Abstract

We present a fast and accurate method for wave propagation through a set of inclined reflecting planes. It is based on the coordinate transformation in reciprocal space leading to a diffraction integral, which can be calculated only by using two 2D Fast Fourier Transforms and one 2D interpolation. The method is numerically tested, and comparisons with standard methods show its superiority in both computational speed and accuracy. The direct application of this method is found in the X-ray phase contrast imaging using the Bragg magnifier—an optics consisting of crystals asymmetrically diffracting in Bragg geometry.

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

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X-ray Bragg magnifier microscope as a linear shift invariant imaging system: image formation and phase retrieval

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References

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  1. S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
    [Crossref]
  2. D. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
    [Crossref]
  3. S. Hrivňak, J. Uličný, L. Mikeš, A. Cecilia, E. Hamann, T. Baumbach, L. Švéda, Z. Zápražný, D. Korytár, E. Gimenez-Navarro, U. H. Wagner, C. Rau, H. Greven, and P. Vagovič, “Single-distance phase retrieval algorithm for Bragg magnifier microscope,” Opt. Express 24, 27753–27762 (2016).
    [Crossref]
  4. P. Vagovič, L. Švéda, A. Cecilia, E. Hamann, D. Pelliccia, E. N. Gimenez, D. Korytár, K. M. Pavlov, Z. Zápražný, M. Zuber, T. Koenig, M. Olbinado, W. Yashiro, A. Momose, M. Fiederle, and T. Baumbach, “X-ray Bragg magnifier microscope as a linear shift invariant imaging system: image formation and phase retrieval,” Opt. Express 22, 21508–21520 (2014).
    [Crossref]
  5. P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74, 054107 (2006).
    [Crossref]
  6. N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast fourier transform approach,” J. Opt. Soc. Am. A 15, 857–867 (1998).
    [Crossref]
  7. N. Delen and B. Hooker, “Verification and comparison of a fast fourier transform-based full diffraction method for tilted and offset planes,” Appl. Opt. 40, 3525–3531 (2001).
    [Crossref]
  8. P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
    [Crossref] [PubMed]
  9. P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
    [Crossref]
  10. S. Hrivňak, A. Hovan, J. Uličný, and P. Vagovič, “Phase retrieval for arbitrary Fresnel-like linear shift-invariant imaging systems suitable for tomography,” Biomed. Opt. Express 9, 4390–4400 (2018).
    [Crossref]
  11. A. Authier, Dynamical Theory of X-ray Diffraction (Oxford University, 2004).
  12. R. D. Spal, “Submicrometer resolution hard x-ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. 86, 3044–3047 (2001).
    [Crossref] [PubMed]
  13. X. Huang and M. Dudley, “A universal computation method for two-beam dynamical X-ray diffraction,” Acta Crystallogr. Sect. A 59, 163–167 (2003).
    [Crossref]

2018 (1)

2016 (1)

2014 (1)

2013 (1)

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

2008 (1)

2006 (1)

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74, 054107 (2006).
[Crossref]

2003 (2)

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

X. Huang and M. Dudley, “A universal computation method for two-beam dynamical X-ray diffraction,” Acta Crystallogr. Sect. A 59, 163–167 (2003).
[Crossref]

2001 (2)

N. Delen and B. Hooker, “Verification and comparison of a fast fourier transform-based full diffraction method for tilted and offset planes,” Appl. Opt. 40, 3525–3531 (2001).
[Crossref]

R. D. Spal, “Submicrometer resolution hard x-ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[Crossref] [PubMed]

1998 (1)

Authier, A.

A. Authier, Dynamical Theory of X-ray Diffraction (Oxford University, 2004).

Baumbach, T.

Cecilia, A.

Chapman, H. N.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Delen, N.

Dolbnya, I.

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

Dudley, M.

X. Huang and M. Dudley, “A universal computation method for two-beam dynamical X-ray diffraction,” Acta Crystallogr. Sect. A 59, 163–167 (2003).
[Crossref]

Fiederle, M.

P. Vagovič, L. Švéda, A. Cecilia, E. Hamann, D. Pelliccia, E. N. Gimenez, D. Korytár, K. M. Pavlov, Z. Zápražný, M. Zuber, T. Koenig, M. Olbinado, W. Yashiro, A. Momose, M. Fiederle, and T. Baumbach, “X-ray Bragg magnifier microscope as a linear shift invariant imaging system: image formation and phase retrieval,” Opt. Express 22, 21508–21520 (2014).
[Crossref]

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

Fleschig, U.

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

Gimenez, E. N.

Gimenez-Navarro, E.

Greven, H.

Hamann, E.

Hanke, M.

Härtwig, J.

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

Hau-Riege, S. P.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

He, H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Hooker, B.

Hovan, A.

Howells, M. R.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Hrivnak, S.

Huang, X.

X. Huang and M. Dudley, “A universal computation method for two-beam dynamical X-ray diffraction,” Acta Crystallogr. Sect. A 59, 163–167 (2003).
[Crossref]

Koenig, T.

Köhler, R.

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74, 054107 (2006).
[Crossref]

Korytár, D.

Lübbert, D.

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74, 054107 (2006).
[Crossref]

Marchesini, S.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Mikeš, L.

Modregger, P.

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74, 054107 (2006).
[Crossref]

Momose, A.

Noy, A.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Oberta, P.

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

Olbinado, M.

Paganin, D.

D. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
[Crossref]

Pavlov, K. M.

Pelliccia, D.

P. Vagovič, L. Švéda, A. Cecilia, E. Hamann, D. Pelliccia, E. N. Gimenez, D. Korytár, K. M. Pavlov, Z. Zápražný, M. Zuber, T. Koenig, M. Olbinado, W. Yashiro, A. Momose, M. Fiederle, and T. Baumbach, “X-ray Bragg magnifier microscope as a linear shift invariant imaging system: image formation and phase retrieval,” Opt. Express 22, 21508–21520 (2014).
[Crossref]

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

Rau, C.

Schäfer, P.

P. Modregger, D. Lübbert, P. Schäfer, R. Köhler, T. Weitkamp, M. Hanke, and T. Baumbach, “Fresnel diffraction in the case of an inclined image plane,” Opt. Express 16, 5141–5149 (2008).
[Crossref] [PubMed]

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74, 054107 (2006).
[Crossref]

Shawney, K.

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

Spal, R. D.

R. D. Spal, “Submicrometer resolution hard x-ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[Crossref] [PubMed]

Spence, J. C. H.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Švéda, L.

Ulicný, J.

Vagovic, P.

Wagner, U. H.

Weierstall, U.

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Weitkamp, T.

Yashiro, W.

Zápražný, Z.

Zuber, M.

Acta Crystallogr. Sect. A (1)

X. Huang and M. Dudley, “A universal computation method for two-beam dynamical X-ray diffraction,” Acta Crystallogr. Sect. A 59, 163–167 (2003).
[Crossref]

Appl. Opt. (1)

Biomed. Opt. Express (1)

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

J. Synchrotron Radiat. (1)

P. Vagovič, D. Korytár, A. Cecilia, E. Hamann, L. Švéda, D. Pelliccia, J. Härtwig, Z. Zápražný, P. Oberta, I. Dolbnya, K. Shawney, U. Fleschig, M. Fiederle, and T. Baumbach, “High-resolution high-efficiency X-ray imaging system based on the in-line Bragg magnifier and the Medipix detector,” J. Synchrotron Radiat. 20, 153–159 (2013).
[Crossref]

Opt. Express (3)

Phys. Rev. B (2)

P. Modregger, D. Lübbert, P. Schäfer, and R. Köhler, “Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory,” Phys. Rev. B 74, 054107 (2006).
[Crossref]

S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diffraction pattern alone,” Phys. Rev. B 68, 140101 (2003).
[Crossref]

Phys. Rev. Lett. (1)

R. D. Spal, “Submicrometer resolution hard x-ray holography with the asymmetric Bragg diffraction microscope,” Phys. Rev. Lett. 86, 3044–3047 (2001).
[Crossref] [PubMed]

Other (2)

A. Authier, Dynamical Theory of X-ray Diffraction (Oxford University, 2004).

D. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Wave propagation through the reflecting inclined plane - crystal (in orange) from sample plane (left) to the detector D. The sketch represents the one-crystal geometry situation from the side view. The axis y0, y1 and y2 are perpendicular to the figure plane. The dashed lines correspond to the active diffraction planes of the crystal.
Fig. 2
Fig. 2 Wave propagation for the situation of 4 crystals forming the Bragg magnifier. Figure taken from [9].
Fig. 3
Fig. 3 The amplitude and the phase of the polystyrene sphere - our phantom object, i.e. the wave-field U0(x0, y0).
Fig. 4
Fig. 4 Simulated intensity patterns |UD(x2, y2)|2 for all three described methods. The yellow lines show the position of the profile lines plotted in Fig. 6.
Fig. 5
Fig. 5 The absolute value of the difference in detector intensities between the results obtained with the ED and RCT methods, in comparison to the reference method MFT. The superiority of the RCT method is clearly visible.
Fig. 6
Fig. 6 The line profiles along the yellow lines in Fig. 4.

Equations (26)

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

f ˜ ( x ˜ , y ˜ ) = 1 ( 2 π ) 2 f ( x , y ) exp [ i ( x ˜ x + y ˜ y ) ] d x d y
f ( x , y ) = 1 [ f ˜ ( x ˜ , y ˜ ) ] = f ˜ ( x ˜ , y ˜ ) exp [ i ( x ˜ x + y ˜ y ) ] d x ˜ d y ˜ .
U C ( x 1 ( i ) , y 1 ) = ( 1 { U ˜ 0 ( x ˜ 0 , y ˜ 0 ) exp [ i z 0 ( i ) 2 K ( x ˜ 0 2 + y ˜ 0 2 ) ] } ) ( i ) ,
U C ( 0 ) i ( x 2 , y 2 ) = { U C ( x 1 ( i ) , y 1 ) i th row , 0 elsewhere .
U C ( x 1 , y 1 ) = i = 1 L U C ( 0 ) i ( x 2 , y 2 )
U D ( x 2 , y 2 ) = i = 1 L 1 { U ˜ C ( 0 ) i ( x 2 , y 2 ) exp [ i z 1 ( i ) 2 K ( x ˜ 2 2 + y ˜ 2 2 ) ] } ,
U D ( x 2 , y 2 ) = 1 { U ˜ 0 ( x ˜ 0 , y ˜ 0 ) exp [ i z 0 2 K ( x ˜ 0 2 + y ˜ 0 2 ) ] exp [ i z 1 2 K ( x ˜ 2 2 + y ˜ 2 2 ) ] } .
U D ( x 2 , y 2 ) = 1 { U ˜ 0 ( x ˜ 0 , y ˜ 0 ) exp [ i [ ( z 0 + z 1 M 2 ) x ˜ 0 2 + ( z 0 + z 1 ) y ˜ 0 2 ] 2 K ] } .
U C ( x 0 , y 0 ) = U ˜ 0 ( x ˜ 0 , y ˜ 0 ) exp [ i z 0 + z ( x 0 ) 2 K ( x ˜ 0 2 + y ˜ 0 2 ) ] exp [ i ( x ˜ 0 x 0 + y ˜ 0 y 0 ) ] d x ˜ 0 d y ˜ 0 ,
U C ( x 1 , y 1 ) = U ˜ 0 ( x ˜ 0 , y ˜ 0 ) exp [ i z 0 2 K ( x ˜ 0 2 + y ˜ 0 2 ) ] × exp { i x 1 [ x ˜ 0 sin α cos α 2 K ( x ˜ 0 2 + y ˜ 0 2 ) ] x ˜ 1 } exp ( i y ˜ 0 y 1 ) d y ˜ 0 d x ˜ 0 .
x ˜ 1 = x ˜ 0 sin α cos α 2 K ( x ˜ 0 2 + y ˜ 0 2 ) , y ˜ 1 = y ˜ 0 ,
x ˜ 0 = K tan α K 2 tan 2 α y ˜ 1 2 2 K cos α x ˜ 1 , y ˜ 0 = y ˜ 1 .
| ( x ˜ 0 , y ˜ 0 ) ( x ˜ 1 , y ˜ 1 ) | = x ˜ 0 x ˜ 1 y ˜ 0 y ˜ 1 x ˜ 0 y ˜ 1 y ˜ 0 x ˜ 1 = 1 sin 2 α y ˜ 1 2 cos 2 α K 2 2 x ˜ 1 cos α K ,
U C ( x 1 , y 1 ) = U ˜ 0 ( x ˜ 0 ( x ˜ 1 , y ˜ 1 ) , y ˜ 0 ( x ˜ 1 , y ˜ 1 ) ) × exp { i z 0 2 K [ x ˜ 0 2 ( x ˜ 1 , y ˜ 1 ) + y ˜ 1 2 ] } sin 2 α y ˜ 1 2 cos 2 α K 2 2 x ˜ 1 cos α K exp { i [ x 1 x ˜ 1 + y 1 y ˜ 1 ] } d y ˜ 1 d x ˜ 1 .
U C ( x 1 , y 1 ) = 1 [ U ˜ 0 ( x ˜ 0 ( x ˜ 1 , y ˜ 1 ) , y ˜ 0 ( x ˜ 1 , y ˜ 1 ) ) P α , z 0 ( 1 ) ( x ˜ 0 ( x ˜ 1 , y ˜ 1 ) , y ˜ 0 ( x ˜ 1 , y ˜ 1 ) ) ] .
P α , z ( 1 ) ( x ˜ i , y ˜ i ) = exp { i z 2 K [ x ˜ i 2 + y ˜ i 2 ] } sin 2 α y ˜ i + 1 2 cos 2 α K 2 2 x ˜ i + 1 cos α K .
U 0 ( x 0 , y 0 ) = 1 [ U ˜ C ( x ˜ 1 ( x ˜ 0 , y ˜ 0 ) , y ˜ 1 ( x ˜ 0 , y ˜ 0 ) ) P α , z 0 ( 1 ) ( x ˜ 1 ( x ˜ 0 , y ˜ 0 ) , y ˜ 1 ( x ˜ 0 , y ˜ 0 ) ) ] .
P β , z ( 2 ) ( x ˜ i , y ˜ i ) = exp { + i z 2 K [ x ˜ i + 1 2 + y ˜ i + 1 2 ] } sin 2 β y ˜ i 2 cos 2 β K 2 2 x ˜ i cos β K ,
U D ( x 2 , y 2 ) = 1 [ U ˜ C ( x ˜ 1 ( x ˜ 2 , y ˜ 2 ) , y ˜ 1 ( x ˜ 2 , y ˜ 2 ) ) P β , z 1 ( 2 ) ( x ˜ 1 ( x ˜ 2 , y ˜ 2 ) , y ˜ 1 ( x ˜ 2 , y ˜ 2 ) ) ] ,
x ˜ 1 = x ˜ 2 sin β cos β 2 K ( x ˜ 2 2 + y ˜ 2 2 ) , y ˜ 1 = y ˜ 2 .
U ˜ D ( x ˜ 2 , y ˜ 2 ) = U ˜ C ( x ˜ 1 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) P β , z 1 ( 2 ) ( x ˜ 1 , ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) = = U ˜ 0 ( x ˜ 0 ( x ˜ 1 ( x ˜ 2 , y ˜ 2 ) , y ˜ 1 ) , y ˜ 2 ) P α , z 0 ( 1 ) ( x ˜ 0 ( x ˜ 1 ( x ˜ 2 , y ˜ 2 ) , y ˜ 1 ) , y ˜ 2 ) P β , z 1 ( 2 ) ( x ˜ 1 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) ,
x ˜ 0 = K tan α K 2 tan 2 α 2 K M x ˜ 2 tan α + cos β cos α ( x ˜ 2 2 + y ˜ 2 2 ) y ˜ 2 2 , y ˜ 0 = y ˜ 2 .
U D ( x 2 , y 2 ) = 1 [ U ˜ 0 ( x ˜ 0 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) P α , z 0 ( 1 ) ( x ˜ 0 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) P β , z 1 ( 2 ) ( x ˜ 1 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) ] .
U C ( x 0 , y 0 ) = U ˜ 0 ( x ˜ 0 , y ˜ 0 ) exp [ i z 0 + z ( x 0 ) 2 K ( x ˜ 0 2 + y ˜ 0 2 ) ] C 1 ( x ˜ 0 , y ˜ 0 ) × exp [ i ( x ˜ 0 x 0 + y ˜ 0 y 0 ) ] d x ˜ 0 d y ˜ 0 .
U D ( x 2 , y 2 ) = 1 [ U ˜ 0 ( x ˜ 0 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) P α , z 0 ( 1 ) ( x ˜ 0 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) P β , z 1 ( 2 ) ( x ˜ 1 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) C 1 ( x ˜ 0 ( x ˜ 2 , y ˜ 2 ) , y ˜ 2 ) ] .
U D ( x 2 N , y 2 N ) = 1 [ U ˜ 0 ( x ˜ 0 * , y ˜ 0 * ) P α 1 , z 0 ( 1 ) ( x ˜ 0 * , y ˜ 0 * ) P β N , z N ( 2 ) ( x ˜ 2 N 1 * , y ˜ 2 N 1 * ) × i = 1 N 1 P α i + 1 , z i / 2 ( 1 ) ( x ˜ 2 i * , y ˜ 2 i * ) P β i , z i / 2 ( 2 ) ( x ˜ 2 i 1 * , y ˜ 2 i 1 * ) i = 1 N C i ( x ˜ 2 i 2 * , y 2 i 2 * ) ] ,

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