Abstract

The aim of this paper is to discuss the possibility of a noninvasive, optical characterization of a transparent (glass) fiber on the basis of scattered light in the vicinity of a primary rainbow. Computational studies show that with the use of a spectrally adjusted incident beam of light, it is possible to form a rainbow with no strong nonlinearities typical for coherent light and that may be interpreted in terms of Airy’s theory of rainbow. An inverse analysis is applied to obtain the fiber diameter with the help of a straightforward mathematical formula based on the Airy integral, corrected by comparison with the solution according to the complex angular momentum method.

© 2014 Optical Society of America

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References

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  1. G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” in Transactions of the Cambridge Philosophical Society (Cambridge University, 1838).
  2. J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Reports 356, 229–365 (2002).
  3. R. L. Lee and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State University, 2001).
  4. P. W. Barber, J. F. Owen, and R. K. Chang, “Resonant scattering for characterization of axisymmetric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
    [Crossref]
  5. J. F. Owen, P. W. Barber, B. J. Messinger, and R. K. Chang, “Determination of optical-fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981).
    [Crossref]
  6. J. A. Lock, “Morphology-dependent resonances of an infinitely long circular cylinder illuminated by a diagonally incident plane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A 14, 653–661 (1997).
    [Crossref]
  7. J. Mroczka, “The cognitive process in metrology,” Measurement 46, 2896–2907 (2013).
    [Crossref]
  8. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  9. A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1535 (1978).
    [Crossref]
  10. A. J. Devaney and G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. 30, 1034–1037 (1982).
    [Crossref]
  11. J. Mroczka and D. Szczuczyński, “Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements,” Metrology Meas. Syst. 16, 333–357 (2009).
  12. J. Mroczka and D. Szczuczyński, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. 49, 4591–4603 (2010).
    [Crossref]
  13. J. Mroczka and D. Szczuczyński, “Simulation research on improved regularized solution of inverse problem in spectral extinction measurements,” Appl. Opt. 51, 1715–1723 (2012).
    [Crossref]
  14. J. Mroczka and D. Szczuczyński, “Improved technique of retrieving particle size distribution from angular scattering measurements,” J. Quant. Spectrosc. Radiat. Transfer 129, 48–59 (2013).
    [Crossref]
  15. J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
    [Crossref]
  16. N. Roth, K. Anders, and A. Frohn, “Refractive-index measurements for the correction of particle sizing methods,” Appl. Opt. 30, 4960–4965 (1991).
    [Crossref]
  17. X. e. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of initial disturbances in a liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
    [Crossref]
  18. L. Mèés, G. Gouesbet, and G. Gréhan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
    [Crossref]
  19. L. Mèés, G. Gréhan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
    [Crossref]
  20. L. Mèés, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: Numerical results,” Appl. Opt. 38, 1867–1876 (1999).
    [Crossref]
  21. G. P. Können and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
    [Crossref]
  22. J. W. Fleming, “Dispersion in GeO2-SiO2 glasses,” Appl. Opt. 23, 4486–4493 (1984).
    [Crossref]
  23. A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
    [Crossref]
  24. E. A. Hovenac and J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
    [Crossref]
  25. R. Li, H. Han, H. Jiang, and K. F. Ren, “Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder,” Appl. Opt. 45, 6255–6262 (2006).
    [Crossref]
  26. L. Kai and A. D’Alessio, “Finely stratified cylinder model for radially inhomogeneous cylinders normally irradiated by electromagnetic plane waves,” Appl. Opt. 34, 5520–5530 (1995).
    [Crossref]
  27. M. Kerker and E. Matijeviĉ, “Scattering of electromagnetic waves from concentric infinite cylinders,” J. Opt. Soc. Am. 51, 506–508 (1961).
    [Crossref]
  28. J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
    [Crossref]
  29. M. R. Vertrano, J. P. A. J. van Beeck, and M. L. Riethmuller, “Global rainbow thermometry: improvements in the data inversion algorithm and validation technique in liquid-liquid suspension,” Appl. Opt. 43, 3600–3607 (2004).
    [Crossref]
  30. T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).
  31. T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).
  32. T. Girasole, J. N. Le Toulouzan, J. Mroczka, and D. Wysoczański, “Fiber orientation and concentration analysis by light scattering: Experimental setup and diagnosis,” Rev. Sci. Instrum. 68, 2805–2811 (1997).
    [Crossref]
  33. F. Onofri, M. Krzysiek, S. Barbosa, V. Messager, K. F. Ren, and J. Mroczka, “Near-critical-angle scattering for the characterization of clouds of bubbles: Particular effects,” Appl. Opt. 50, 5759–5769 (2011).
    [Crossref]
  34. V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
    [Crossref]
  35. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
    [Crossref]
  36. P. Laven, “How glories are formed?” Appl. Opt. 44, 5675–5683 (2005).
    [Crossref]
  37. 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]
  38. C. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: Twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548–1558 (2001).
    [Crossref]
  39. C. L. Adler, J. A. Lock, I. P. Rafferty, and W. Hickok, “Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity,” Appl. Opt. 42, 6584–6594 (2003).
    [Crossref]
  40. J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory,” Appl. Opt. 39, 5040–5051 (2000).
    [Crossref]
  41. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (National Bureau of Standards, 1970).
  42. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).
  43. H. M. Nussenzveig, “Complex angular momentum of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
    [Crossref]
  44. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering, Montroll Memorial Lecture Series in Mathematical Physics: 1 (Cambridge University, 1992).
  45. R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
    [Crossref]
  46. A. Keppens, W. R. Ryckaert, G. Deconinck, and P. Hanselaer, “Modeling high power light-emitting diode spectra and their variation with junction temperature,” J. Appl. Phys. 108, 043104 (2010).
    [Crossref]
  47. J. Mroczka, “Temperature stabilisation of light-emitting diode radiation,” J. Phys. E 21, 306–309 (1988).
  48. J. Mroczka and M. Parol, “Methods of temperature stabilization of light-emitting diode radiation,” Rev. Sci. Instrum. 65, 803–806 (1994).
    [Crossref]

2013 (2)

J. Mroczka, “The cognitive process in metrology,” Measurement 46, 2896–2907 (2013).
[Crossref]

J. Mroczka and D. Szczuczyński, “Improved technique of retrieving particle size distribution from angular scattering measurements,” J. Quant. Spectrosc. Radiat. Transfer 129, 48–59 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (2)

J. Mroczka and D. Szczuczyński, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. 49, 4591–4603 (2010).
[Crossref]

A. Keppens, W. R. Ryckaert, G. Deconinck, and P. Hanselaer, “Modeling high power light-emitting diode spectra and their variation with junction temperature,” J. Appl. Phys. 108, 043104 (2010).
[Crossref]

2009 (1)

J. Mroczka and D. Szczuczyński, “Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements,” Metrology Meas. Syst. 16, 333–357 (2009).

2006 (1)

2005 (1)

2004 (1)

2003 (1)

2002 (1)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Reports 356, 229–365 (2002).

2001 (4)

L. Mèés, G. Gouesbet, and G. Gréhan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
[Crossref]

L. Mèés, G. Gréhan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[Crossref]

J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[Crossref]

C. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: Twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548–1558 (2001).
[Crossref]

2000 (1)

1999 (1)

1998 (1)

1997 (4)

J. A. Lock, “Morphology-dependent resonances of an infinitely long circular cylinder illuminated by a diagonally incident plane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A 14, 653–661 (1997).
[Crossref]

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).

T. Girasole, J. N. Le Toulouzan, J. Mroczka, and D. Wysoczański, “Fiber orientation and concentration analysis by light scattering: Experimental setup and diagnosis,” Rev. Sci. Instrum. 68, 2805–2811 (1997).
[Crossref]

1996 (1)

1995 (1)

1994 (1)

J. Mroczka and M. Parol, “Methods of temperature stabilization of light-emitting diode radiation,” Rev. Sci. Instrum. 65, 803–806 (1994).
[Crossref]

1992 (1)

1991 (2)

1988 (1)

J. Mroczka, “Temperature stabilisation of light-emitting diode radiation,” J. Phys. E 21, 306–309 (1988).

1984 (1)

1982 (2)

A. J. Devaney and G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. 30, 1034–1037 (1982).
[Crossref]

P. W. Barber, J. F. Owen, and R. K. Chang, “Resonant scattering for characterization of axisymmetric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[Crossref]

1981 (1)

1979 (2)

1978 (1)

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1535 (1978).
[Crossref]

1974 (2)

1969 (1)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

1961 (1)

1951 (1)

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (National Bureau of Standards, 1970).

Adam, J. A.

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Reports 356, 229–365 (2002).

Aden, A. L.

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
[Crossref]

Adler, C.

Adler, C. L.

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” in Transactions of the Cambridge Philosophical Society (Cambridge University, 1838).

Anders, K.

Barber, P. W.

P. W. Barber, J. F. Owen, and R. K. Chang, “Resonant scattering for characterization of axisymmetric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[Crossref]

J. F. Owen, P. W. Barber, B. J. Messinger, and R. K. Chang, “Determination of optical-fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981).
[Crossref]

Barbosa, S.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Bultynck, H.

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

Chang, R. K.

P. W. Barber, J. F. Owen, and R. K. Chang, “Resonant scattering for characterization of axisymmetric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[Crossref]

J. F. Owen, P. W. Barber, B. J. Messinger, and R. K. Chang, “Determination of optical-fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981).
[Crossref]

Corbin, F.

D’Alessio, A.

de Boer, J. H.

Deconinck, G.

A. Keppens, W. R. Ryckaert, G. Deconinck, and P. Hanselaer, “Modeling high power light-emitting diode spectra and their variation with junction temperature,” J. Appl. Phys. 108, 043104 (2010).
[Crossref]

Devaney, A. J.

A. J. Devaney and G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. 30, 1034–1037 (1982).
[Crossref]

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1535 (1978).
[Crossref]

Fleming, J. W.

Fraser, A. B.

R. L. Lee and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State University, 2001).

Frohn, A.

Girasole, T.

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).

T. Girasole, J. N. Le Toulouzan, J. Mroczka, and D. Wysoczański, “Fiber orientation and concentration analysis by light scattering: Experimental setup and diagnosis,” Rev. Sci. Instrum. 68, 2805–2811 (1997).
[Crossref]

Gouesbet, G.

L. Mèés, G. Gréhan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[Crossref]

L. Mèés, G. Gouesbet, and G. Gréhan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
[Crossref]

L. Mèés, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: Numerical results,” Appl. Opt. 38, 1867–1876 (1999).
[Crossref]

X. e. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of initial disturbances in a liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
[Crossref]

T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

Gréhan, G.

L. Mèés, G. Gréhan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[Crossref]

L. Mèés, G. Gouesbet, and G. Gréhan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
[Crossref]

L. Mèés, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: Numerical results,” Appl. Opt. 38, 1867–1876 (1999).
[Crossref]

X. e. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of initial disturbances in a liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
[Crossref]

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).

Han, H.

Han, X. e.

Hanselaer, P.

A. Keppens, W. R. Ryckaert, G. Deconinck, and P. Hanselaer, “Modeling high power light-emitting diode spectra and their variation with junction temperature,” J. Appl. Phys. 108, 043104 (2010).
[Crossref]

Hickok, W.

Hovenac, E. A.

Jiang, H.

Kai, L.

Keppens, A.

A. Keppens, W. R. Ryckaert, G. Deconinck, and P. Hanselaer, “Modeling high power light-emitting diode spectra and their variation with junction temperature,” J. Appl. Phys. 108, 043104 (2010).
[Crossref]

Kerker, M.

Khare, V.

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

Können, G. P.

Krzysiek, M.

Laven, P.

Le Meur, F.

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

Le Toulouzan, J. N.

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).

T. Girasole, J. N. Le Toulouzan, J. Mroczka, and D. Wysoczański, “Fiber orientation and concentration analysis by light scattering: Experimental setup and diagnosis,” Rev. Sci. Instrum. 68, 2805–2811 (1997).
[Crossref]

Lee, R. L.

R. L. Lee and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State University, 2001).

Li, R.

Lock, J. A.

Marcuse, D.

Matijevic, E.

Mèés, L.

Messager, V.

Messinger, B. J.

Mroczka, J.

J. Mroczka and D. Szczuczyński, “Improved technique of retrieving particle size distribution from angular scattering measurements,” J. Quant. Spectrosc. Radiat. Transfer 129, 48–59 (2013).
[Crossref]

J. Mroczka, “The cognitive process in metrology,” Measurement 46, 2896–2907 (2013).
[Crossref]

J. Mroczka and D. Szczuczyński, “Simulation research on improved regularized solution of inverse problem in spectral extinction measurements,” Appl. Opt. 51, 1715–1723 (2012).
[Crossref]

F. Onofri, M. Krzysiek, S. Barbosa, V. Messager, K. F. Ren, and J. Mroczka, “Near-critical-angle scattering for the characterization of clouds of bubbles: Particular effects,” Appl. Opt. 50, 5759–5769 (2011).
[Crossref]

J. Mroczka and D. Szczuczyński, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. 49, 4591–4603 (2010).
[Crossref]

J. Mroczka and D. Szczuczyński, “Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements,” Metrology Meas. Syst. 16, 333–357 (2009).

T. Girasole, J. N. Le Toulouzan, J. Mroczka, and D. Wysoczański, “Fiber orientation and concentration analysis by light scattering: Experimental setup and diagnosis,” Rev. Sci. Instrum. 68, 2805–2811 (1997).
[Crossref]

T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

J. Mroczka and M. Parol, “Methods of temperature stabilization of light-emitting diode radiation,” Rev. Sci. Instrum. 65, 803–806 (1994).
[Crossref]

J. Mroczka, “Temperature stabilisation of light-emitting diode radiation,” J. Phys. E 21, 306–309 (1988).

Nash, J. K.

Nussenzveig, H. M.

H. M. Nussenzveig, “Complex angular momentum of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
[Crossref]

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering, Montroll Memorial Lecture Series in Mathematical Physics: 1 (Cambridge University, 1992).

Onofri, F.

Owen, J. F.

P. W. Barber, J. F. Owen, and R. K. Chang, “Resonant scattering for characterization of axisymmetric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[Crossref]

J. F. Owen, P. W. Barber, B. J. Messinger, and R. K. Chang, “Determination of optical-fiber diameter from resonances in the elastic scattering spectrum,” Opt. Lett. 6, 272–274 (1981).
[Crossref]

Parol, M.

J. Mroczka and M. Parol, “Methods of temperature stabilization of light-emitting diode radiation,” Rev. Sci. Instrum. 65, 803–806 (1994).
[Crossref]

Presby, H. M.

Rafferty, I. P.

Ren, K. F.

Riethmuller, M. L.

Roth, N.

Roze, C.

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

Ryckaert, W. R.

A. Keppens, W. R. Ryckaert, G. Deconinck, and P. Hanselaer, “Modeling high power light-emitting diode spectra and their variation with junction temperature,” J. Appl. Phys. 108, 043104 (2010).
[Crossref]

Saunders, K. W.

Sherman, G. C.

A. J. Devaney and G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. 30, 1034–1037 (1982).
[Crossref]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (National Bureau of Standards, 1970).

Szczuczynski, D.

J. Mroczka and D. Szczuczyński, “Improved technique of retrieving particle size distribution from angular scattering measurements,” J. Quant. Spectrosc. Radiat. Transfer 129, 48–59 (2013).
[Crossref]

J. Mroczka and D. Szczuczyński, “Simulation research on improved regularized solution of inverse problem in spectral extinction measurements,” Appl. Opt. 51, 1715–1723 (2012).
[Crossref]

J. Mroczka and D. Szczuczyński, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. 49, 4591–4603 (2010).
[Crossref]

J. Mroczka and D. Szczuczyński, “Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements,” Metrology Meas. Syst. 16, 333–357 (2009).

van Beeck, J.

J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[Crossref]

van Beeck, J. P. A. J.

van de Hulst, H. C.

Vertrano, M. R.

Wang, R. T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Wu, Z.

Wysoczanski, D.

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).

T. Girasole, J. N. Le Toulouzan, J. Mroczka, and D. Wysoczański, “Fiber orientation and concentration analysis by light scattering: Experimental setup and diagnosis,” Rev. Sci. Instrum. 68, 2805–2811 (1997).
[Crossref]

Zimmer, L.

J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[Crossref]

Appl. Opt. (19)

J. Mroczka and D. Szczuczyński, “Improved regularized solution of the inverse problem in turbidimetric measurements,” Appl. Opt. 49, 4591–4603 (2010).
[Crossref]

J. Mroczka and D. Szczuczyński, “Simulation research on improved regularized solution of inverse problem in spectral extinction measurements,” Appl. Opt. 51, 1715–1723 (2012).
[Crossref]

J. P. A. J. van Beeck and M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[Crossref]

N. Roth, K. Anders, and A. Frohn, “Refractive-index measurements for the correction of particle sizing methods,” Appl. Opt. 30, 4960–4965 (1991).
[Crossref]

X. e. Han, K. F. Ren, Z. Wu, F. Corbin, G. Gouesbet, and G. Gréhan, “Characterization of initial disturbances in a liquid jet by rainbow sizing,” Appl. Opt. 37, 8498–8503 (1998).
[Crossref]

L. Mèés, G. Gouesbet, and G. Gréhan, “Scattering of laser pulses (plane wave and focused Gaussian beam) by spheres,” Appl. Opt. 40, 2546–2550 (2001).
[Crossref]

L. Mèés, K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation: Numerical results,” Appl. Opt. 38, 1867–1876 (1999).
[Crossref]

G. P. Können and J. H. de Boer, “Polarized rainbow,” Appl. Opt. 18, 1961–1965 (1979).
[Crossref]

J. W. Fleming, “Dispersion in GeO2-SiO2 glasses,” Appl. Opt. 23, 4486–4493 (1984).
[Crossref]

R. Li, H. Han, H. Jiang, and K. F. Ren, “Debye series of normally incident plane-wave scattering by an infinite multilayered cylinder,” Appl. Opt. 45, 6255–6262 (2006).
[Crossref]

L. Kai and A. D’Alessio, “Finely stratified cylinder model for radially inhomogeneous cylinders normally irradiated by electromagnetic plane waves,” Appl. Opt. 34, 5520–5530 (1995).
[Crossref]

M. R. Vertrano, J. P. A. J. van Beeck, and M. L. Riethmuller, “Global rainbow thermometry: improvements in the data inversion algorithm and validation technique in liquid-liquid suspension,” Appl. Opt. 43, 3600–3607 (2004).
[Crossref]

P. Laven, “How glories are formed?” Appl. Opt. 44, 5675–5683 (2005).
[Crossref]

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]

C. Adler, J. A. Lock, J. K. Nash, and K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: Twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548–1558 (2001).
[Crossref]

C. L. Adler, J. A. Lock, I. P. Rafferty, and W. Hickok, “Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity,” Appl. Opt. 42, 6584–6594 (2003).
[Crossref]

J. A. Lock, “Supernumerary spacing of rainbows produced by an elliptical-cross-section cylinder. I. Theory,” Appl. Opt. 39, 5040–5051 (2000).
[Crossref]

F. Onofri, M. Krzysiek, S. Barbosa, V. Messager, K. F. Ren, and J. Mroczka, “Near-critical-angle scattering for the characterization of clouds of bubbles: Particular effects,” Appl. Opt. 50, 5759–5769 (2011).
[Crossref]

R. T. Wang and H. C. van de Hulst, “Rainbows: Mie computations and the Airy approximation,” Appl. Opt. 30, 106–117 (1991).
[Crossref]

IEEE Trans. Antennas Propag. (2)

A. J. Devaney and G. C. Sherman, “Nonuniqueness in inverse source and scattering problems,” IEEE Trans. Antennas Propag. 30, 1034–1037 (1982).
[Crossref]

P. W. Barber, J. F. Owen, and R. K. Chang, “Resonant scattering for characterization of axisymmetric objects,” IEEE Trans. Antennas Propag. 30, 168–172 (1982).
[Crossref]

J. Appl. Phys. (2)

A. L. Aden, “Electromagnetic scattering from spheres with sizes comparable to the wavelength,” J. Appl. Phys. 22, 601–605 (1951).
[Crossref]

A. Keppens, W. R. Ryckaert, G. Deconinck, and P. Hanselaer, “Modeling high power light-emitting diode spectra and their variation with junction temperature,” J. Appl. Phys. 108, 043104 (2010).
[Crossref]

J. Math. Phys. (2)

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. II. Theory of the rainbow and the glory,” J. Math. Phys. 10, 125–176 (1969).
[Crossref]

A. J. Devaney, “Nonuniqueness in the inverse scattering problem,” J. Math. Phys. 19, 1526–1535 (1978).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Phys. E (1)

J. Mroczka, “Temperature stabilisation of light-emitting diode radiation,” J. Phys. E 21, 306–309 (1988).

J. Quant. Spectrosc. Radiat. Transfer (1)

J. Mroczka and D. Szczuczyński, “Improved technique of retrieving particle size distribution from angular scattering measurements,” J. Quant. Spectrosc. Radiat. Transfer 129, 48–59 (2013).
[Crossref]

Measurement (1)

J. Mroczka, “The cognitive process in metrology,” Measurement 46, 2896–2907 (2013).
[Crossref]

Metrology Meas. Syst. (1)

J. Mroczka and D. Szczuczyński, “Inverse problems formulated in terms of first-kind Fredholm integral equations in indirect measurements,” Metrology Meas. Syst. 16, 333–357 (2009).

Opt. Commun. (1)

L. Mèés, G. Gréhan, and G. Gouesbet, “Time-resolved scattering diagrams for a sphere illuminated by plane wave and focused short pulses,” Opt. Commun. 194, 59–65 (2001).
[Crossref]

Opt. Lett. (1)

Part. Part. Syst. Charact. (3)

J. van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[Crossref]

T. Girasole, H. Bultynck, G. Gouesbet, G. Gréhan, F. Le Meur, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, C. Roze, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 1: Numerical aspects,” Part. Part. Syst. Charact. 14, 163–174 (1997).

T. Girasole, G. Gouesbet, G. Gréhan, J. N. Le Toulouzan, J. Mroczka, K. F. Ren, and D. Wysoczański, “Cylindrical fibre orientation analysis by light scattering: Part 2: Experimental aspects,” Part. Part. Syst. Charact. 14, 211–218 (1997).

Phys. Reports (1)

J. A. Adam, “The mathematical physics of rainbows and glories,” Phys. Reports 356, 229–365 (2002).

Phys. Rev. Lett. (1)

V. Khare and H. M. Nussenzveig, “Theory of the rainbow,” Phys. Rev. Lett. 33, 976–980 (1974).
[Crossref]

Rev. Sci. Instrum. (2)

J. Mroczka and M. Parol, “Methods of temperature stabilization of light-emitting diode radiation,” Rev. Sci. Instrum. 65, 803–806 (1994).
[Crossref]

T. Girasole, J. N. Le Toulouzan, J. Mroczka, and D. Wysoczański, “Fiber orientation and concentration analysis by light scattering: Experimental setup and diagnosis,” Rev. Sci. Instrum. 68, 2805–2811 (1997).
[Crossref]

Other (6)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. (National Bureau of Standards, 1970).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

R. L. Lee and A. B. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State University, 2001).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” in Transactions of the Cambridge Philosophical Society (Cambridge University, 1838).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering, Montroll Memorial Lecture Series in Mathematical Physics: 1 (Cambridge University, 1992).

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Figures (8)

Fig. 1.
Fig. 1. (a) Far-field intensity versus scattering angle over the vicinity at the primary rainbow for a SiO2 fiber with diameter of 125 μm. (b) Same as (a) but as a function of fiber diameter (λ=0.6328μm, TM polarization, m(λ)=1.45702+i1E7.
Fig. 2.
Fig. 2. Scattering geometry.
Fig. 3.
Fig. 3. Far-field scattered intensity Isca versus scattering angle θ in the vicinity of the primary rainbow for a SiO2 fiber with diameter d of 10, 12–200 μm. The case of incident light with the spectral line half-width fwhm and the peak wavelength of 0.6328 μm is considered: (a) fwhm=0.1nm, (b) 1 nm, and (c) 15 nm.
Fig. 4.
Fig. 4. Scattered intensity over the first rainbow peak as a function of very small diameter changes (Δd=0.0001μm) of a SiO2 fiber for the case of: (a) incident monochromatic light (λ=0.6328μm) and (b) incident low-coherent light (λ=0.6328μm, fwhm=20nm).
Fig. 5.
Fig. 5. Angular position of the first and second dark (θ1,θ2) and bright (θ1+,θ2+) rainbow fringes versus SiO2 fiber diameter for the case of incident light with the spectral line half-width fwhm and the peak wavelength of 0.6328 μm.
Fig. 6.
Fig. 6. Normalized scattering intensity around the rainbow angle for a SiO2 fiber of 125 μm diameter according to (—) Lorenz-Mie theory calculations for low-coherent light with the spectral half-width of 20 nm, (···) Debye series calculations for monochromatic incident beam due to contribution of p=2 order rays (see inset).
Fig. 7.
Fig. 7. Normalized scattering intensity around the rainbow angle for a SiO2 fiber with diameter of 125 μm according to: (—) Lorenz-Mie theory calculations for low-coherent light with spectral half-width of 20 nm, (- - -) Airy theory with correction, for monochromatic incident beam, and (···) classical Airy theory, monochromatic incident beam.
Fig. 8.
Fig. 8. Fiber diameter limiting error versus (a) spectral line half-width of the incident light with the peak wavelength 0.6328 μm for three measurement ranges (see inset) and (b) measurement range for a fixed spectral line half-width. Empty symbols refer to the inversion formula related to the Airy theory; the filled ones, to the Airy theory with correction.

Tables (2)

Tables Icon

Table 1. Angular Position of the First and Second Dark (θ1,θ2) and Bright (θ1+,θ2+) Rainbow Fringes for Different Fibers and the Case of Low-Coherent Incident Beam (λ0=0.6328μm, fwhm=15nm)

Tables Icon

Table 2. Angular Position of the First and Second Dark (θ1,θ2) and Bright (θ1+,θ2+) Rainbow Fringes for Different Spectral Distributions of the Incident Light (λ0=0.6328μm, fwhm=14.5nm)

Equations (18)

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

I(λ)=I0exp[0.5((λλ0)/σ)2],
σ=fwhm/[2(2ln2)1/2]fwhm/2.355.
m(λ)=[1+i=13Aiλ2/(λ2i2)]1/2+iκ.
Isca(θ,x,m)=(2/πkr)|b0+2n=1bncos(nθ)|2Iinc,
bn=mJn(mx)Jn(x)Jn(mx)Jn(x)mJn(mx)Hn(1)(x)Jn(mx)Hn(1)(x),
Isca(θ,x,n)=(πI0F/r)(dx1/3/h2/3)×Ai2[x2/3Δ/h1/3],
ΔθθD,
θD=π+2θiD4θtD,cos(θiD)=[(n21)/3]1/2,sin(θtD)=n1sin(θiD),
F=T21(θiD)·[R11(θiD)]p1·T12(θiD),
h=9(4n2)1/2/4(n21)3/2.
Esca(θ,x,n)w(Δ,x)·Ai[(x2/3Δ/h1/3)u(Δ)]ix1/3v(Δ,x)·Ai[(x2/3Δ/h1/3)u(Δ)].
u(Δ)=1+BΔ,
B=[(875c61257c4+657c2+45)/8640(cs)3],s=[(4n2)/3]1/2,c=[(n21)/3]1/2,
Isca(θ,x,n)Ai2[(x2/3Δ/h1/3)(1+BΔ)].
zizj=(x2/3/h1/3)[Δi(1+BΔi)Δj(1+BΔj)],
ΔiθiθD,ΔjθjθD.
d^(θi,θj,λ,n)=λπh1/2[zizjΔi(1+BΔi)Δj(1+BΔj)]3/2.
δd=max{|(dd^)/d|·100}(%).

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