Abstract

We demonstrate a new approach to the transport of intensity equation (TIE) phase retrieval method which uses structured illumination to improve low-frequency noise performance. In the hybrid scheme, two phase images are acquired: one with uniform and one with sinusoidal grating illumination intensity. The former preserves the high spatial frequency features of the phase best, whereas the latter dramatically increase the response at low spatial frequencies (where traditional TIE notoriously suffers). We then theoretically prove the design of a spectral filter that optimally combines the two phase results while suppressing noise. The combination of uniformly and structured illuminated TIE (hybrid TIE) phase imaging is experimentally demonstrated optically with a calibrated pure phase object.

© 2014 Optical Society of America

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2014 (3)

2013 (3)

2012 (2)

2010 (2)

J. Mertz and J. Kim, “Scanning light-sheet microscopy in the whole mouse brain with HiLo background rejection,” J. Biomed Opt. 15(1), 016027 (2010).
[Crossref] [PubMed]

L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
[Crossref] [PubMed]

2009 (1)

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (1)

P. Bach, J. Jett, U. Pastorino, M. Tockman, S. Swensen, and C. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297(9), 953–961 (2007).
[Crossref] [PubMed]

2006 (2)

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

2005 (3)

2004 (5)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).
[Crossref]

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573 (2004).
[Crossref] [PubMed]

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase–amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

L. Turner, B. Dhal, J. Hayes, A. Mancuso, K. Nugent, D. Paterson, R. Scholten, C. Tran, and A. Peele, “X-ray phase imaging: Demonstration of extended conditions for homogeneous objects,” Opt. Express 12(13), 2960–2965 (2004).
[Crossref] [PubMed]

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214(1), 7–12 (2004).
[Crossref] [PubMed]

2002 (1)

E. D. Barone-Nugent, A. N. T. O. N. Barty, and K. A. Nugent, “Quantitative phase–amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref]

2001 (1)

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

1998 (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586 (1998).
[Crossref]

1995 (1)

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. 12(9), 1942–1946 (1995).
[Crossref]

1990 (1)

1988 (1)

1984 (1)

1983 (1)

1978 (1)

1971 (1)

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34(3), 275–284 (1971).

1969 (1)

R. K. Crane, “Interference phase measurement,” Appl. Opt. 8, 538 (1969).

Agour, M.

Almoro, P. F.

Arnison, M. R.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214(1), 7–12 (2004).
[Crossref] [PubMed]

Asundi, A.

Bach, P.

P. Bach, J. Jett, U. Pastorino, M. Tockman, S. Swensen, and C. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297(9), 953–961 (2007).
[Crossref] [PubMed]

Barbastathis, G.

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. N. T. O. N. Barty, and K. A. Nugent, “Quantitative phase–amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref]

Barty, A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase–amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

Barty, A. N. T. O. N.

E. D. Barone-Nugent, A. N. T. O. N. Barty, and K. A. Nugent, “Quantitative phase–amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref]

Begg, C.

P. Bach, J. Jett, U. Pastorino, M. Tockman, S. Swensen, and C. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297(9), 953–961 (2007).
[Crossref] [PubMed]

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

Chen, Q.

Chowdhury, S.

Claus, R. A.

Cloetens, P.

Cogswell, C. J.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214(1), 7–12 (2004).
[Crossref] [PubMed]

Cohen, M.

Crane, R. K.

R. K. Crane, “Interference phase measurement,” Appl. Opt. 8, 538 (1969).

Dauwels, J.

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

Dhal, B.

Diaz, A.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

Ettl, S.

Falldorf, C.

Faulkner, H. M.

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).
[Crossref]

Fienup, J. R.

Gao, P.

Gerchberg, R.

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34(3), 275–284 (1971).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

Guérineau, N.

Gureyev, T. E.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[Crossref]

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. 12(9), 1942–1946 (1995).
[Crossref]

Gustafsson, M. G.

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[Crossref] [PubMed]

Hanson, S. G.

Häusler, G.

Hayes, J.

Horstmeyer, R.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photon. 7(9), 739–745 (2013).
[Crossref]

Ichikawa, K.

Izatt, J.

Jett, J.

P. Bach, J. Jett, U. Pastorino, M. Tockman, S. Swensen, and C. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297(9), 953–961 (2007).
[Crossref] [PubMed]

Jingshan, Z.

Kaminski, J.

Kim, J.

J. Mertz and J. Kim, “Scanning light-sheet microscopy in the whole mouse brain with HiLo background rejection,” J. Biomed Opt. 15(1), 016027 (2010).
[Crossref] [PubMed]

Knauer, M. C.

Kwon, O. Y.

Larkin, K. G.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214(1), 7–12 (2004).
[Crossref] [PubMed]

Lewis, R.

R. Lewis, “Medical phase contrast x-ray imaging: current status and future prospects,” Phys. Med. Biol. 49(16), 3573 (2004).
[Crossref] [PubMed]

Lohmann, A. W.

Mancuso, A.

McMahon, P. J.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase–amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

Menzel, A.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

Mertz, J.

J. Mertz and J. Kim, “Scanning light-sheet microscopy in the whole mouse brain with HiLo background rejection,” J. Biomed Opt. 15(1), 016027 (2010).
[Crossref] [PubMed]

Nesterets, Y. I.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[Crossref]

Nugent, K.

Nugent, K. A.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase–amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

E. D. Barone-Nugent, A. N. T. O. N. Barty, and K. A. Nugent, “Quantitative phase–amplitude microscopy I: optical microscopy,” J. Microsc. 206(3), 194–203 (2002).
[Crossref]

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586 (1998).
[Crossref]

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. 12(9), 1942–1946 (1995).
[Crossref]

Osten, W.

Paganin, D.

D. Paganin, A. Barty, P. J. McMahon, and K. A. Nugent, “Quantitative phase–amplitude microscopy. III. The effects of noise,” J. Microsc. 214(1), 51–61 (2004).
[Crossref] [PubMed]

K. A. Nugent, D. Paganin, and T. E. Gureyev, “A phase odyssey,” Phys. Today 54(8), 27–32 (2001).
[Crossref]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586 (1998).
[Crossref]

Paganin, D. M.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[Crossref]

Pan, A.

Y. Zhu, A. Pan, and G. Barbastathis, “Low-noise TIE phase imaging by structured illumination,” in Computational Optical Sensing and Imaging, Optical Society of America (2014), CTh3C-5.
[Crossref]

Pastorino, U.

P. Bach, J. Jett, U. Pastorino, M. Tockman, S. Swensen, and C. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297(9), 953–961 (2007).
[Crossref] [PubMed]

Paterson, D.

Pedrini, G.

Peele, A.

Petruccelli, J. C.

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

Pogany, A.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[Crossref]

Primot, J.

Roberts, A.

T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. 12(9), 1942–1946 (1995).
[Crossref]

Roddier, F.

Rodenburg, J. M.

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).
[Crossref]

Saxton, W.

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34(3), 275–284 (1971).

Scholten, R.

Shanker, A.

A. Shanker, L. Tian, and L. Waller, “Defocus-based quantitative phase imaging by coded illumination,” Proc. SPIE 8949, 89490R (2014).

Sheppard, C. J.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214(1), 7–12 (2004).
[Crossref] [PubMed]

Smith, N. I.

M. R. Arnison, K. G. Larkin, C. J. Sheppard, N. I. Smith, and C. J. Cogswell, “Linear phase imaging using differential interference contrast microscopy,” J. Microsc. 214(1), 7–12 (2004).
[Crossref] [PubMed]

Stampanoni, M.

Swensen, S.

P. Bach, J. Jett, U. Pastorino, M. Tockman, S. Swensen, and C. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297(9), 953–961 (2007).
[Crossref] [PubMed]

Takeda, M.

Teague, M. R.

Thibault, P.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

Tian, L.

Tockman, M.

P. Bach, J. Jett, U. Pastorino, M. Tockman, S. Swensen, and C. Begg, “Computed tomography screening and lung cancer outcomes,” Jama 297(9), 953–961 (2007).
[Crossref] [PubMed]

Tran, C.

Turner, L.

Velghe, S.

Waller, L.

Wattellier, B.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2(4), 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

Wilkins, S. W.

T. E. Gureyev, Y. I. Nesterets, D. M. Paganin, A. Pogany, and S. W. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region. 2. Partially coherent illumination,” Opt. Commun. 259(2), 569–580 (2006).
[Crossref]

Yang, C.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photon. 7(9), 739–745 (2013).
[Crossref]

Yu, Y.

Zheng, G.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photon. 7(9), 739–745 (2013).
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Figures (7)

Fig. 1
Fig. 1

Conceptual setup of uniformly (a) and structure-illuminated (b) TIE.

Fig. 2
Fig. 2

Transfer function (a) and inverse transfer function (b) of uniformly–illuminated TIE (solid line) and structure–illuminated TIE (dashed line). The singular point at spatial frequency u = 0 and low values near low spatial frequency results in noise amplification in this region. The figure also shows increased response and suppressed low–frequency noise amplification at u < fc (see Eq. (18)), when comparing the structure-illuminated TIE to the uniformly–illuminated TIE.

Fig. 3
Fig. 3

Procedures of phase differential measurement using structure–illuminated TIE.

Fig. 4
Fig. 4

Power spectral density errors of phase retrieval via uniform illuminated TIE (a), sinusoidal-illuminated TIE along one direction (b), sinusoidal-illuminated TIE with two orthogonal directions (c),and the optimized hybrid TIE (d) in logarithmic scale.

Fig. 5
Fig. 5

Simulation results of phase retrieval using various TIE methods. (a) original phase profile, (b)–(e) phase recovered using uniformly-illuminated TIE ϕ̂1, structure-illuminated TIE via using direct integral ϕ̂2 and via Poisson equation construction ϕ̂3, and hybrid TIE reconstruction ϕ̂h, (h)–(i) corresponding calculated phase PSD error, plotted in log scale.

Fig. 6
Fig. 6

Experimental setup for structure-illuminated TIE. The object (a lens) is placed in approximately the same plane as a sinusoidal amplitude grating for obtaining the illumination structure.

Fig. 7
Fig. 7

Phase retrieval of (a) (failed) uniformly-, (b) structure-illuminated TIE, and (c) the hybrid reconstruction of a lens. (d) shows an extracted line at y = 2.112 mm. Note that an air bubble is visible in the direct TIE and hybrid reconstruction, but not captured in the structured illuminated case. The bubble area is labeled with a small black box in (b) and (c).

Equations (38)

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k I ( x , y ; z ) z | z = 0 = ( I ( x , y ; 0 ) ϕ ( x , y ) ) ,
1 k I 0 2 ϕ = I z I ( d ) I 0 d Δ ( d ) d ,
ϕ ( x , y ) = 1 [ 1 H ( u , v ) ( Δ ( d ) I 0 ) ] ,
H ( u , v ) = 4 π 2 d k ( u 2 + v 2 )
I m , X ( x , y , 0 ) = a I 0 ( 1 + η sin ( 2 π f m x ) ) ,
k I m ( x , y ; z ) z | z = 0 k I m ( d ) I m ( 0 ) d k Δ m ( d ) d = = ( I m ( 0 ) ϕ ( x , y ) ) = I m ϕ ( x , y ) + I m 2 ϕ ( x , y ) ,
I m ϕ = k d ( Δ m ( d ) I m I 0 Δ ( d ) ) k d s m ( d ) .
( I m ) ( ϕ ( x , y ) ) = k d ( s m ( d ) ) .
( ϕ x ) ( u , v ) π f m a η I 0 i ( δ ( u f m ) δ ( u + f m ) ) = = π f m a η I 0 i ( ( ϕ x ) ( u + f m , v ) ( ϕ x ) ( u f m , v ) ) = k d ( s m , X ( d ) ) ( u , v ) ,
( ϕ x ) ( u , v ) ( ϕ x ) ( u + 2 f m , v ) = i k d 1 π f m a η I 0 ( s m , X ( d ) ) ( u + f m , v ) .
ϕ x = 1 [ i k d 1 π f m a η I 0 ( s m , X ( d ) ) ( u + f m , v ) T x ( u , v ) ] .
ϕ = 1 [ H m 1 ( u , v ) ( s m , X ( d ) I 0 ) ( u + f m , v ) T x ( u , v ) ] ,
H m ( u , v ) = 2 π 2 a η f m d k u
I q , X ( x , y , 0 ) = a I 0 ( 1 + η cos ( 2 π f m x ) ) .
( ϕ x ) ( u , v ) + ( ϕ x ) ( u + 2 f m , v ) = k π f m d a I 0 η ( s q , X ( d ) ) ( u + f m , v )
ϕ x = 1 [ k d 1 2 π f m a η I 0 ( ( i s m , X ( d ) s q , X ( d ) ) ( u + f m , v ) ) ] .
H m ( u , v ) = 4 π 2 a η f m d k u
f c = f m / a η .
ϕ y = 1 [ k d 1 2 π f m a η I 0 ( ( i s m , Y ( d ) s q , Y ( d ) ) ( u , v + f m ) ) ] ,
I m , Y ( x , y , 0 ) = a I 0 ( 1 + η sin ( 2 π f m y ) )
I q , Y ( x , y , 0 ) = a I 0 ( 1 + η cos ( 2 π f m y ) )
2 ϕ = ϕ x x + ϕ y y ,
ϕ = 1 [ 1 4 π 2 ( u 2 + v 2 ) ( ϕ x x + ϕ y y ) ] .
Δ ^ ( d ) Δ ( d ) + n ( x , y ) ,
S n ( u , v ) = σ 2 ,
S ϕ ^ 1 ( u , v ) S ϕ ( u , v ) E ϕ ^ 1 ( u , v ) = | H ( u , v ) | 2 S n ( u , v ) ,
E ϕ ^ 1 ( u , v ) = [ k d 1 4 π 2 ( u 2 + v 2 ) I 0 ] 2 σ 2 ,
E ϕ ^ x ( u , v ) = ( k d 1 4 π f m a η I 0 ) 2 4 σ 2 = ( k d 1 2 π f m a η I 0 ) 2 σ 2 ,
E ϕ ^ 2 ( u , v ) = ( 1 2 π u ) 2 E ϕ ^ x .
E ϕ ^ 3 ( u , v ) = [ 1 4 π 2 ( u 2 + v 2 ) ] 2 [ ( 2 π u ) 2 E ϕ ^ x + ( 2 π v ) 2 E ϕ ^ y ] = 1 4 π 2 ( u 2 + v 2 ) E ϕ ^ x .
E ϕ ^ 2 ( u , v ) = 1 u 2 ( k d 1 4 π 2 f m a η I 0 ) 2 σ 2
E ϕ ^ 3 ( u , v ) = 1 u 2 + v 2 ( k d 1 4 π 2 f m a η I 0 ) 2 σ 2 .
ϕ ^ h ( x , y ) = ϕ ^ 1 ( x , y ) t ( x , y ) + ϕ ^ 3 ( x , y ) ( 1 t ( x , y ) ) ,
S ϕ ^ h ( u , v ) S ϕ ( u , v ) E ϕ ^ h ( u , v ) = | T ( u , v ) | 2 E ϕ ^ 1 + | 1 T ( u , v ) | 2 E ϕ ^ 3 .
P n = [ | T ( u , v ) | 2 E ϕ ^ 1 + | 1 T ( u , v ) | 2 E ϕ ^ 3 ] d u d v ,
δ P n = 2 ( T E ϕ ^ 1 + ( T 1 ) E ϕ ^ 3 ) δ T d u d v = 0 ,
T ( u , v ) = E ϕ ^ 3 ( u , v ) ( u , v ) E ϕ ^ 1 + E ϕ ^ 3 ( u , v ) = u 2 + v 2 ( f m a η ) 2 + ( u 2 + v 2 ) .
E ϕ ^ h ( u , v ) = E ϕ ^ 1 ( u , v ) E ϕ ^ 3 ( u , v ) E ϕ ^ 1 ( u , v ) + E ϕ ^ 3 ( u , v ) .

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