Abstract

A method for measurement of temperature in laminar free convection flow of water is presented using digital holographic interferometry. The method is relatively simple and fast because the method uses lensless Fourier transform digital holography, for which the reconstruction algorithm is simple and fast, and also the method does not require use of any extra experimental efforts as in phase shifting. The quantitative unwrapped phase difference is calculated experimentally from two digital holograms recorded in two different states of water—one in the quiescent state, the other in the laminar free convection. Unknown temperature in laminar free convection is measured quantitatively using a known value of temperature in the quiescent state from the unwrapped phase difference, where the equation by Tilton and Taylor describing the variation of refractive index of water with temperature is used to connect the phase with temperature. Experiments are also performed to visualize the turbulent free convection flow.

© 2009 Optical Society of America

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    [CrossRef]
  10. P. Singh, S. Mirza, S. K. Angra, A. Kumar, C. Shakher, and R. S. Sirohi, “Measurement of temperature and temperature profile of an axisymmetric flame and a 2-D flame using digital speckle photography,” J. Hologr. Speckle 2, 84-89 (2005).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  37. W. Merzkirch, Flow Visualization (Academic, 1987).

2007

C. Shakher and R. S. Sirohi, “Flame temperature measurement using speckle techniques,” in New Directions in Holography and Speckle, H. Caulfield and C. S. Vikram, eds. (American Scientific, 2007), pp. 1-16.

M. M. Hossain, G. Sheoran, D. S. Mehta, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

J. Colombani and J. Bert, “Holographic interferometry for the study of liquids,” J. Mol. Liq. 134, 8-14 (2007).
[CrossRef]

F. Charrière, B. Rappaz, J. Kühn, T. Colomb, P. Marquetand, and C. Depeursinge, “Influence of shot noise on phase measurement accuracy in digital holographic microscopy,” Opt. Express 15, 8818-8831 (2007).
[CrossRef] [PubMed]

2006

M. M. Hossain, D. S. Mehta, and C. Shakher, “Refractive index determination: an application of lensless Fourier digital holography,” Opt. Eng. 45, 106203 (2006).
[CrossRef]

2005

2002

2001

R. Tauscher, “Holographic interferometry,” in Optical Measurements Techniques and Applications, 2nd ed., F.Mayinger and O.Feldmann, eds. (Springer, 2001), pp. 39-53.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126(2001).
[CrossRef]

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417-428 (2001).
[CrossRef]

2000

1999

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]

A. Asseban, N. A. Fomin, M. Lallemand, E. Lavinskaja, J. B. Saulnier, and D. E. Vitkin, “Temperature field determination in free and forced convective flows by speckle photography,” Proc. SPIE 3783, 80-88 (1999).
[CrossRef]

B. Skarman, K. Wozniac, and J. Becker, “Digital in-line holography for the analysis of Bénard convection,” Flow Meas. Instrum. 10, 91-97 (1999).
[CrossRef]

1998

1997

G. S. Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, “Temperature measurement in laninar free convection using electro-optic holography,” J. Phys. III 7, 1893-1898 (1997).

G. Pedrini, Ph. Froening, H. Fessler, and H. Tiziani, “Transient vibration measurements using multi-pulse digital holography,” Opt. Laser Technol. 29, 505-511 (1997).
[CrossRef]

1996

B. Skarman, J. Becker, and K. Wozniac, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6(1996).
[CrossRef]

M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, “Single-transform Fourier--Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z.Fuzessy, W.Juptner, and W.Osten, eds. (Akademie Verlag, 1996), pp. 67-73.

1994

1993

H. Phillipp, H. Funch, and E. Winkhofer, “A flame diagnosis by light sheet imaging and by shearing interferometry,” Opt. Eng. 32, 1025-1032 (1993).
[CrossRef]

1987

W. Merzkirch, Flow Visualization (Academic, 1987).

1986

1984

1983

D. L. Reuss, “Temperature measurement in a radially symmetric flame using holographic interferometry,” Combust. Flame 49, 207-219 (1983).
[CrossRef]

1981

1979

C. M. Vest, Holographic Interferometry (Wiley, 1979).

1977

M. N. Özisik, Basic Heat Transfer (McGraw-Hill, 1977).

C. M. Vest and P. T. Radulovic, “Measurement of three-dimensional temperature fields by holographic interferometry,” in Applications of Holography and Optical Data Processing, E. Marom and A. A. Friesem, eds. (Pergamon, 1977), pp. 241-249.

1974

D. W. Sweeney and C. M. Vest, “Measurement of three-dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Transfer 17, 1443-1454 (1974).
[CrossRef]

1963

F. J. Weinberg, Optics of Flame (Butterworth, 1963).

Ambrosini, D.

G. S. Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, “Temperature measurement in laninar free convection using electro-optic holography,” J. Phys. III 7, 1893-1898 (1997).

Angra, S. K.

P. Singh, S. Mirza, S. K. Angra, A. Kumar, C. Shakher, and R. S. Sirohi, “Measurement of temperature and temperature profile of an axisymmetric flame and a 2-D flame using digital speckle photography,” J. Hologr. Speckle 2, 84-89 (2005).
[CrossRef]

Asseban, A.

A. Asseban, N. A. Fomin, M. Lallemand, E. Lavinskaja, J. B. Saulnier, and D. E. Vitkin, “Temperature field determination in free and forced convective flows by speckle photography,” Proc. SPIE 3783, 80-88 (1999).
[CrossRef]

Baumbach, T.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126(2001).
[CrossRef]

Becker, J.

B. Skarman, K. Wozniac, and J. Becker, “Digital in-line holography for the analysis of Bénard convection,” Flow Meas. Instrum. 10, 91-97 (1999).
[CrossRef]

B. Skarman, J. Becker, and K. Wozniac, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6(1996).
[CrossRef]

Bert, J.

J. Colombani and J. Bert, “Holographic interferometry for the study of liquids,” J. Mol. Liq. 134, 8-14 (2007).
[CrossRef]

Boileau, J. P.

Breteau, J. M.

Carlsson, T. E.

Charrière, F.

Chu, H.

H. Chu, “Laminar flow over an isothermal plate,” http://www.efunda.com/formulae/heat_transfer/convection_forced/isothermalplate_lamflow.cfm.

Colomb, T.

Colombani, J.

J. Colombani and J. Bert, “Holographic interferometry for the study of liquids,” J. Mol. Liq. 134, 8-14 (2007).
[CrossRef]

Cuche, E.

Daniel, A. J. P.

De la Torre-Ibarra, M.

Depeursinge, C.

Fessler, H.

G. Pedrini, Ph. Froening, H. Fessler, and H. Tiziani, “Transient vibration measurements using multi-pulse digital holography,” Opt. Laser Technol. 29, 505-511 (1997).
[CrossRef]

Fomin, N. A.

A. Asseban, N. A. Fomin, M. Lallemand, E. Lavinskaja, J. B. Saulnier, and D. E. Vitkin, “Temperature field determination in free and forced convective flows by speckle photography,” Proc. SPIE 3783, 80-88 (1999).
[CrossRef]

Froening, Ph.

G. Pedrini, Ph. Froening, H. Fessler, and H. Tiziani, “Transient vibration measurements using multi-pulse digital holography,” Opt. Laser Technol. 29, 505-511 (1997).
[CrossRef]

Funch, H.

H. Phillipp, H. Funch, and E. Winkhofer, “A flame diagnosis by light sheet imaging and by shearing interferometry,” Opt. Eng. 32, 1025-1032 (1993).
[CrossRef]

Gautier, B.

Gillet, S.

Glatt, I.

Grill, M.

Hirayama, T.

M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, “Single-transform Fourier--Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z.Fuzessy, W.Juptner, and W.Osten, eds. (Akademie Verlag, 1996), pp. 67-73.

Hossain, M. M.

C. Shakher, M. M. Hossain, D. S. Mehta, and G. Sheoran, “Measurement of temperature field in steady laminar free convection flow using digital holography,” presented at the LM2008-9th International Symposium on Laser Metrology, Singapore, 30 June-2 July 2008.

M. M. Hossain, G. Sheoran, D. S. Mehta, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

M. M. Hossain, D. S. Mehta, and C. Shakher, “Refractive index determination: an application of lensless Fourier digital holography,” Opt. Eng. 45, 106203 (2006).
[CrossRef]

Juptner, W.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126(2001).
[CrossRef]

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]

Kafri, O.

Kato, J.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417-428 (2001).
[CrossRef]

Keren Bar-.Ziv, E. E.

Kohgo, H.

M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, “Single-transform Fourier--Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z.Fuzessy, W.Juptner, and W.Osten, eds. (Akademie Verlag, 1996), pp. 67-73.

Kreis, T.

Kreis, T. M.

T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829-1839 (2002).
[CrossRef]

Kühn, J.

Kumar, A.

P. Singh, S. Mirza, S. K. Angra, A. Kumar, C. Shakher, and R. S. Sirohi, “Measurement of temperature and temperature profile of an axisymmetric flame and a 2-D flame using digital speckle photography,” J. Hologr. Speckle 2, 84-89 (2005).
[CrossRef]

Lallemand, M.

A. Asseban, N. A. Fomin, M. Lallemand, E. Lavinskaja, J. B. Saulnier, and D. E. Vitkin, “Temperature field determination in free and forced convective flows by speckle photography,” Proc. SPIE 3783, 80-88 (1999).
[CrossRef]

Lavinskaja, E.

A. Asseban, N. A. Fomin, M. Lallemand, E. Lavinskaja, J. B. Saulnier, and D. E. Vitkin, “Temperature field determination in free and forced convective flows by speckle photography,” Proc. SPIE 3783, 80-88 (1999).
[CrossRef]

Leval, J.

Marquet, P.

Marquetand, P.

Mehta, D. S.

C. Shakher, M. M. Hossain, D. S. Mehta, and G. Sheoran, “Measurement of temperature field in steady laminar free convection flow using digital holography,” presented at the LM2008-9th International Symposium on Laser Metrology, Singapore, 30 June-2 July 2008.

M. M. Hossain, G. Sheoran, D. S. Mehta, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

M. M. Hossain, D. S. Mehta, and C. Shakher, “Refractive index determination: an application of lensless Fourier digital holography,” Opt. Eng. 45, 106203 (2006).
[CrossRef]

Mendoza-Santoyo, F.

Merzkirch, W.

W. Merzkirch, Flow Visualization (Academic, 1987).

Mirza, S.

P. Singh, S. Mirza, S. K. Angra, A. Kumar, C. Shakher, and R. S. Sirohi, “Measurement of temperature and temperature profile of an axisymmetric flame and a 2-D flame using digital speckle photography,” J. Hologr. Speckle 2, 84-89 (2005).
[CrossRef]

Murata, K.

Nakano, Y.

Nilsson, B.

Ohta, S.

I. Yamaguchi, S. Ohta, and J. Kato, “Surface contouring by phase-shifting digital holography,” Opt. Lasers Eng. 36, 417-428 (2001).
[CrossRef]

Osten, W.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126(2001).
[CrossRef]

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]

Özisik, M. N.

M. N. Özisik, Basic Heat Transfer (McGraw-Hill, 1977).

Paoletti, D.

G. S. Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, “Temperature measurement in laninar free convection using electro-optic holography,” J. Phys. III 7, 1893-1898 (1997).

Pascal, J. C.

Pedrini, G.

G. Pedrini, Ph. Froening, H. Fessler, and H. Tiziani, “Transient vibration measurements using multi-pulse digital holography,” Opt. Laser Technol. 29, 505-511 (1997).
[CrossRef]

Perez-Lopez, C.

Phillipp, H.

H. Phillipp, H. Funch, and E. Winkhofer, “A flame diagnosis by light sheet imaging and by shearing interferometry,” Opt. Eng. 32, 1025-1032 (1993).
[CrossRef]

Picart, P.

Ponticiello, A.

G. S. Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, “Temperature measurement in laninar free convection using electro-optic holography,” J. Phys. III 7, 1893-1898 (1997).

Radulovic, P. T.

C. M. Vest and P. T. Radulovic, “Measurement of three-dimensional temperature fields by holographic interferometry,” in Applications of Holography and Optical Data Processing, E. Marom and A. A. Friesem, eds. (Pergamon, 1977), pp. 241-249.

Rappaz, B.

Reuss, D. L.

D. L. Reuss, “Temperature measurement in a radially symmetric flame using holographic interferometry,” Combust. Flame 49, 207-219 (1983).
[CrossRef]

Saucedo-A, T.

Saulnier, J. B.

A. Asseban, N. A. Fomin, M. Lallemand, E. Lavinskaja, J. B. Saulnier, and D. E. Vitkin, “Temperature field determination in free and forced convective flows by speckle photography,” Proc. SPIE 3783, 80-88 (1999).
[CrossRef]

Schnars, U.

Seebacher, S.

S. Seebacher, W. Osten, T. Baumbach, and W. Juptner, “The determination of material parameters of microcomponents using digital holography,” Opt. Lasers Eng. 36, 103-126(2001).
[CrossRef]

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]

Shakher, C.

C. Shakher, M. M. Hossain, D. S. Mehta, and G. Sheoran, “Measurement of temperature field in steady laminar free convection flow using digital holography,” presented at the LM2008-9th International Symposium on Laser Metrology, Singapore, 30 June-2 July 2008.

C. Shakher and R. S. Sirohi, “Flame temperature measurement using speckle techniques,” in New Directions in Holography and Speckle, H. Caulfield and C. S. Vikram, eds. (American Scientific, 2007), pp. 1-16.

M. M. Hossain, G. Sheoran, D. S. Mehta, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

M. M. Hossain, D. S. Mehta, and C. Shakher, “Refractive index determination: an application of lensless Fourier digital holography,” Opt. Eng. 45, 106203 (2006).
[CrossRef]

P. Singh, S. Mirza, S. K. Angra, A. Kumar, C. Shakher, and R. S. Sirohi, “Measurement of temperature and temperature profile of an axisymmetric flame and a 2-D flame using digital speckle photography,” J. Hologr. Speckle 2, 84-89 (2005).
[CrossRef]

M. Thakur, A. L. Vyas, and C. Shakher, “Measurement of temperature profile of a gaseous flames using Lau phase interferometer with circular gratings,” Appl. Opt. 41, 654-657 (2002).
[CrossRef] [PubMed]

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

Sheoran, G.

C. Shakher, M. M. Hossain, D. S. Mehta, and G. Sheoran, “Measurement of temperature field in steady laminar free convection flow using digital holography,” presented at the LM2008-9th International Symposium on Laser Metrology, Singapore, 30 June-2 July 2008.

M. M. Hossain, G. Sheoran, D. S. Mehta, and C. Shakher, “Contouring of diffused objects by using digital holography,” Opt. Lasers Eng. 45, 684-689 (2007).
[CrossRef]

Singh, P.

P. Singh, S. Mirza, S. K. Angra, A. Kumar, C. Shakher, and R. S. Sirohi, “Measurement of temperature and temperature profile of an axisymmetric flame and a 2-D flame using digital speckle photography,” J. Hologr. Speckle 2, 84-89 (2005).
[CrossRef]

Sirohi, R. S.

C. Shakher and R. S. Sirohi, “Flame temperature measurement using speckle techniques,” in New Directions in Holography and Speckle, H. Caulfield and C. S. Vikram, eds. (American Scientific, 2007), pp. 1-16.

P. Singh, S. Mirza, S. K. Angra, A. Kumar, C. Shakher, and R. S. Sirohi, “Measurement of temperature and temperature profile of an axisymmetric flame and a 2-D flame using digital speckle photography,” J. Hologr. Speckle 2, 84-89 (2005).
[CrossRef]

Skarman, B.

B. Skarman, K. Wozniac, and J. Becker, “Digital in-line holography for the analysis of Bénard convection,” Flow Meas. Instrum. 10, 91-97 (1999).
[CrossRef]

B. Skarman, J. Becker, and K. Wozniac, “Simultaneous 3D-PIV and temperature measurement using a new CCD based holographic interferometer,” Flow Meas. Instrum. 7, 1-6(1996).
[CrossRef]

Spagnolo, G. S.

G. S. Spagnolo, D. Ambrosini, A. Ponticiello, and D. Paoletti, “Temperature measurement in laninar free convection using electro-optic holography,” J. Phys. III 7, 1893-1898 (1997).

Sweeney, D. W.

D. W. Sweeney and C. M. Vest, “Measurement of three-dimensional temperature fields above heated surfaces by holographic interferometry,” Int. J. Heat Mass Transfer 17, 1443-1454 (1974).
[CrossRef]

Takeda, M.

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M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, “Single-transform Fourier--Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z.Fuzessy, W.Juptner, and W.Osten, eds. (Akademie Verlag, 1996), pp. 67-73.

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[CrossRef]

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F. J. Weinberg, Optics of Flame (Butterworth, 1963).

C. Shakher and R. S. Sirohi, “Flame temperature measurement using speckle techniques,” in New Directions in Holography and Speckle, H. Caulfield and C. S. Vikram, eds. (American Scientific, 2007), pp. 1-16.

C. Shakher, M. M. Hossain, D. S. Mehta, and G. Sheoran, “Measurement of temperature field in steady laminar free convection flow using digital holography,” presented at the LM2008-9th International Symposium on Laser Metrology, Singapore, 30 June-2 July 2008.

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M. Takeda, K. Taniguchi, T. Hirayama, and H. Kohgo, “Single-transform Fourier--Hartley fringe analysis for holographic interferometry,” in Simulation and Experiment in Laser Metrology, Z.Fuzessy, W.Juptner, and W.Osten, eds. (Akademie Verlag, 1996), pp. 67-73.

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Figures (9)

Fig. 1
Fig. 1

Schematic of the lensless Fourier transform digital holography setup used for temperature measurement.

Fig. 2
Fig. 2

Flow chart of the algorithm used for measurement of temperature in laminar free convection flow using digital holographic interferometry.

Fig. 3
Fig. 3

Schematic of the arrangement with the heater shown inside the test section.

Fig. 4
Fig. 4

(a) Interference phase map obtained in a state approaching the steady laminar free convection and (b) interference phase map obtained in the steady laminar free convection flow.

Fig. 5
Fig. 5

Two-dimensional unwrapped phase map of Fig. 4b.

Fig. 6
Fig. 6

One-dimensional interference phase drawn along the line AB of Fig. 4b.

Fig. 7
Fig. 7

One-dimensional unwrapped phase map obtained from Fig. 5 along the line CD.

Fig. 8
Fig. 8

Variation of temperature with distance from plane of the heater. Temperature is calculated on ten points marked in the figure.

Fig. 9
Fig. 9

(a), (b) Interference phase map showing the turbulent free convection flow fields at two different times.

Equations (18)

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I ( X H , Y H ) = R ( X H , Y H ) R * ( X H , Y H ) + O ( X H , Y H ) O ( X H , Y H ) * + R ( X H , Y H ) * O ( X H , Y H ) + R ( X H , Y H ) O ( X H , Y H ) * ,
I ( r Δ X H , s Δ Y H ) = | R ( r Δ X H , s Δ Y H ) | 2 + | O ( r Δ X H , s Δ Y H ) | 2 + R ( r Δ X H , s Δ Y H ) * O ( r Δ X H , s Δ Y H ) + R ( r Δ X H , s Δ Y H ) O ( r Δ X H , s Δ Y H ) * ,
O ( X I , Y I ) = exp ( i k d ) i λ d exp [ i k 2 d ( X I 2 + Y I 2 ) ] × ( X H , Y H ) I ( X H , Y H ) R ( X H , Y H ) exp [ k 2 d ( X H 2 + Y 2 H ) ] exp [ i 2 π λ d ( X I X H + Y I Y H ) ] d X H d Y H ,
O ( X I , Y I ) = exp ( i k d ) i λ d exp [ i ( k 2 d ) ( X I 2 + Y I 2 ) ] F λ d 1 { I ( X H , Y H ) R ( X H , Y H ) exp [ i ( k 2 d ) ( X H 2 + Y H 2 ) ] } ,
R ( X H , Y H ) = ( const ) exp [ i ( k 2 d ) ( X H 2 + Y H 2 ) ] .
O ( X I , Y I ) = ( const ) exp ( i k d ) i λ d exp [ i ( k 2 d ) ( X I 2 + Y I 2 ) ] F 1 λ d { I ( X H , Y H ) } .
O ( m , n ) = const exp ( i k d ) i λ d exp [ i π λ d ( m 2 N 2 Δ X H 2 + n 2 N 2 Δ Y H 2 ) ] × IDFT { I ( r Δ X H , s Δ Y H ) } ,
Δ X I = λ d / ( N Δ X H ) , Δ Y I = λ d / ( N Δ Y H ) .
O ( m , n ) = const exp ( i k d ) i λ d exp [ i π λ d ( m 2 N 2 Δ X H 2 + n 2 N 2 Δ Y H 2 ) ] × IFFT { I ( r Δ X H , s Δ Y H ) } .
I ( m , n ) = | O ( m , n ) | 2 = Re 2 | O ( m , n ) | + Im 2 | O ( m , n ) | ,
ϕ ( m , n ) = arctan Im [ O ( m , n ) ] Re [ O ( m , n ) ] ,
Δ Opt ( x , y ) = [ n ( x , y ) n c ] ,
Δ Opt ( y ) = [ n ( y ) n c ] ,
Δ Φ ( y ) = 2 π λ Δ Opt ( y ) = 2 π λ [ n ( y ) n c ] .
n ( y ) = 1.33711 9.3784 × 10 6 T ( y ) 2.1726 × 10 6 T ( y ) 2 .
A T ( y ) 2 + B T ( y ) + C = 0 ,
C = [ λ Δ Φ 2 π A T c 2 B T c ] ,
Δ Φ ( m , n ) = { Φ 2 ( m , n ) Φ 1 ( m , n ) + 2 π if     Φ 2 ( m , n ) Φ 1 ( m , n ) < π Φ 2 ( m , n ) Φ 1 ( m , n ) 2 π if     Φ 2 ( m , n ) Φ 1 ( m , n ) + π Φ 2 ( m , n ) Φ 1 ( m , n ) else .

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