Abstract

We have investigated the utility of a Lau phase interferometer with circular gratings and a white-light source to measure the temperature and temperature profile of an axisymmetric flame. The temperature measured with a Lau phase interferometer is in good agreement with the temperature measured with a thermocouple and multilogger. Detailed theoretical analysis and experimental investigation are presented.

© 2002 Optical Society of America

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References

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  1. R. Peyturaux, J. C. Renault, R. Vitry, “An automatic precise polychromatic pyrometer,” J. Opt. (Paris) 24, 107–112 (1993).
    [CrossRef]
  2. E. Pungor, Flame Photometry (Van Nostrand, London, 1976), pp. 147–152.
  3. A. A. Boiarski, R. H. Barnes, J. F. Kircher, “Flame measurements utilizing Raman scattering,” Combust Flame 32, 111–114 (1978).
    [CrossRef]
  4. S. Kampmann, A. Leipertz, K. Dobbling, J. Hanumann, T. Sattelmayer, “Two-dimensional temperature measurement in a technical combustor with laser Rayleigh scattering,” Appl. Opt. 32, 6167–6172 (1993).
    [CrossRef] [PubMed]
  5. M. Giglio, S. Musazzi, U. Perini, “A white light speckle Schlieren technique,” Opt. Commun. 36, 117–120 (1981).
    [CrossRef]
  6. E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, “Measurement of temperature distribution of flame by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
    [CrossRef] [PubMed]
  7. D. K. Sharma, S. Stephen, R. Natarajan, “Structure of burning n-hexane droplet by moiré deflectometry,” Combust. Sci. Tech. 13, 305–321 (1998).
    [CrossRef]
  8. P. V. Farrel, G. S. Springer, C. M. Vest, “Heterodyne holographic interferometry: concentration and temperature measurement in gas mixtures,” Appl. Opt. 21, 1624–1627 (1982).
    [CrossRef]
  9. D. L. Reuss, “Temperature measurement in a radially symmetric flame using holographic interferometry,” Combust. Flame 49, 207–209 (1983).
    [CrossRef]
  10. P. V. Farrel, D. L. Hofeldt, “Temperature measurement in gases using speckle photography,” Appl. Opt. 23, 1055–1059 (1984).
    [CrossRef]
  11. C. Shakher, A. Kumar Nirala, J. Pramila, S. Kumar Verma, “Use of speckle technique for temperature measurement in gaseous flame,” J. Opt. (Paris) 23, 35–39 (1992).
    [CrossRef]
  12. S. Yokozeki, T. Suzuki, “Shearing interferometry using the grating as the beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
    [CrossRef] [PubMed]
  13. C. Shakher, A. Kumar Nirala, “Measurement of temperature using speckle shearing interferometry,” Appl. Opt. 33, 2125–2127 (1994).
    [CrossRef] [PubMed]
  14. D. E. Silva, “Talbot interferometry for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
    [CrossRef] [PubMed]
  15. C. Shakher, A. J. Pramila Daniel, “Talbot interferometry with circular grating for the measurement of temperature in axisymmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
    [CrossRef] [PubMed]
  16. A. J. Parmila Daniel, “Diffractive optical element in speckle metrology and temperature measurement,” Ph.D. dissertation (Instrument Design and Development Centre, Indian Institute of Technology, Dehli, 1994).
  17. C. Shakher, A. J. Pramila Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (pre-mixed laminar flow slot burner) using Talbot interferometer,” Opt. Eng. 33, 2663–2669 (1994).
    [CrossRef]
  18. Y.-S. Cheng, R.-C. Chang, “Theory of image formation using the Talbot effect,” Appl. Opt. 33, 1863–1874 (1994).
    [CrossRef] [PubMed]
  19. Y.-S. Cheng, R.-C. Chang, “Image formation foe two dimensional periodic object using Talbot effect,” Opt. Commun. 20, 335–347 (1995).
    [CrossRef]
  20. C. Shakher, A. Kumar Nirala, “A review on refractive index and temperature profile measurement using laser-based interferometric techniques,” Opt. Laser Eng. 31, 455–491 (1999).
    [CrossRef]
  21. J. Jahns, A. W. Lohman, “The Lau effect (A diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  22. F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  23. R. Sudol, B. J. Thompson, “A explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
    [CrossRef]
  24. R. J. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef] [PubMed]
  25. H. O. Bartelt, J. Jahns, “Inteferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
    [CrossRef]
  26. G. J. Swanson, E. N. Leith, “Lau effect and grating imaging,” J. Opt. Soc. Am. 72, 552–555 (1982).
    [CrossRef]
  27. K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–748 (1983).
    [CrossRef]
  28. K. Patorski, “Incoherent superposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2403 (1986).
    [CrossRef]
  29. H. Bartlet, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
    [CrossRef]
  30. C. Colautti, L. M. Zerbino, E. E. Sicre, M. Garavaglia, “Lau effect using circular gratings,” Appl. Opt. 26, 2061–2062 (1987).
    [CrossRef] [PubMed]
  31. R. Bracewell, The Fourier Transform and Application (McGraw Hill, New York, 1978), Chap. 5, p. 262.
  32. M. Born, E. Wolf, Principle of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 2, pp. 87–90; Chap. 3, pp. 122–124.

1999 (1)

C. Shakher, A. Kumar Nirala, “A review on refractive index and temperature profile measurement using laser-based interferometric techniques,” Opt. Laser Eng. 31, 455–491 (1999).
[CrossRef]

1998 (1)

D. K. Sharma, S. Stephen, R. Natarajan, “Structure of burning n-hexane droplet by moiré deflectometry,” Combust. Sci. Tech. 13, 305–321 (1998).
[CrossRef]

1995 (1)

Y.-S. Cheng, R.-C. Chang, “Image formation foe two dimensional periodic object using Talbot effect,” Opt. Commun. 20, 335–347 (1995).
[CrossRef]

1994 (4)

1993 (2)

1992 (1)

C. Shakher, A. Kumar Nirala, J. Pramila, S. Kumar Verma, “Use of speckle technique for temperature measurement in gaseous flame,” J. Opt. (Paris) 23, 35–39 (1992).
[CrossRef]

1987 (1)

1986 (1)

1984 (1)

1983 (3)

D. L. Reuss, “Temperature measurement in a radially symmetric flame using holographic interferometry,” Combust. Flame 49, 207–209 (1983).
[CrossRef]

H. Bartlet, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–748 (1983).
[CrossRef]

1982 (2)

1981 (3)

1979 (4)

H. O. Bartelt, J. Jahns, “Inteferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

J. Jahns, A. W. Lohman, “The Lau effect (A diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “A explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

1978 (1)

A. A. Boiarski, R. H. Barnes, J. F. Kircher, “Flame measurements utilizing Raman scattering,” Combust Flame 32, 111–114 (1978).
[CrossRef]

1972 (1)

1971 (1)

Angra, S. K.

C. Shakher, A. J. Pramila Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (pre-mixed laminar flow slot burner) using Talbot interferometer,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

Barnes, R. H.

A. A. Boiarski, R. H. Barnes, J. F. Kircher, “Flame measurements utilizing Raman scattering,” Combust Flame 32, 111–114 (1978).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, J. Jahns, “Inteferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

Bartlet, H.

H. Bartlet, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
[CrossRef]

Bar-Ziv, E.

Boiarski, A. A.

A. A. Boiarski, R. H. Barnes, J. F. Kircher, “Flame measurements utilizing Raman scattering,” Combust Flame 32, 111–114 (1978).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principle of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 2, pp. 87–90; Chap. 3, pp. 122–124.

Bracewell, R.

R. Bracewell, The Fourier Transform and Application (McGraw Hill, New York, 1978), Chap. 5, p. 262.

Chang, R.-C.

Y.-S. Cheng, R.-C. Chang, “Image formation foe two dimensional periodic object using Talbot effect,” Opt. Commun. 20, 335–347 (1995).
[CrossRef]

Y.-S. Cheng, R.-C. Chang, “Theory of image formation using the Talbot effect,” Appl. Opt. 33, 1863–1874 (1994).
[CrossRef] [PubMed]

Cheng, Y.-S.

Y.-S. Cheng, R.-C. Chang, “Image formation foe two dimensional periodic object using Talbot effect,” Opt. Commun. 20, 335–347 (1995).
[CrossRef]

Y.-S. Cheng, R.-C. Chang, “Theory of image formation using the Talbot effect,” Appl. Opt. 33, 1863–1874 (1994).
[CrossRef] [PubMed]

Colautti, C.

Dobbling, K.

Farrel, P. V.

Garavaglia, M.

Giglio, M.

M. Giglio, S. Musazzi, U. Perini, “A white light speckle Schlieren technique,” Opt. Commun. 36, 117–120 (1981).
[CrossRef]

Glatt, I.

Gori, F.

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Hanumann, J.

Hofeldt, D. L.

Jahns, J.

J. Jahns, A. W. Lohman, “The Lau effect (A diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

H. O. Bartelt, J. Jahns, “Inteferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

Kafri, O.

Kampmann, S.

Keren, E.

Kircher, J. F.

A. A. Boiarski, R. H. Barnes, J. F. Kircher, “Flame measurements utilizing Raman scattering,” Combust Flame 32, 111–114 (1978).
[CrossRef]

Kumar Nirala, A.

C. Shakher, A. Kumar Nirala, “A review on refractive index and temperature profile measurement using laser-based interferometric techniques,” Opt. Laser Eng. 31, 455–491 (1999).
[CrossRef]

C. Shakher, A. Kumar Nirala, “Measurement of temperature using speckle shearing interferometry,” Appl. Opt. 33, 2125–2127 (1994).
[CrossRef] [PubMed]

C. Shakher, A. Kumar Nirala, J. Pramila, S. Kumar Verma, “Use of speckle technique for temperature measurement in gaseous flame,” J. Opt. (Paris) 23, 35–39 (1992).
[CrossRef]

Kumar Verma, S.

C. Shakher, A. Kumar Nirala, J. Pramila, S. Kumar Verma, “Use of speckle technique for temperature measurement in gaseous flame,” J. Opt. (Paris) 23, 35–39 (1992).
[CrossRef]

Leipertz, A.

Leith, E. N.

Li, Y.

H. Bartlet, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
[CrossRef]

Lohman, A. W.

J. Jahns, A. W. Lohman, “The Lau effect (A diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Musazzi, S.

M. Giglio, S. Musazzi, U. Perini, “A white light speckle Schlieren technique,” Opt. Commun. 36, 117–120 (1981).
[CrossRef]

Natarajan, R.

D. K. Sharma, S. Stephen, R. Natarajan, “Structure of burning n-hexane droplet by moiré deflectometry,” Combust. Sci. Tech. 13, 305–321 (1998).
[CrossRef]

Parmila Daniel, A. J.

A. J. Parmila Daniel, “Diffractive optical element in speckle metrology and temperature measurement,” Ph.D. dissertation (Instrument Design and Development Centre, Indian Institute of Technology, Dehli, 1994).

Patorski, K.

K. Patorski, “Incoherent superposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2403 (1986).
[CrossRef]

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–748 (1983).
[CrossRef]

Perini, U.

M. Giglio, S. Musazzi, U. Perini, “A white light speckle Schlieren technique,” Opt. Commun. 36, 117–120 (1981).
[CrossRef]

Peyturaux, R.

R. Peyturaux, J. C. Renault, R. Vitry, “An automatic precise polychromatic pyrometer,” J. Opt. (Paris) 24, 107–112 (1993).
[CrossRef]

Pramila, J.

C. Shakher, A. Kumar Nirala, J. Pramila, S. Kumar Verma, “Use of speckle technique for temperature measurement in gaseous flame,” J. Opt. (Paris) 23, 35–39 (1992).
[CrossRef]

Pramila Daniel, A. J.

C. Shakher, A. J. Pramila Daniel, “Talbot interferometry with circular grating for the measurement of temperature in axisymmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
[CrossRef] [PubMed]

C. Shakher, A. J. Pramila Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (pre-mixed laminar flow slot burner) using Talbot interferometer,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

Pungor, E.

E. Pungor, Flame Photometry (Van Nostrand, London, 1976), pp. 147–152.

Renault, J. C.

R. Peyturaux, J. C. Renault, R. Vitry, “An automatic precise polychromatic pyrometer,” J. Opt. (Paris) 24, 107–112 (1993).
[CrossRef]

Reuss, D. L.

D. L. Reuss, “Temperature measurement in a radially symmetric flame using holographic interferometry,” Combust. Flame 49, 207–209 (1983).
[CrossRef]

Sattelmayer, T.

Shakher, C.

C. Shakher, A. Kumar Nirala, “A review on refractive index and temperature profile measurement using laser-based interferometric techniques,” Opt. Laser Eng. 31, 455–491 (1999).
[CrossRef]

C. Shakher, A. J. Pramila Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (pre-mixed laminar flow slot burner) using Talbot interferometer,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

C. Shakher, A. Kumar Nirala, “Measurement of temperature using speckle shearing interferometry,” Appl. Opt. 33, 2125–2127 (1994).
[CrossRef] [PubMed]

C. Shakher, A. J. Pramila Daniel, “Talbot interferometry with circular grating for the measurement of temperature in axisymmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
[CrossRef] [PubMed]

C. Shakher, A. Kumar Nirala, J. Pramila, S. Kumar Verma, “Use of speckle technique for temperature measurement in gaseous flame,” J. Opt. (Paris) 23, 35–39 (1992).
[CrossRef]

Sharma, D. K.

D. K. Sharma, S. Stephen, R. Natarajan, “Structure of burning n-hexane droplet by moiré deflectometry,” Combust. Sci. Tech. 13, 305–321 (1998).
[CrossRef]

Sicre, E. E.

Silva, D. E.

Springer, G. S.

Stephen, S.

D. K. Sharma, S. Stephen, R. Natarajan, “Structure of burning n-hexane droplet by moiré deflectometry,” Combust. Sci. Tech. 13, 305–321 (1998).
[CrossRef]

Sudol, R.

R. Sudol, B. J. Thompson, “A explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Sudol, R. J.

Suzuki, T.

Swanson, G. J.

Thompson, B. J.

R. J. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef] [PubMed]

R. Sudol, B. J. Thompson, “A explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

Vest, C. M.

Vitry, R.

R. Peyturaux, J. C. Renault, R. Vitry, “An automatic precise polychromatic pyrometer,” J. Opt. (Paris) 24, 107–112 (1993).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principle of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 2, pp. 87–90; Chap. 3, pp. 122–124.

Yokozeki, S.

Zerbino, L. M.

Appl. Opt. (12)

S. Kampmann, A. Leipertz, K. Dobbling, J. Hanumann, T. Sattelmayer, “Two-dimensional temperature measurement in a technical combustor with laser Rayleigh scattering,” Appl. Opt. 32, 6167–6172 (1993).
[CrossRef] [PubMed]

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, “Measurement of temperature distribution of flame by moiré deflectometry,” Appl. Opt. 20, 4263–4266 (1981).
[CrossRef] [PubMed]

P. V. Farrel, G. S. Springer, C. M. Vest, “Heterodyne holographic interferometry: concentration and temperature measurement in gas mixtures,” Appl. Opt. 21, 1624–1627 (1982).
[CrossRef]

S. Yokozeki, T. Suzuki, “Shearing interferometry using the grating as the beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
[CrossRef] [PubMed]

C. Shakher, A. Kumar Nirala, “Measurement of temperature using speckle shearing interferometry,” Appl. Opt. 33, 2125–2127 (1994).
[CrossRef] [PubMed]

D. E. Silva, “Talbot interferometry for radial and lateral derivatives,” Appl. Opt. 11, 2613–2624 (1972).
[CrossRef] [PubMed]

C. Shakher, A. J. Pramila Daniel, “Talbot interferometry with circular grating for the measurement of temperature in axisymmetric gaseous flames,” Appl. Opt. 33, 6068–6072 (1994).
[CrossRef] [PubMed]

P. V. Farrel, D. L. Hofeldt, “Temperature measurement in gases using speckle photography,” Appl. Opt. 23, 1055–1059 (1984).
[CrossRef]

Y.-S. Cheng, R.-C. Chang, “Theory of image formation using the Talbot effect,” Appl. Opt. 33, 1863–1874 (1994).
[CrossRef] [PubMed]

R. J. Sudol, B. J. Thompson, “Lau effect: theory and experiment,” Appl. Opt. 20, 1107–1116 (1981).
[CrossRef] [PubMed]

K. Patorski, “Incoherent superposition of multiple self-imaging under plane wave-front illumination,” Appl. Opt. 25, 2396–2403 (1986).
[CrossRef]

C. Colautti, L. M. Zerbino, E. E. Sicre, M. Garavaglia, “Lau effect using circular gratings,” Appl. Opt. 26, 2061–2062 (1987).
[CrossRef] [PubMed]

Combust Flame (1)

A. A. Boiarski, R. H. Barnes, J. F. Kircher, “Flame measurements utilizing Raman scattering,” Combust Flame 32, 111–114 (1978).
[CrossRef]

Combust. Flame (1)

D. L. Reuss, “Temperature measurement in a radially symmetric flame using holographic interferometry,” Combust. Flame 49, 207–209 (1983).
[CrossRef]

Combust. Sci. Tech. (1)

D. K. Sharma, S. Stephen, R. Natarajan, “Structure of burning n-hexane droplet by moiré deflectometry,” Combust. Sci. Tech. 13, 305–321 (1998).
[CrossRef]

J. Opt. (Paris) (2)

R. Peyturaux, J. C. Renault, R. Vitry, “An automatic precise polychromatic pyrometer,” J. Opt. (Paris) 24, 107–112 (1993).
[CrossRef]

C. Shakher, A. Kumar Nirala, J. Pramila, S. Kumar Verma, “Use of speckle technique for temperature measurement in gaseous flame,” J. Opt. (Paris) 23, 35–39 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

K. Patorski, “Incoherent superposition of multiple self-imaging Lau effect and moiré fringe explanation,” Opt. Acta 30, 745–748 (1983).
[CrossRef]

Opt. Commun. (7)

H. O. Bartelt, J. Jahns, “Inteferometry based on the Lau effect,” Opt. Commun. 30, 268–274 (1979).
[CrossRef]

Y.-S. Cheng, R.-C. Chang, “Image formation foe two dimensional periodic object using Talbot effect,” Opt. Commun. 20, 335–347 (1995).
[CrossRef]

J. Jahns, A. W. Lohman, “The Lau effect (A diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau effect and coherence theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “A explanation of the Lau effect based on the coherence theory,” Opt. Commun. 31, 105–110 (1979).
[CrossRef]

H. Bartlet, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
[CrossRef]

M. Giglio, S. Musazzi, U. Perini, “A white light speckle Schlieren technique,” Opt. Commun. 36, 117–120 (1981).
[CrossRef]

Opt. Eng. (1)

C. Shakher, A. J. Pramila Daniel, S. K. Angra, “Measurement of temperature profile of atomic absorption spectrophotometer burner (pre-mixed laminar flow slot burner) using Talbot interferometer,” Opt. Eng. 33, 2663–2669 (1994).
[CrossRef]

Opt. Laser Eng. (1)

C. Shakher, A. Kumar Nirala, “A review on refractive index and temperature profile measurement using laser-based interferometric techniques,” Opt. Laser Eng. 31, 455–491 (1999).
[CrossRef]

Other (4)

R. Bracewell, The Fourier Transform and Application (McGraw Hill, New York, 1978), Chap. 5, p. 262.

M. Born, E. Wolf, Principle of Optics, 4th ed. (Pergamon, New York, 1970), Chap. 2, pp. 87–90; Chap. 3, pp. 122–124.

A. J. Parmila Daniel, “Diffractive optical element in speckle metrology and temperature measurement,” Ph.D. dissertation (Instrument Design and Development Centre, Indian Institute of Technology, Dehli, 1994).

E. Pungor, Flame Photometry (Van Nostrand, London, 1976), pp. 147–152.

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Figures (3)

Fig. 1
Fig. 1

Schematic of the experimental setup of Lau phase interferometer with circular gratings.

Fig. 2
Fig. 2

Photograph of the fringes formed with an axisymmetric flame using circular gratings.

Fig. 3
Fig. 3

Variation of temperature with distance from the center of the flame.

Equations (3)

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

ϕ=dydx=1n0-+nryrrdrr2+y21/2.
n-n0n0=-1πrϕdyy2+r21/2.
T=T0n-n0n03PA-2RT3PA+1,

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