Abstract

In this paper the effect of uniform magnetic field B on the temperature and temperature profile of the diffusion flame is investigated using lensless Fourier transform digital holographic interferometry. The evaluation of temperature profile reveals that the width of flame as well as the maximum value of temperature inside the flame is increased.

© 2012 Optical Society of America

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  1. M. Faraday, “On the diamagnetic conditions of flames and gases,” London Edinburgh Dublin Philos. Mag. J. Sci. 31, 401–421 (1847).
    [CrossRef]
  2. H. Hayashi, “The external magnetic field effect on the emission intensity of the A2∑+→X2Π(0−0) transition of the OH radical in flames,” Chem. Phys. Lett. 87, 113–116 (1982).
    [CrossRef]
  3. A. V. Engle and J. R. Cozens, “Flames plasmas,” Adv. Electron. Electron Phys. 20, 99–146 (1964).
  4. S. Ueno, H. Esaki, and K. Harada, “Magnetic field effects on combustion,” IEEE Transl. J. Magn. Jpn. TJMJ-2, 861–862 (1987).
    [CrossRef]
  5. S. Ueno and K. Harada, “Effects of magnetic fields on flames and gas flow,” IEEE Trans. Magn. 23, 2752–2754 (1987).
    [CrossRef]
  6. T. Aoki, “Radical emissions and butane diffusion flames exposed to uniform magnetic fields encircled by magnetic gradient fields,” Jpn. J. Appl. Phys. 29, 952–957 (1990).
    [CrossRef]
  7. N. I. Wakayama, “Effect of a gradient magnetic field on the combustion reaction of methane in air,” Chem. Phys. Lett. 188, 279–281 (1992).
    [CrossRef]
  8. N. I. Wakayama, “Magnetic promotion of combustion in diffusion flames,” Combust. Flame 93, 207–214 (1993).
    [CrossRef]
  9. E. Yamada, M. Shinoda, H. Yamashita, and K. Kitagawa, “Numerical analysis of a hydrogen-oxygen diffusion flame in vertical or horizontal gradient of magnetic field,” Combust. Sci. Technol. 174, 149–164 (2002).
    [CrossRef]
  10. J. Baker and M. E. Calvert, “A study of the characteristics of slotted laminar jet diffusion flames in the presence of nonuniform magnetic fields,” Combust. Flame 133, 345–357 (2003).
    [CrossRef]
  11. S. Kinoshita, T. Takagi, H. Kotera, and N. I. Wakayama, “Numerical simulation of diffusion flames with and without magnetic field,” IEEE Trans. Appl. Supercond. 14, 1685–1688 (2004).
    [CrossRef]
  12. A. Gupta and J. Baker, “Uniform magnetic fields and equilibrium flame temperatures,” J. Thermophys. Heat Transfer 21, 520–525 (2007).
    [CrossRef]
  13. V. Gilard, P. Gillon, J.-N. Blanchard, and B. Sarh, “Influence of a horizontal magnetic field on a coflow methane/air diffusion flame,” Combust. Sci. Technol. 180, 1920–1935 (2008).
    [CrossRef]
  14. F. Khaldi, K. Messadek, and A. M. Benselama, “Isolation of gravity effects on diffusion flames by magnetic field,” Microgravity Sci. Technol. 22, 1–5 (2010).
    [CrossRef]
  15. P. Gillon, J. N. Blachard, and V. Gilard, “Magnetic field influence on coflow laminar diffusion flames,” Russ. J. Phys. Chem. B 4, 279–285 (2010).
    [CrossRef]
  16. S. Sharma, G. Sheoran, and C. Shakher, “Digital holographic interferometry for measurement of temperature in axisymmetric flames,” Appl. Opt. 51, 3228–3235 (2012).
  17. U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
    [CrossRef]
  18. C. Wagner, S. Seebacher, W. Osten, and W. Juptner, “Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology,” Appl. Opt. 38, 4812–4820 (1999).
    [CrossRef]
  19. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Academic, 2005).
  20. M. Born and E. F. Wolf, Principles of Optics, 4th ed. (Academic, 1978), Chap. 2, p. 102.
  21. Y. T. Cho and S.-J. Na, “Application of Abel inversion in real-time calculations for circularly and elliptically symmetric radiation sources,” Meas. Sci. Technol. 16, 878–884 (2005).
    [CrossRef]
  22. R. M. Goldstein, H. A. Zebker, and C. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radiosci. 23, 713–720 (1988).
    [CrossRef]
  23. F. Charrière, B. Rappaz, J. Kühn, T. Colomb, P. Marquet, and C. Depeursinge, “Influence of shot noise on phase measurement accuracy in digital holographic microscopy,” Opt. Express 15, 8818–8831 (2007).
    [CrossRef]

2012 (1)

2010 (2)

F. Khaldi, K. Messadek, and A. M. Benselama, “Isolation of gravity effects on diffusion flames by magnetic field,” Microgravity Sci. Technol. 22, 1–5 (2010).
[CrossRef]

P. Gillon, J. N. Blachard, and V. Gilard, “Magnetic field influence on coflow laminar diffusion flames,” Russ. J. Phys. Chem. B 4, 279–285 (2010).
[CrossRef]

2008 (1)

V. Gilard, P. Gillon, J.-N. Blanchard, and B. Sarh, “Influence of a horizontal magnetic field on a coflow methane/air diffusion flame,” Combust. Sci. Technol. 180, 1920–1935 (2008).
[CrossRef]

2007 (2)

2005 (1)

Y. T. Cho and S.-J. Na, “Application of Abel inversion in real-time calculations for circularly and elliptically symmetric radiation sources,” Meas. Sci. Technol. 16, 878–884 (2005).
[CrossRef]

2004 (1)

S. Kinoshita, T. Takagi, H. Kotera, and N. I. Wakayama, “Numerical simulation of diffusion flames with and without magnetic field,” IEEE Trans. Appl. Supercond. 14, 1685–1688 (2004).
[CrossRef]

2003 (1)

J. Baker and M. E. Calvert, “A study of the characteristics of slotted laminar jet diffusion flames in the presence of nonuniform magnetic fields,” Combust. Flame 133, 345–357 (2003).
[CrossRef]

2002 (2)

E. Yamada, M. Shinoda, H. Yamashita, and K. Kitagawa, “Numerical analysis of a hydrogen-oxygen diffusion flame in vertical or horizontal gradient of magnetic field,” Combust. Sci. Technol. 174, 149–164 (2002).
[CrossRef]

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

1999 (1)

1993 (1)

N. I. Wakayama, “Magnetic promotion of combustion in diffusion flames,” Combust. Flame 93, 207–214 (1993).
[CrossRef]

1992 (1)

N. I. Wakayama, “Effect of a gradient magnetic field on the combustion reaction of methane in air,” Chem. Phys. Lett. 188, 279–281 (1992).
[CrossRef]

1990 (1)

T. Aoki, “Radical emissions and butane diffusion flames exposed to uniform magnetic fields encircled by magnetic gradient fields,” Jpn. J. Appl. Phys. 29, 952–957 (1990).
[CrossRef]

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radiosci. 23, 713–720 (1988).
[CrossRef]

1987 (2)

S. Ueno, H. Esaki, and K. Harada, “Magnetic field effects on combustion,” IEEE Transl. J. Magn. Jpn. TJMJ-2, 861–862 (1987).
[CrossRef]

S. Ueno and K. Harada, “Effects of magnetic fields on flames and gas flow,” IEEE Trans. Magn. 23, 2752–2754 (1987).
[CrossRef]

1982 (1)

H. Hayashi, “The external magnetic field effect on the emission intensity of the A2∑+→X2Π(0−0) transition of the OH radical in flames,” Chem. Phys. Lett. 87, 113–116 (1982).
[CrossRef]

1964 (1)

A. V. Engle and J. R. Cozens, “Flames plasmas,” Adv. Electron. Electron Phys. 20, 99–146 (1964).

1847 (1)

M. Faraday, “On the diamagnetic conditions of flames and gases,” London Edinburgh Dublin Philos. Mag. J. Sci. 31, 401–421 (1847).
[CrossRef]

Aoki, T.

T. Aoki, “Radical emissions and butane diffusion flames exposed to uniform magnetic fields encircled by magnetic gradient fields,” Jpn. J. Appl. Phys. 29, 952–957 (1990).
[CrossRef]

Baker, J.

A. Gupta and J. Baker, “Uniform magnetic fields and equilibrium flame temperatures,” J. Thermophys. Heat Transfer 21, 520–525 (2007).
[CrossRef]

J. Baker and M. E. Calvert, “A study of the characteristics of slotted laminar jet diffusion flames in the presence of nonuniform magnetic fields,” Combust. Flame 133, 345–357 (2003).
[CrossRef]

Benselama, A. M.

F. Khaldi, K. Messadek, and A. M. Benselama, “Isolation of gravity effects on diffusion flames by magnetic field,” Microgravity Sci. Technol. 22, 1–5 (2010).
[CrossRef]

Blachard, J. N.

P. Gillon, J. N. Blachard, and V. Gilard, “Magnetic field influence on coflow laminar diffusion flames,” Russ. J. Phys. Chem. B 4, 279–285 (2010).
[CrossRef]

Blanchard, J.-N.

V. Gilard, P. Gillon, J.-N. Blanchard, and B. Sarh, “Influence of a horizontal magnetic field on a coflow methane/air diffusion flame,” Combust. Sci. Technol. 180, 1920–1935 (2008).
[CrossRef]

Born, M.

M. Born and E. F. Wolf, Principles of Optics, 4th ed. (Academic, 1978), Chap. 2, p. 102.

Calvert, M. E.

J. Baker and M. E. Calvert, “A study of the characteristics of slotted laminar jet diffusion flames in the presence of nonuniform magnetic fields,” Combust. Flame 133, 345–357 (2003).
[CrossRef]

Charrière, F.

Cho, Y. T.

Y. T. Cho and S.-J. Na, “Application of Abel inversion in real-time calculations for circularly and elliptically symmetric radiation sources,” Meas. Sci. Technol. 16, 878–884 (2005).
[CrossRef]

Colomb, T.

Cozens, J. R.

A. V. Engle and J. R. Cozens, “Flames plasmas,” Adv. Electron. Electron Phys. 20, 99–146 (1964).

Depeursinge, C.

Engle, A. V.

A. V. Engle and J. R. Cozens, “Flames plasmas,” Adv. Electron. Electron Phys. 20, 99–146 (1964).

Esaki, H.

S. Ueno, H. Esaki, and K. Harada, “Magnetic field effects on combustion,” IEEE Transl. J. Magn. Jpn. TJMJ-2, 861–862 (1987).
[CrossRef]

Faraday, M.

M. Faraday, “On the diamagnetic conditions of flames and gases,” London Edinburgh Dublin Philos. Mag. J. Sci. 31, 401–421 (1847).
[CrossRef]

Gilard, V.

P. Gillon, J. N. Blachard, and V. Gilard, “Magnetic field influence on coflow laminar diffusion flames,” Russ. J. Phys. Chem. B 4, 279–285 (2010).
[CrossRef]

V. Gilard, P. Gillon, J.-N. Blanchard, and B. Sarh, “Influence of a horizontal magnetic field on a coflow methane/air diffusion flame,” Combust. Sci. Technol. 180, 1920–1935 (2008).
[CrossRef]

Gillon, P.

P. Gillon, J. N. Blachard, and V. Gilard, “Magnetic field influence on coflow laminar diffusion flames,” Russ. J. Phys. Chem. B 4, 279–285 (2010).
[CrossRef]

V. Gilard, P. Gillon, J.-N. Blanchard, and B. Sarh, “Influence of a horizontal magnetic field on a coflow methane/air diffusion flame,” Combust. Sci. Technol. 180, 1920–1935 (2008).
[CrossRef]

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radiosci. 23, 713–720 (1988).
[CrossRef]

Gupta, A.

A. Gupta and J. Baker, “Uniform magnetic fields and equilibrium flame temperatures,” J. Thermophys. Heat Transfer 21, 520–525 (2007).
[CrossRef]

Harada, K.

S. Ueno and K. Harada, “Effects of magnetic fields on flames and gas flow,” IEEE Trans. Magn. 23, 2752–2754 (1987).
[CrossRef]

S. Ueno, H. Esaki, and K. Harada, “Magnetic field effects on combustion,” IEEE Transl. J. Magn. Jpn. TJMJ-2, 861–862 (1987).
[CrossRef]

Hayashi, H.

H. Hayashi, “The external magnetic field effect on the emission intensity of the A2∑+→X2Π(0−0) transition of the OH radical in flames,” Chem. Phys. Lett. 87, 113–116 (1982).
[CrossRef]

Juptner, W.

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Khaldi, F.

F. Khaldi, K. Messadek, and A. M. Benselama, “Isolation of gravity effects on diffusion flames by magnetic field,” Microgravity Sci. Technol. 22, 1–5 (2010).
[CrossRef]

Kinoshita, S.

S. Kinoshita, T. Takagi, H. Kotera, and N. I. Wakayama, “Numerical simulation of diffusion flames with and without magnetic field,” IEEE Trans. Appl. Supercond. 14, 1685–1688 (2004).
[CrossRef]

Kitagawa, K.

E. Yamada, M. Shinoda, H. Yamashita, and K. Kitagawa, “Numerical analysis of a hydrogen-oxygen diffusion flame in vertical or horizontal gradient of magnetic field,” Combust. Sci. Technol. 174, 149–164 (2002).
[CrossRef]

Kotera, H.

S. Kinoshita, T. Takagi, H. Kotera, and N. I. Wakayama, “Numerical simulation of diffusion flames with and without magnetic field,” IEEE Trans. Appl. Supercond. 14, 1685–1688 (2004).
[CrossRef]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Academic, 2005).

Kühn, J.

Marquet, P.

Messadek, K.

F. Khaldi, K. Messadek, and A. M. Benselama, “Isolation of gravity effects on diffusion flames by magnetic field,” Microgravity Sci. Technol. 22, 1–5 (2010).
[CrossRef]

Na, S.-J.

Y. T. Cho and S.-J. Na, “Application of Abel inversion in real-time calculations for circularly and elliptically symmetric radiation sources,” Meas. Sci. Technol. 16, 878–884 (2005).
[CrossRef]

Osten, W.

Rappaz, B.

Sarh, B.

V. Gilard, P. Gillon, J.-N. Blanchard, and B. Sarh, “Influence of a horizontal magnetic field on a coflow methane/air diffusion flame,” Combust. Sci. Technol. 180, 1920–1935 (2008).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Seebacher, S.

Shakher, C.

Sharma, S.

Sheoran, G.

Shinoda, M.

E. Yamada, M. Shinoda, H. Yamashita, and K. Kitagawa, “Numerical analysis of a hydrogen-oxygen diffusion flame in vertical or horizontal gradient of magnetic field,” Combust. Sci. Technol. 174, 149–164 (2002).
[CrossRef]

Takagi, T.

S. Kinoshita, T. Takagi, H. Kotera, and N. I. Wakayama, “Numerical simulation of diffusion flames with and without magnetic field,” IEEE Trans. Appl. Supercond. 14, 1685–1688 (2004).
[CrossRef]

Ueno, S.

S. Ueno and K. Harada, “Effects of magnetic fields on flames and gas flow,” IEEE Trans. Magn. 23, 2752–2754 (1987).
[CrossRef]

S. Ueno, H. Esaki, and K. Harada, “Magnetic field effects on combustion,” IEEE Transl. J. Magn. Jpn. TJMJ-2, 861–862 (1987).
[CrossRef]

Wagner, C.

Wakayama, N. I.

S. Kinoshita, T. Takagi, H. Kotera, and N. I. Wakayama, “Numerical simulation of diffusion flames with and without magnetic field,” IEEE Trans. Appl. Supercond. 14, 1685–1688 (2004).
[CrossRef]

N. I. Wakayama, “Magnetic promotion of combustion in diffusion flames,” Combust. Flame 93, 207–214 (1993).
[CrossRef]

N. I. Wakayama, “Effect of a gradient magnetic field on the combustion reaction of methane in air,” Chem. Phys. Lett. 188, 279–281 (1992).
[CrossRef]

Werner, C.

R. M. Goldstein, H. A. Zebker, and C. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radiosci. 23, 713–720 (1988).
[CrossRef]

Wolf, E. F.

M. Born and E. F. Wolf, Principles of Optics, 4th ed. (Academic, 1978), Chap. 2, p. 102.

Yamada, E.

E. Yamada, M. Shinoda, H. Yamashita, and K. Kitagawa, “Numerical analysis of a hydrogen-oxygen diffusion flame in vertical or horizontal gradient of magnetic field,” Combust. Sci. Technol. 174, 149–164 (2002).
[CrossRef]

Yamashita, H.

E. Yamada, M. Shinoda, H. Yamashita, and K. Kitagawa, “Numerical analysis of a hydrogen-oxygen diffusion flame in vertical or horizontal gradient of magnetic field,” Combust. Sci. Technol. 174, 149–164 (2002).
[CrossRef]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radiosci. 23, 713–720 (1988).
[CrossRef]

Adv. Electron. Electron Phys. (1)

A. V. Engle and J. R. Cozens, “Flames plasmas,” Adv. Electron. Electron Phys. 20, 99–146 (1964).

Appl. Opt. (2)

Chem. Phys. Lett. (2)

H. Hayashi, “The external magnetic field effect on the emission intensity of the A2∑+→X2Π(0−0) transition of the OH radical in flames,” Chem. Phys. Lett. 87, 113–116 (1982).
[CrossRef]

N. I. Wakayama, “Effect of a gradient magnetic field on the combustion reaction of methane in air,” Chem. Phys. Lett. 188, 279–281 (1992).
[CrossRef]

Combust. Flame (2)

N. I. Wakayama, “Magnetic promotion of combustion in diffusion flames,” Combust. Flame 93, 207–214 (1993).
[CrossRef]

J. Baker and M. E. Calvert, “A study of the characteristics of slotted laminar jet diffusion flames in the presence of nonuniform magnetic fields,” Combust. Flame 133, 345–357 (2003).
[CrossRef]

Combust. Sci. Technol. (2)

V. Gilard, P. Gillon, J.-N. Blanchard, and B. Sarh, “Influence of a horizontal magnetic field on a coflow methane/air diffusion flame,” Combust. Sci. Technol. 180, 1920–1935 (2008).
[CrossRef]

E. Yamada, M. Shinoda, H. Yamashita, and K. Kitagawa, “Numerical analysis of a hydrogen-oxygen diffusion flame in vertical or horizontal gradient of magnetic field,” Combust. Sci. Technol. 174, 149–164 (2002).
[CrossRef]

IEEE Trans. Appl. Supercond. (1)

S. Kinoshita, T. Takagi, H. Kotera, and N. I. Wakayama, “Numerical simulation of diffusion flames with and without magnetic field,” IEEE Trans. Appl. Supercond. 14, 1685–1688 (2004).
[CrossRef]

IEEE Trans. Magn. (1)

S. Ueno and K. Harada, “Effects of magnetic fields on flames and gas flow,” IEEE Trans. Magn. 23, 2752–2754 (1987).
[CrossRef]

IEEE Transl. J. Magn. Jpn. (1)

S. Ueno, H. Esaki, and K. Harada, “Magnetic field effects on combustion,” IEEE Transl. J. Magn. Jpn. TJMJ-2, 861–862 (1987).
[CrossRef]

J. Thermophys. Heat Transfer (1)

A. Gupta and J. Baker, “Uniform magnetic fields and equilibrium flame temperatures,” J. Thermophys. Heat Transfer 21, 520–525 (2007).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Aoki, “Radical emissions and butane diffusion flames exposed to uniform magnetic fields encircled by magnetic gradient fields,” Jpn. J. Appl. Phys. 29, 952–957 (1990).
[CrossRef]

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

M. Faraday, “On the diamagnetic conditions of flames and gases,” London Edinburgh Dublin Philos. Mag. J. Sci. 31, 401–421 (1847).
[CrossRef]

Meas. Sci. Technol. (2)

Y. T. Cho and S.-J. Na, “Application of Abel inversion in real-time calculations for circularly and elliptically symmetric radiation sources,” Meas. Sci. Technol. 16, 878–884 (2005).
[CrossRef]

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Microgravity Sci. Technol. (1)

F. Khaldi, K. Messadek, and A. M. Benselama, “Isolation of gravity effects on diffusion flames by magnetic field,” Microgravity Sci. Technol. 22, 1–5 (2010).
[CrossRef]

Opt. Express (1)

Radiosci. (1)

R. M. Goldstein, H. A. Zebker, and C. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radiosci. 23, 713–720 (1988).
[CrossRef]

Russ. J. Phys. Chem. B (1)

P. Gillon, J. N. Blachard, and V. Gilard, “Magnetic field influence on coflow laminar diffusion flames,” Russ. J. Phys. Chem. B 4, 279–285 (2010).
[CrossRef]

Other (2)

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Academic, 2005).

M. Born and E. F. Wolf, Principles of Optics, 4th ed. (Academic, 1978), Chap. 2, p. 102.

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Figures (10)

Fig. 1.
Fig. 1.

Coordinate system used in Fresnel transform reconstruction method.

Fig. 2.
Fig. 2.

Cross section of an axisymmetric flame at a particular height.

Fig. 3.
Fig. 3.

Schematic of the experimental setup for measurement of temperature and temperature profile of an axisymmetric flame under uniform magnetic field using LLFTDH.

Fig. 4.
Fig. 4.

Flow chart of the experimental and analytical process.

Fig. 5.
Fig. 5.

(a) Phase difference map of the air without flame and with flame. (b) Phase difference map of the air without flame and with flame in the presence of uniform magnetic field (0.35 T).

Fig. 6.
Fig. 6.

(a) 3D phase map profile corresponding to Fig. 5(a). (b) 3D phase map profile corresponding to Fig. 5(b).

Fig. 7.
Fig. 7.

(a) Refractive index profile corresponding to Fig. 6(a). (b) Refractive index profile corresponding to Fig. 6(b).

Fig. 8.
Fig. 8.

(a) 3D radial distribution of refractive index corresponding to Fig. 6(a). (b) 3D radial distribution of refractive index corresponding to Fig. 6(b).

Fig. 9.
Fig. 9.

(a) Temperature distribution at a height of 10 mm above the burner corresponding to Fig. 7(a). (b) Temperature distribution at a height of 10 mm above the burner corresponding to Fig. 7(b).

Fig. 10.
Fig. 10.

(a) 3D temperature profile corresponding to Fig. 8(a). (b) 3D Temperature profile corresponding to Fig. 8(b).

Equations (20)

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

O(XI,YI)=iλdexp[i2πλd]exp[iπλd(XI2+YI2)]×FFT{R(XH,YH)h(XH,YH)exp[+iπλd(XH2+YH2)]},
R(XH,YH)=1dexp[i2πλd]exp[iπλd(XH2+YH2)].
O(XI,YI)=iλdexp[i2πλ2d]exp[iπλd(XI2+YI2)]×FFT{h(XH,YH)}.
ϕ(XI,YI)=tan1[Im{O(XI,YI)}Re{O(XI,YI)}],
ϕ1(k,l)=arctanIm[O1(k,l)]Re[O1(k,l)],
ϕ2(k,l)=arctanIm[O2(k,l)]Re[O2(k,l)].
ϕ(x)=2πλ0Ln(r)dz,
Δϕ(x)=ϕ1(x)ϕ2(x)=2πλ0L[nr(r)n0(r)]dz,
Δϕ(x)=22πλxR[nr(r)n0(r)]r(r2x2)12dr,
Δn(r)=λ2π2rRd(Δϕ)dx(x2r2)12dx.
T=T0(nn0n0)(3PA+2RT03PA)+1,
Δϕ(x)ΔxΔy=z0+z0Δn(r)ΔxΔyΔz.
Δϕ(x)Δx=z0+z0Δn(r)ΔxΔz.
Δϕ1=2(S11Δn1+S12Δn2+S1nΔnn)/d,Δϕ2=2(S22Δn2+S23Δn3+S2nΔnn)/d,Δϕn=2(SnnΔnn)/d,
[S11S12S1n0S22S2n00Snn][n1n2nn]=d2[Δϕ1Δϕ2Δϕn].
θij={cos1((i1)/j)(ij)0(i>j),
pij={12(jd)2θij12((i1)d)2tan(θij)(ij)0(i>j),
Sij={(Pi,jPi+1,j)(Pi,j1Pi+1,j1)(ij)0(i>j).
[Δn]=d2[S]1[Δϕ].
Δϕ(kΔx,lΔy)={ϕ1(kΔx,lΔy)ϕ2(kΔx,lΔy)ifϕ1(kΔx,lΔy)ϕ2(kΔx,lΔy)ϕ1(kΔx,lΔy)ϕ2(kΔx,lΔy)+2πif  ϕ1(kΔx,lΔy)<ϕ2(kΔx,lΔy).

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