Abstract

Differential Phase Contrast (DPC) microscopy is a practical method for recovering quantitative phase from intensity images captured with different source patterns in an LED array microscope. Being a partially coherent imaging method, DPC does not suffer from speckle artifacts and achieves 2× better resolution than coherent methods. Like all imaging systems, however, DPC is susceptible to aberrations. Here, we propose a method of algorithmic self-calibration for DPC where we simultaneously recover the complex-field of the sample and the spatially-variant aberrations of the system, using 4 images with different illumination source patterns. The resulting phase reconstructions are digitally aberration-corrected.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (2)

Z. F. Phillips, M. Chen, and L. Waller, “Single-shot quantitative phase microscopy with color-multiplexed differential phase contrast (cDPC),” PLOS ONE 12, e0171228 (2017).
[Crossref] [PubMed]

G. Satat, M. Tancik, O. Gupta, B. Heshmat, and R. Raskar, “Object classification through scattering media with deep learning on time resolved measurement,” Opt. Express 25, 17466–17479 (2017).
[Crossref] [PubMed]

2016 (5)

2015 (6)

2014 (6)

2013 (3)

2012 (2)

2011 (2)

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19, 1016–1026 (2011).
[Crossref] [PubMed]

S. Boyd, N. Parikh, B. P. E Chu, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

2009 (2)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

S. B. Mehta and C. J. Sheppard, “Quantitative phase-gradient imaging at high resolution with asymmetric illumination-based differential phase contrast,” Opt. Lett. 34, 1924–1926 (2009).
[Crossref] [PubMed]

2008 (1)

2006 (1)

1994 (1)

1989 (1)

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[Crossref]

1985 (1)

B. Kachar, “Asymmetric illumination contrast: a method of image formation for video light microscopy,” Science 227, 766–768 (1985).
[Crossref] [PubMed]

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345 (1955).
[Crossref] [PubMed]

Alieva, J. A. R. T.

Bian, L.

Bian, Z.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

Boyd, S.

S. Boyd, N. Parikh, B. P. E Chu, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Chen, F.

Chen, M.

Chu, B. P. E

S. Boyd, N. Parikh, B. P. E Chu, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Chung, J.

Claus, R. A.

Dai, Q.

Dauwels, J.

Ding, H.

Dong, J.

Dong, S.

Eckstein, J.

S. Boyd, N. Parikh, B. P. E Chu, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Fienup, J. R.

Garcia-Sucerquia, J.

Gillette, M. U.

Guizar-Sicairos, M.

Guo, K.

Gupta, O.

Heshmat, B.

Horstmeyer, R.

Humphry, M. J.

Jericho, M. H.

Jericho, S. K.

Kachar, B.

B. Kachar, “Asymmetric illumination contrast: a method of image formation for video light microscopy,” Science 227, 766–768 (1985).
[Crossref] [PubMed]

Kim, J.

Kim, M.K.

Klages, P.

Kreuzer, H. J.

Lang, W.

W. Lang, Nomarski Differential Interference-Contrast Microscopy (Carl Zeiss, 1982).

Li, X.

Liu, C.

Liu, D. C.

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[Crossref]

Liu, S.

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref] [PubMed]

Liu, Z.

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref] [PubMed]

Lu, H.

Mahajan, V. N.

Maiden, A. M.

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

McNulty, I.

Mehta, S. B.

Millet, L.

Mir, M.

Naulleau, P. P.

Neureuther, A. R.

Nocedal, J.

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[Crossref]

Ou, X.

Parikh, N.

S. Boyd, N. Parikh, B. P. E Chu, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Phillips, Z. F.

Z. F. Phillips, M. Chen, and L. Waller, “Single-shot quantitative phase microscopy with color-multiplexed differential phase contrast (cDPC),” PLOS ONE 12, e0171228 (2017).
[Crossref] [PubMed]

Popescu, G.

Ramchandran, K.

Raskar, R.

Rodenburg, J. M.

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Rogers, J.

Satat, G.

Sheppard, C. J.

Shpyrko, O. G.

Soltanolkotabi, M.

Suo, J.

Tancik, M.

Tang, G.

Tian, L.

M. Chen, L. Tian, and L. Waller, “3D differential phase contrast microscopy,” Biomed. Opt. Express 7, 3940–3950 (2016).
[Crossref] [PubMed]

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear Optimization Algorithm for Partially Coherent Phase Retrieval and Source Recovery,” IEEE Trans. Comput. Imag. 2, 310–322 (2016).
[Crossref]

J. Zhong, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase imaging with simultaneous source recovery,” Opt. Express 6, 257–265 (2015).
[Crossref]

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

L. Tian and L. Waller, “Quantitative differential phase contrast imaging in an LED array microscope,” Opt. Express 23, 11394–11403 (2015).
[Crossref] [PubMed]

L. Tian, X. Li, K. Ramchandran, and L. Waller, “Multiplexed coded illumination for Fourier ptychography with an LED array microscope,” Biomed. Opt. Express 5, 2376–2389 (2014).
[Crossref] [PubMed]

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref] [PubMed]

Tripathi, A.

Unarunotai, S.

Varma, P.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear Optimization Algorithm for Partially Coherent Phase Retrieval and Source Recovery,” IEEE Trans. Comput. Imag. 2, 310–322 (2016).
[Crossref]

Waller, L.

Z. F. Phillips, M. Chen, and L. Waller, “Single-shot quantitative phase microscopy with color-multiplexed differential phase contrast (cDPC),” PLOS ONE 12, e0171228 (2017).
[Crossref] [PubMed]

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear Optimization Algorithm for Partially Coherent Phase Retrieval and Source Recovery,” IEEE Trans. Comput. Imag. 2, 310–322 (2016).
[Crossref]

M. Chen, L. Tian, and L. Waller, “3D differential phase contrast microscopy,” Biomed. Opt. Express 7, 3940–3950 (2016).
[Crossref] [PubMed]

L. Tian and L. Waller, “Quantitative differential phase contrast imaging in an LED array microscope,” Opt. Express 23, 11394–11403 (2015).
[Crossref] [PubMed]

L. Tian, Z. Liu, L.-H. Yeh, M. Chen, J. Zhong, and L. Waller, “Computational illumination for high-speed in vitro Fourier ptychographic microscopy,” Optica 2, 904–911 (2015).
[Crossref]

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

R. A. Claus, P. P. Naulleau, A. R. Neureuther, and L. Waller, “Quantitative phase retrieval with arbitrary pupil and illumination,” Opt. Express 23, 26672–26682 (2015).
[Crossref] [PubMed]

J. Zhong, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase imaging with simultaneous source recovery,” Opt. Express 6, 257–265 (2015).
[Crossref]

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref] [PubMed]

L. Tian, X. Li, K. Ramchandran, and L. Waller, “Multiplexed coded illumination for Fourier ptychography with an LED array microscope,” Biomed. Opt. Express 5, 2376–2389 (2014).
[Crossref] [PubMed]

Wang, Z.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

Xu, W.

Yang, C.

J. Chung, J. Kim, X. Ou, R. Horstmeyer, and C. Yang, “Wide field-of-view fluorescence image deconvolution with aberration-estimation from Fourier ptychography,” Biomed. Opt. Express 7, 352–368 (2016).
[Crossref] [PubMed]

H. Lu, J. Chung, X. Ou, and C. Yang, “Quantitative phase imaging and complex field reconstruction by pupil modulation differential phase contrast,” Opt. Express 24, 25345–25361 (2016).
[Crossref] [PubMed]

L. Bian, G. Zheng, K. Guo, J. Suo, C. Yang, F. Chen, and Q. Dai, “Motion-corrected fourier ptychography,” Biomed. Opt. Express 7, 4543–4553 (2016).
[Crossref] [PubMed]

X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “High numerical aperture Fourier ptychography: principle, implementation and characterization,” Opt. Express 23, 3472–3491 (2015).
[Crossref] [PubMed]

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Opt. Express 22, 24062–24080 (2014).
[Crossref] [PubMed]

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

G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. express 21, 15131–15143 (2013).
[Crossref] [PubMed]

Yeh, L.-H.

Yu, X.

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345 (1955).
[Crossref] [PubMed]

Zheng, G.

Zhong, J.

Appl. Opt. (3)

Biomed. Opt. Express (4)

Foundations Trends Mach. Learn. (1)

S. Boyd, N. Parikh, B. P. E Chu, and J. Eckstein, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Foundations Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

IEEE Trans. Comput. Imag. (1)

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear Optimization Algorithm for Partially Coherent Phase Retrieval and Source Recovery,” IEEE Trans. Comput. Imag. 2, 310–322 (2016).
[Crossref]

J. Biomed. Opt. (1)

Z. Liu, L. Tian, S. Liu, and L. Waller, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (1)

Math. Program. (1)

D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[Crossref]

Nat. Photon. (1)

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

Opt. Express (14)

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19, 1016–1026 (2011).
[Crossref] [PubMed]

J. A. R. T. Alieva, “Rapid Quantitative Phase Imaging for Partially Coherent Light Microscopy,” Opt. Express 22, 13472–13483 (2014).
[Crossref] [PubMed]

J. Zhong, L. Tian, J. Dauwels, and L. Waller, “Partially coherent phase imaging with simultaneous source recovery,” Opt. Express 6, 257–265 (2015).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16, 7264–7278 (2008).
[Crossref] [PubMed]

L. Tian and L. Waller, “Quantitative differential phase contrast imaging in an LED array microscope,” Opt. Express 23, 11394–11403 (2015).
[Crossref] [PubMed]

R. A. Claus, P. P. Naulleau, A. R. Neureuther, and L. Waller, “Quantitative phase retrieval with arbitrary pupil and illumination,” Opt. Express 23, 26672–26682 (2015).
[Crossref] [PubMed]

X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “High numerical aperture Fourier ptychography: principle, implementation and characterization,” Opt. Express 23, 3472–3491 (2015).
[Crossref] [PubMed]

H. Lu, J. Chung, X. Ou, and C. Yang, “Quantitative phase imaging and complex field reconstruction by pupil modulation differential phase contrast,” Opt. Express 24, 25345–25361 (2016).
[Crossref] [PubMed]

A. Tripathi, I. McNulty, and O. G. Shpyrko, “Ptychographic overlap constraint errors and the limits of their numerical recovery using conjugate gradient descent methods,” Opt. Express 22, 1452–1466 (2014).
[Crossref] [PubMed]

Z. Bian, S. Dong, and G. Zheng, “Adaptive system correction for robust Fourier ptychographic imaging,” Opt. Express 21, 32400–32410 (2013).
[Crossref]

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

G. Satat, M. Tancik, O. Gupta, B. Heshmat, and R. Raskar, “Object classification through scattering media with deep learning on time resolved measurement,” Opt. Express 25, 17466–17479 (2017).
[Crossref] [PubMed]

G. Zheng, X. Ou, R. Horstmeyer, and C. Yang, “Characterization of spatially varying aberrations for wide field-of-view microscopy,” Opt. express 21, 15131–15143 (2013).
[Crossref] [PubMed]

R. Horstmeyer, X. Ou, J. Chung, G. Zheng, and C. Yang, “Overlapped Fourier coding for optical aberration removal,” Opt. Express 22, 24062–24080 (2014).
[Crossref] [PubMed]

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]

Opt. Lett. (1)

Optica (1)

PLOS ONE (1)

Z. F. Phillips, M. Chen, and L. Waller, “Single-shot quantitative phase microscopy with color-multiplexed differential phase contrast (cDPC),” PLOS ONE 12, e0171228 (2017).
[Crossref] [PubMed]

Science (2)

B. Kachar, “Asymmetric illumination contrast: a method of image formation for video light microscopy,” Science 227, 766–768 (1985).
[Crossref] [PubMed]

F. Zernike, “How I discovered phase contrast,” Science 121, 345 (1955).
[Crossref] [PubMed]

Ultramicroscopy (1)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109, 1256–1262 (2009).
[Crossref] [PubMed]

Other (2)

W. Lang, Nomarski Differential Interference-Contrast Microscopy (Carl Zeiss, 1982).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, 1999), 7th ed.
[Crossref]

Supplementary Material (2)

NameDescription
» Visualization 1       Validation of quantitative accuracy of Differential Phase Contrast (DPC) phase imaging method using 6 USAF 1951 phase target of heights ranging from 50nm to 300nm.
» Visualization 2       Quantitative phase reconstructions of a USAF 1951 phase resolution target at various defocus distances with and without aberration correction, and the corresponding recovered pupil aberrations.

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Figures (5)

Fig. 1
Fig. 1 Our LED array microscope captures 4 images with different illumination source patterns (three half-circles and one single LED). The intensity images are used to simultaneously reconstruct both amplitude and phase of the sample, and to estimate the pupil aberrations at each spatial location, which are then digitally corrected for. We show reconstructions for 4 regions with different spatially-varying aberrations.
Fig. 2
Fig. 2 Performance of joint phase and aberrations estimation on a simulated dataset. (a) Simulated FPM and DPC measurements. Red dashed circles indicate the NA of the objective lens. (b) Joint estimation of optical field and pupil aberrations, comparing ground truth, FPM and DPC measurements. (c) Errors for complex-field and aberrations at each iteration.
Fig. 3
Fig. 3 (a) Experimental FPM and DPC measurements for different LED source patterns. Zoomed regions at different orientations for coherent illumination are marked in cyan and pink boxes, respectively. (b) Quantitative phase of a star target using FPM and DPC, along with 1D cutlines for FPM (red) and DPC (blue) along the dashed lines. (c) Reconstructed wavefront error function and the weights of each Zernike mode up to the 4th radial degree.
Fig. 4
Fig. 4 (a) Quantitative phase reconstructions of a USAF 1951 resolution target at various defocus distances with and without aberration correction, and the corresponding recovered aberrations (see Visualization 2). (b) Zoomed-in reconstructions at defocus of 10µm. (c) Known and experimentally-estimated defocus values from the 4th Zernike mode over time.
Fig. 5
Fig. 5 (a) Reconstructed absorption, phase and spatially-varying aberrations (recovered pupil wavefronts for different regions of the field-of-view). (b) Comparison of results with and without pupil estimation for central and edge regions of the field-of-view.

Equations (13)

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I n = F 1 ( diag ( H μ ) F μ + i diag ( H ϕ ) F ϕ ) .
H μ = 1 I o [ F 1 diag ( O * ) F + F 1 diag ( F * P * ) F diag ( S ) ] P
H ϕ = 1 I o [ F 1 diag ( O * ) F F 1 diag ( F * P * ) F diag ( S ) ] P ,
P ( c ) = Circ ( λ u N A ) m = 0 M e i c m Z m ,
min μ , ϕ , c s = 1 N s F I n , s diag ( H μ , s ( c ) ) F μ i diag ( H ϕ , s ( c ) ) F ϕ 2 2 + τ R ( μ , ϕ ) ,
μ k + 1 , ϕ k + 1 = arg min μ , ϕ s = 1 N s F I n , s diag ( H μ , s ( c k ) ) F μ i diag ( H ϕ , s ( c k ) ) F ϕ 2 2 + τ R ( μ , ϕ ) .
c k + 1 = arg min c s = 1 N s F I n , s diag ( H μ , s ( c ) ) F μ k + 1 i diag ( H ϕ , s ( c ) ) F ϕ k + 1 2 2 .
c f = i I o Z T diag ( P * ) s = 1 N s [ F 1 diag ( O s ) F [ diag ( F * μ k + 1 * ) idiag ( F * ϕ k + 1 * ) ] + diag ( S s ) F 1 diag ( F P ) F [ diag ( F * μ k + 1 * ) + idiag ( F * ϕ k + 1 * ) ] ] ε s .
I ( r , u ) = S ( u ) | e Φ ( r , u ) h ( r , u ) | 2 .
I ( r , u ) = S ( u ) e Φ ( r 1 , u ) + Φ * ( r 2 , u ) h ( r r 1 , u ) h * ( r r 2 , u ) d 2 r 1 d 2 r 2 .
I ( r , u ) = S ( u ) ( 1 h ( r , u ) ) ( 1 h * ( r , u ) ) + S ( u ) [ ( Φ ( r , u ) h ( r , u ) ) ( 1 h * ( r , u ) ) + ( Φ * ( r , u ) h * ( r , u ) ) ( 1 h ( r , u ) ) ] .
I ˜ ( u ) = S ( u ) | P ( u ) | 2 d 2 u δ ( u ) + S ( u ) [ Φ ˜ ( u , u ) P ( u + u ) P * ( u + u ) δ ( u ) ] d 2 u + S ( u ) [ Φ ˜ * ( u , u ) P * ( u + u ) P ( u + u ) δ ( u ) ] d 2 u .
I ˜ n ( u ) = λ 2 | u | 2 I o S ( u ) [ P ( u + u ) P * ( u ) λ 2 | u + u | 2 + P ( u ) P * ( u + u ) λ 2 | u + u | 2 ] d 2 u μ ˜ ( u ) + i λ 2 | u | 2 I o S ( u ) [ P ( u + u ) P * ( u ) λ 2 | u + u | 2 P ( u ) P * ( u + u ) λ 2 | u + u | 2 ] d 2 u ϕ ˜ ( u ) .

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