Abstract

A novel non-iterative phase retrieval method is proposed and demonstrated with a proof-of-principle experiment. The method uses a fixed specially designed mask and through-focus intensity measurements. It is demonstrated that this method is robust to spatial partial coherence in the illumination, making it suitable for coherent diffractive imaging using spatially partially coherent light, as well as for coherence characterization.

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

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References

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  1. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3(1), 27 (1978).
    [Crossref] [PubMed]
  2. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758 (1982).
    [Crossref] [PubMed]
  3. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795 (2004).
    [Crossref]
  4. J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
    [Crossref]
  5. P. Gao, B. Yao, I. Harder, N. Lindlein, and F. J. Torcal-Milla, “Phase-shifting Zernike phase contrast microscopy for quantitative phase measurement,” Opt. Lett. 36(21), 4305 (2011).
    [Crossref] [PubMed]
  6. S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
    [Crossref] [PubMed]
  7. A.P. Konijnenberg, W.M.J. Coene, and H.P. Urbach, “Non-iterative phase retrieval by phase modulation through a single parameter,” Ultramicroscopy 174, 70–78 (2017).
    [Crossref] [PubMed]
  8. M. op de Beeck, D. van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64(1–4), 167–183 (1996).
    [Crossref]
  9. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434 (1983).
    [Crossref]
  10. N. Nakajima, “Phase retrieval from Fresnel zone intensity measurements by use of Gaussian filtering,” Appl. Opt. 37(26), 6219 (1998).
    [Crossref]
  11. E. Wolf, “New theory of partial coherence in the space–frequency domain part i: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. A,  72(3), 343 (1982).
    [Crossref]
  12. N. Burdet, X. Shi, D. Parks, J. N. Clark, X. Huang, S. D. Kevan, and I. K. Robinson, “Evaluation of partial coherence correction in x-ray ptychography,” Opt. Express 23(5), 5452 (2015).
    [Crossref] [PubMed]
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  14. C. Q. Tran, D. Paterson, I. McNulty, A. G. Peele, A. Roberts, and K. A. Nugent, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A,  22(8), 1691 (2005).
    [Crossref]
  15. F. Wang, X. Liu, Y. Yuan, and Y. Cai, “Experimental generation of partially coherent beams with different complex degrees of coherence,” Opt. Lett. 38(11), 1814–1816 (2013).
    [Crossref] [PubMed]
  16. 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]
  17. Y. Chen, L. Liu, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
    [Crossref] [PubMed]

2017 (1)

A.P. Konijnenberg, W.M.J. Coene, and H.P. Urbach, “Non-iterative phase retrieval by phase modulation through a single parameter,” Ultramicroscopy 174, 70–78 (2017).
[Crossref] [PubMed]

2016 (1)

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

2015 (1)

2014 (1)

2013 (1)

2011 (1)

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]

2005 (1)

2004 (2)

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

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

1998 (1)

1996 (1)

M. op de Beeck, D. van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64(1–4), 167–183 (1996).
[Crossref]

1983 (1)

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434 (1983).
[Crossref]

1982 (2)

E. Wolf, “New theory of partial coherence in the space–frequency domain part i: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. A,  72(3), 343 (1982).
[Crossref]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758 (1982).
[Crossref] [PubMed]

1978 (1)

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]

Burdet, N.

Cai, Y.

Chen, Y.

Clark, J. N.

Coene, W.

M. op de Beeck, D. van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64(1–4), 167–183 (1996).
[Crossref]

Coene, W.M.J.

A.P. Konijnenberg, W.M.J. Coene, and H.P. Urbach, “Non-iterative phase retrieval by phase modulation through a single parameter,” Ultramicroscopy 174, 70–78 (2017).
[Crossref] [PubMed]

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]

Eberhardt, W.

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Eisebitt, S.

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Faulkner, H. M. L.

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

Fienup, J. R.

Gao, P.

Harder, I.

Hellwig, O.

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Huang, X.

Kevan, S. D.

Konijnenberg, A.P.

A.P. Konijnenberg, W.M.J. Coene, and H.P. Urbach, “Non-iterative phase retrieval by phase modulation through a single parameter,” Ultramicroscopy 174, 70–78 (2017).
[Crossref] [PubMed]

Y. Shao, X. Lu, A.P. Konijnenberg, C. Zhao, H.P. Urbach, and Y. Cai, “Spatial coherence measurement and diffractive imaging using partially spatially coherent illumination,” arXiv:1706.07203 [physics.optics], (2017).

Lindlein, N.

Liu, L.

Liu, X.

Lörgen, M

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Lu, X.

Y. Shao, X. Lu, A.P. Konijnenberg, C. Zhao, H.P. Urbach, and Y. Cai, “Spatial coherence measurement and diffractive imaging using partially spatially coherent illumination,” arXiv:1706.07203 [physics.optics], (2017).

Lüning, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

McNulty, I.

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]

Nakajima, N.

Nugent, K. A.

op de Beeck, M.

M. op de Beeck, D. van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64(1–4), 167–183 (1996).
[Crossref]

Parks, D.

Paterson, D.

Peele, A. G.

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]

Roberts, A.

Robinson, I. K.

Rodenburg, J. M.

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

Schlotter, W. F.

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Shao, Y.

Y. Shao, X. Lu, A.P. Konijnenberg, C. Zhao, H.P. Urbach, and Y. Cai, “Spatial coherence measurement and diffractive imaging using partially spatially coherent illumination,” arXiv:1706.07203 [physics.optics], (2017).

Shi, X.

Stöhr, J.

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Teague, M. R.

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434 (1983).
[Crossref]

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.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Torcal-Milla, F. J.

Tran, C. Q.

Urbach, H.P.

A.P. Konijnenberg, W.M.J. Coene, and H.P. Urbach, “Non-iterative phase retrieval by phase modulation through a single parameter,” Ultramicroscopy 174, 70–78 (2017).
[Crossref] [PubMed]

Y. Shao, X. Lu, A.P. Konijnenberg, C. Zhao, H.P. Urbach, and Y. Cai, “Spatial coherence measurement and diffractive imaging using partially spatially coherent illumination,” arXiv:1706.07203 [physics.optics], (2017).

van Dyck, D.

M. op de Beeck, D. van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64(1–4), 167–183 (1996).
[Crossref]

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. Imaging 2(3), 310–322 (2016).
[Crossref]

Waller, L.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Wang, F.

Wolf, E.

E. Wolf, “New theory of partial coherence in the space–frequency domain part i: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. A,  72(3), 343 (1982).
[Crossref]

Yao, B.

Yuan, Y.

Zhao, C.

Y. Chen, L. Liu, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014).
[Crossref] [PubMed]

Y. Shao, X. Lu, A.P. Konijnenberg, C. Zhao, H.P. Urbach, and Y. Cai, “Spatial coherence measurement and diffractive imaging using partially spatially coherent illumination,” arXiv:1706.07203 [physics.optics], (2017).

Zhong, J.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

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

IEEE Trans. Comput. Imaging (1)

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

J. Opt. Soc. Am. A (3)

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434 (1983).
[Crossref]

E. Wolf, “New theory of partial coherence in the space–frequency domain part i: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. A,  72(3), 343 (1982).
[Crossref]

C. Q. Tran, D. Paterson, I. McNulty, A. G. Peele, A. Roberts, and K. A. Nugent, “X-ray imaging: a generalized approach using phase-space tomography,” J. Opt. Soc. Am. A,  22(8), 1691 (2005).
[Crossref]

Nature (1)

S. Eisebitt, J. Lüning, W. F. Schlotter, M Lörgen, O. Hellwig, W. Eberhardt, and J. Stöhr, “Lensless imaging of magnetic nanostructures by X-ray spectro-holography,” Nature 432(7019), 885–888 (2004).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Ultramicroscopy (3)

A.P. Konijnenberg, W.M.J. Coene, and H.P. Urbach, “Non-iterative phase retrieval by phase modulation through a single parameter,” Ultramicroscopy 174, 70–78 (2017).
[Crossref] [PubMed]

M. op de Beeck, D. van Dyck, and W. Coene, “Wave function reconstruction in HRTEM: the parabola method,” Ultramicroscopy 64(1–4), 167–183 (1996).
[Crossref]

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]

Other (1)

Y. Shao, X. Lu, A.P. Konijnenberg, C. Zhao, H.P. Urbach, and Y. Cai, “Spatial coherence measurement and diffractive imaging using partially spatially coherent illumination,” arXiv:1706.07203 [physics.optics], (2017).

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Figures (8)

Fig. 1
Fig. 1 Demonstration that a star-shaped mask is suitable for non-iterative focus-variation reconstruction. If P is chosen to be one of the star’s points, then we see that for any x for which O(x + P) is non-zero, all y for which Py is perpendicular to x intersect the object only in P.
Fig. 2
Fig. 2 Plots showing the contribution of each y to the integral, superimposed on the mask of the object. The yellow arrow indicates P, and the red arrow indicates a certain choice of x. The size of the object mask and the values for A are the same as the ones used in the experiment (see Section 3). For larger |x|, the line consisting of the y that contribute to the integral becomes narrower. By applying a Hamming window function, the effects of the sidelobes are reduced significantly, at the expense of broadening the central lobe.
Fig. 3
Fig. 3 Illustration of the experimental setup. RGGD=Rotating Ground Glass Disk, GAF=Gaussian Amplitude Filter, BS=Beam Splitter, SLM=Spatial Light Modulator.
Fig. 4
Fig. 4 The image that is assigned to the SLM, which serves as the object that is to be reconstructed. The grayscale values denote the phase shift in radians.
Fig. 5
Fig. 5 Non-iterative reconstructions for different degrees of coherence. The simulated amplitudes are obtained by multiplying the shifted object amplitude with a Gaussian function e−|x|2/2σ2. Because the amplitude of the non-iterative reconstruction blows up for small |x|, their colorbars have been truncated. To ensure a fair comparison, the cut-off value for each plot is determined by its value at the position of the red dot as indicated in the top image.
Fig. 6
Fig. 6 Non-iterative reconstructions for low coherence for various choices of the reference point P. It is illustrated how the coherence width affects the field of view, and how using multiple reference points P can help in creating a more complete picture of the object.
Fig. 7
Fig. 7 Demonstration of how reconstructions for different reference points Pj can be synthesized into a reconstruction of O(x) with an extended field of view. In the top two rows are the amplitude and phase of the non-iterative reconstructions ψj(x) for four different Pj that correspond to the four different protrusions of the star-shaped mask as shown in Fig. 6. In the bottom row are the amplitude and phase of the reconstructed object O(x) and coherence function J0(x). Note that the coherence function is reconstructed only in the regions that are covered by O(x + Pj).
Fig. 8
Fig. 8 Top row: reconstructions for different reference points P. The lines indicate the cross-sections that are used to find σ, which are plotted in the bottom left figure, in which also the cross-section for the fitted function is plotted, which is shown in its entirety in the bottom right figure.

Equations (34)

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

I ( u , A ) = J ( x 1 , x 2 ) e 2 π i A ( | x 1 | 2 | x 2 | 2 ) e 2 π i ( x 1 x 2 ) u d x 1 d x 2 ,
A = z f 2 λ f 2 ,
u = ( x f λ f , y f λ f ) .
u , A 1 { I } ( x , ω A ) = J ( x 1 , x 2 ) δ ( ω A ( | x 1 | 2 | x 2 | 2 ) ) δ ( x ( x 1 x 2 ) ) d x 1 d x 2 = J ( x + x 2 , x 2 ) δ ( ω A ( | x + x 2 | 2 | x 2 | 2 ) ) d x 2 .
u , A 1 { I } ( x , ω A ) = J ( x + y , y ) δ ( ω A | x | 2 2 x y ) d y .
u , A 1 { I } ( x , | x | 2 + 2 x P ) = J ( x + y , y ) δ ( 2 x ( P y ) ) d y .
u , A 1 { I ( u , A ) } ( x , | x | 2 + 2 x P ) J ( x + P , P ) .
J ( x + P , P ) A u 1 { I ( u , A ) } ( x , A ) e 2 π i A ( | x | 2 + 2 x P ) ,
J ( x + P , P ) A u 1 { I ( u , A ) } ( x , A ) H ( A ) e 2 π i A ( | x | 2 + 2 x P ) .
A 1 { H ( A ) e 2 π i A ( | x + y | 2 | y | 2 ) } ( | x | 2 + 2 x P ) = W sinc ( π W ( 2 x ( P y ) ) )
A = 1 2 λ z ,
u = ( x λ z , y λ z ) ,
I ( u , A ) = ( J ( x 1 , x 2 ) * e 2 π i ( A ) ( | x 1 | 2 | x 2 | 2 ) e 2 π i ( x 1 x 2 ) ( u ) d x 1 d x 2 ) * = ( J ( x 1 , x 2 ) e 2 π i ( A ) ( | x 1 | 2 | x 2 | 2 ) e 2 π i ( x 1 x 2 ) ( u ) d x 1 d x 2 ) * = I ( u , A ) * = I ( u , A ) .
J ( x + P , P ) = O ( x + P ) O ( P ) * e | ( x + P ) P | 2 / 2 σ 2 O ( x + P ) e | x | 2 / 2 σ 2
ψ j ( x ) = O ( x ) J 0 ( x P j ) .
π O [ J 0 ( x ) ] Ψ = j ψ j ( x ) J 0 ( x P j ) * j | J 0 ( x P j ) | 2 + , π J [ O ( x ) ] Ψ = j ψ j ( x + P j ) O ( x + P j ) * j | O ( x + P j ) | 2 + ,
O ( k + 1 ) ( x ) = ( 1 μ ) O ( k ) ( x ) + μ π O [ J 0 ( k ) ( x ) ] Ψ , J 0 ( k + 1 ) ( x ) = ( 1 μ ) J 0 ( k ) ( x ) + μ π J [ O ( k + 1 ) ( x ) ] Ψ .
J 0 ( x ) = e | x | 2 / 2 σ 2 L n 0 ( | x | 2 / 2 σ 2 ) ,
U ^ ( x f , y f ) = { U ( x , y ) } ( x f λ f , y f λ f ) ,
{ U ^ ( x f , y f ) } ( f x , f y ) = U ( x , y ) e 2 π i ( x x f λ f + y y f λ f ) e 2 π i ( x f f x + y f f y ) d x d y d x f d y f = ( λ f ) 2 U ( λ f f x λ f f y ) .
e i z k 2 k x 2 k y 2 e i z f k e i z f k x 2 + k y 2 2 k , = e i z f k e i z f ( 2 π ) 2 2 k ( f x 2 + f y 2 ) = e i z f k e i z f π λ ( f x 2 + f y 2 ) .
U ^ ( x f , y f , z f ) e i z f k ( λ f ) 2 U ( λ f f x , λ f f y ) e i z f π λ ( f x 2 + f y 2 ) e 2 π i ( f x x f + f y y f ) d f x d f y = e i z f k U ( x , y ) e i z f π λ f 2 ( x 2 + y 2 ) e 2 π i ( x x f λ f + y y f λ f ) d x d y = { U ( x , y ) e i z f π λ f 2 ( x 2 + y 2 ) } ( x f λ f , y f λ f ) .
U ^ ( x f , y f , z f ) R 2 { U ( X , Y ) e i z f π λ ( R f ) 2 ( X 2 + Y 2 ) } ( x f λ f , y f λ f ) .
u , A 1 { I } ( x , ω A ) = J ( x + y , y ) δ ( ω A | x | 2 2 x y ) d y = J ( x + y , y ) δ ( x v ) d y ,
v = μ x 2 y μ = ω A | x | 2 1 .
x v = 0 ,
v = λ [ x 2 x 1 ]
v = μ [ x 1 x 2 ] 2 [ P 1 P 2 ] .
[ x 1 x 2 x 2 x 1 ] [ μ λ ] = 2 [ P 1 P 2 ] .
μ = 2 P x | x | 2 ,
ω A = | x | 2 + 2 P x .
v = ( μ + 2 ) [ x 1 x 2 ] 2 [ P 1 P 2 ] .
μ = 2 P x | x | 2 2 ,
ω A = | x | 2 + 2 P x .

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