, “Measurement of small dispersion values in optical components [WDM networks],” Electron. Lett. 35, 409–411 (1999).
[Crossref]

Peucheret, C.

C. Peucheret, F. Liu, and R. J. S. Pedersen, “Measurement of small dispersion values in optical components [WDM networks],” Electron. Lett. 35, 409–411 (1999).
[Crossref]

Ramadan, W.

A. Hamza, T. Sokkar, and W. Ramadan, “On the microinterferometric determination of refractive indices and birefringence of fibres,” Pure Appl. Opt. 1, 321 (1992).
[Crossref]

Raslan, M.

A. Hamza, T. Sokkar, K. El-Farahaty, and M. Raslan, “A novel double-image Fizeau system for accurate investigation of anisotropic polymer fibres,” J. Microsc. 254, 84–94 (2014).
[Crossref]

T. Sokkar, K. El-Farahaty, and M. Raslan, “Online double-arm of a multiple-beam Fizeau system: I. Optical setup for simultaneous recording of two interferometric patterns in the same frame,” Polym. Test. 29, 1065–1074 (2010).
[Crossref]

Riande, E.

E. Riande, J. Guzmán, M. P. Tarazona, and E. Saiz, “Random-coil configurations of aromatic polyesters: stress-optical behavior of poly (diethylene glycol terephthalate),” J. Polym. Sci., Polym. Phys. Ed. 22, 917–929 (1984).
[Crossref]

Saiz, E.

E. Riande, J. Guzmán, M. P. Tarazona, and E. Saiz, “Random-coil configurations of aromatic polyesters: stress-optical behavior of poly (diethylene glycol terephthalate),” J. Polym. Sci., Polym. Phys. Ed. 22, 917–929 (1984).
[Crossref]

Shahin, M.

A. Hamza, T. Sokkar, and M. Shahin, “Interferometric determination of optical anisotropy in fibers III: multilayer fibers,” J. Appl. Phys. 70, 4480–4484 (1991).
[Crossref]

Sirohi, R. S.

R. S. Sirohi, Introduction to Optical Metrology (CRC Press, 2015).

Sokkar, T.

A. Hamza, T. Sokkar, K. El-Farahaty, and M. Raslan, “A novel double-image Fizeau system for accurate investigation of anisotropic polymer fibres,” J. Microsc. 254, 84–94 (2014).
[Crossref]

T. Sokkar, K. El-Farahaty, and M. Raslan, “Online double-arm of a multiple-beam Fizeau system: I. Optical setup for simultaneous recording of two interferometric patterns in the same frame,” Polym. Test. 29, 1065–1074 (2010).
[Crossref]

A. Hamza, I. Fouda, T. Sokkar, and M. El-Bakary, “Determination of spectral dispersion curves of polypropylene fibres,” J. Opt. A 1, 359 (1999).
[Crossref]

A. Hamza, T. Sokkar, and W. Ramadan, “On the microinterferometric determination of refractive indices and birefringence of fibres,” Pure Appl. Opt. 1, 321 (1992).
[Crossref]

A. Hamza, T. Sokkar, and M. Shahin, “Interferometric determination of optical anisotropy in fibers III: multilayer fibers,” J. Appl. Phys. 70, 4480–4484 (1991).
[Crossref]

Tarazona, M. P.

E. Riande, J. Guzmán, M. P. Tarazona, and E. Saiz, “Random-coil configurations of aromatic polyesters: stress-optical behavior of poly (diethylene glycol terephthalate),” J. Polym. Sci., Polym. Phys. Ed. 22, 917–929 (1984).
[Crossref]

Tolansky, S.

S. Tolansky, “XXXI: New contributions to interferometry. Part V—new multiple beam white light interference fringes and their applications,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 36, 225–236 (1945).
[Crossref]

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Oxford University, 1948).

White, E. H.

F. A. Jenkins and E. H. White, Fundamentals of Optics (McGraw-Hill, 1976).

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Colloid Polym. Sci. (1)

H. de Vries, “A new approach to the continuum theory of birefringence of oriented polymers,” Colloid Polym. Sci. 257, 226–238 (1979).
[Crossref]

Electron. Lett. (1)

C. Peucheret, F. Liu, and R. J. S. Pedersen, “Measurement of small dispersion values in optical components [WDM networks],” Electron. Lett. 35, 409–411 (1999).
[Crossref]

J. Appl. Phys. (1)

A. Hamza, T. Sokkar, and M. Shahin, “Interferometric determination of optical anisotropy in fibers III: multilayer fibers,” J. Appl. Phys. 70, 4480–4484 (1991).
[Crossref]

J. Colloid Interface Sci. (1)

R. Gupta and J. Fréchette, “Interferometry of surfaces with well-defined topography in the surface force apparatus,” J. Colloid Interface Sci. 412, 82–88 (2013).
[Crossref]

J. Microsc. (1)

A. Hamza, T. Sokkar, K. El-Farahaty, and M. Raslan, “A novel double-image Fizeau system for accurate investigation of anisotropic polymer fibres,” J. Microsc. 254, 84–94 (2014).
[Crossref]

J. Opt. A (1)

A. Hamza, I. Fouda, T. Sokkar, and M. El-Bakary, “Determination of spectral dispersion curves of polypropylene fibres,” J. Opt. A 1, 359 (1999).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

J. Polym. Sci., Polym. Phys. Ed. (1)

E. Riande, J. Guzmán, M. P. Tarazona, and E. Saiz, “Random-coil configurations of aromatic polyesters: stress-optical behavior of poly (diethylene glycol terephthalate),” J. Polym. Sci., Polym. Phys. Ed. 22, 917–929 (1984).
[Crossref]

London, Edinburgh, Dublin Philos. Mag. J. Sci. (1)

S. Tolansky, “XXXI: New contributions to interferometry. Part V—new multiple beam white light interference fringes and their applications,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 36, 225–236 (1945).
[Crossref]

Opt. Laser Eng. (1)

F. El-Diasty, “Characterization of optical fibers by two- and multiple-beam interferometry,” Opt. Laser Eng. 46, 291–305 (2008).
[Crossref]

Polym. Bull. (1)

F. Hernández-Sánchez and P. Herrera-Franco, “Electrical and thermal properties of recycled polypropylene-carbon black composites,” Polym. Bull. 45, 509–516 (2001).
[Crossref]

Polym. Test. (1)

T. Sokkar, K. El-Farahaty, and M. Raslan, “Online double-arm of a multiple-beam Fizeau system: I. Optical setup for simultaneous recording of two interferometric patterns in the same frame,” Polym. Test. 29, 1065–1074 (2010).
[Crossref]

Pure Appl. Opt. (1)

A. Hamza, T. Sokkar, and W. Ramadan, “On the microinterferometric determination of refractive indices and birefringence of fibres,” Pure Appl. Opt. 1, 321 (1992).
[Crossref]

Textile Res. J. (2)

N. Barakat and A. Hindeleh, “Determination of the refractive indices, birefringence, and tensile properties of normal viscose rayon fibers,” Textile Res. J. 34, 581–584 (1964).
[Crossref]

N. Barakat and A. Hindeleh, “Interferometric determination of the refractive indices and birefringence of mohair wool fibers,” Textile Res. J. 34, 357–362 (1964).
[Crossref]

Other (5)

E. Cohen, Quantities, Units and Symbols in Physical Chemistry (RSC Publishing, 2007), p. 28.

F. A. Jenkins and E. H. White, Fundamentals of Optics (McGraw-Hill, 1976).

R. S. Sirohi, Introduction to Optical Metrology (CRC Press, 2015).

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).

S. Tolansky, Multiple-Beam Interferometry of Surfaces and Films (Oxford University, 1948).

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Figures (10)

Fig. 1.
Fig. 1. Example of FECO extending from $\lambda = {435}\,\,{\rm nm}$ to $\lambda = {639}\,\,{\rm nm}$.
Fig. 2.
Fig. 2. Introduced optical setup for producing multiple-beam FECO patterns of parallel and perpendicular polarizations, simultaneously. a, halogen lamp; b, condenser lens; c, slit diaphragm; d, collimating convex lens; e, reflector; f, test sample; g, wedge interferometer; h, objective lens; i, reflector; j, constant deviation spectrometer; k, beam splitter; l, polarizers; m, CCD camera.
Fig. 3.
Fig. 3. Flow chart of the procedures executed by the designed software during the capturing, processing, and analysis of FECO patterns.
Fig. 4.
Fig. 4. Multiple-beam FECO patterns captured simultaneously using the modified system for (a) parallel and (b) perpendicular polarizations for a sample of undrawn isotactic polypropylene fiber.
Fig. 5.
Fig. 5. Cross section of the undrawn isotactic polypropylene fiber: (a) micrograph and (b) schematic diagram.
Fig. 6.
Fig. 6. (a) Calculated dispersion curves of the refractive indices for parallel and perpendicular polarizations for a sample of undrawn isotactic polypropylene fiber. (b) Calculated dispersion curve of birefringence for a sample of undrawn isotactic polypropylene fiber.
Fig. 7.
Fig. 7. Multiple-beam Fizeau patterns captured simultaneously using the double-image Fizeau technique for (a) parallel and (b) perpendicular polarizations for a sample of undrawn isotactic polypropylene fiber.
Fig. 8.
Fig. 8. Relation between ${({n^2}-{1})^{ - 1}}$ and ${E^2}$.
Fig. 9.
Fig. 9. Multiple-beam FECO patterns of parallel and perpendicular polarizations, captured dynamically during stretching of isotactic polypropylene fiber, captured at different strains 0.00, 0.69, 1.38, 2.07, 2.76, 3.44, 4.13, 4.82, and 5.51.
Fig. 10.
Fig. 10. Dispersion curves for parallel and perpendicular polarizations, measured dynamically during stretching of isotactic polypropylene fiber depicted as a function of strain.

Tables (4)

Tables Icon

Table 1. Cauchy Parameters for Undrawn Isotactic Polypropylene Fibers in Cases of Parallel and Perpendicular Polarizations Obtained Using the Modified FECO Technique

Tables Icon

Table 2. Values of the Refractive Indices of the Undrawn Isotactic Polypropylene Fiber in Cases of Parallel and Perpendicular Polarizations and the Birefringence Value Measured Using the Multiple-Beam Fizeau System

Tables Icon

Table 3. Some Parameters Calculated Using the Synchronized Measurement of n I I and n

Tables Icon

Table 4. Fitting Parameters of the Model Given by Eq. (6) to the Measured Dispersion Data, Depicted in Fig. 8

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

S t r a i n = 2 π r 200 × N L o ,
n ( λ ) = A + B λ 2 .
C s = 2 π 45 K B T ( n 2 + 2 ) 2 n 2 ( α 1 α 2 ) ,
α 1 = 3 ϵ 0 m N A ρ ( n 1 2 1 n 1 2 + 2 ) a n d α 2 = 3 ϵ 0 m N A ρ ( n 2 2 1 n 2 2 + 2 ) ,
ρ = n 0.9374 0.6273 .
( n 2 1 ) 1 = E o E d E 2 E o E d ,

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