Abstract

We present an efficient method to perform x-ray optics simulation with high or partially coherent x-ray sources using Gaussian superposition technique. In a previous paper, we have demonstrated that full characterization of optical systems, diffractive and geometric, is possible by using the Fresnel Gaussian Shape Invariant (FGSI) previously reported in the literature. The complex amplitude distribution in the object plane is represented by a linear superposition of complex Gaussians wavelets and then propagated through the optical system by means of the referred Gaussian invariant. This allows ray tracing through the optical system and at the same time allows calculating with high precision the complex wave-amplitude distribution at any plane of observation. This technique can be applied in a wide spectral range where the Fresnel diffraction integral applies including visible, x-rays, acoustic waves, etc. We describe the technique and include some computer simulations as illustrative examples for x-ray optical component. We show also that this method can be used to study partial or total coherence illumination problem.

© 2011 OSA

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    [CrossRef]
  25. A. Snigirev, V. Kohn, I. Snigireva, A. Souvorov, and B. Lengeler, “Focusing high-energy x rays by compound refractive lenses,” Appl. Opt. 37(4), 653–662 (1998).
    [CrossRef] [PubMed]
  26. C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
    [CrossRef] [PubMed]

2011 (1)

2010 (4)

M. Cywiak, A. Morales, M. Servín, and R. Gómez-Medina, “A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling,” Opt. Express 18(18), 19141–19155 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19141 .
[CrossRef] [PubMed]

I. A. Vartanyants, A. P. Mancuso, A. Singer, O. M. Yefanov, and J. Gulden, “Coherence measurements and coherent diffractive imaging at FLASH,” J. Phys. At. Mol. Opt. Phys. 43(19), 194016 (2010).
[CrossRef]

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

T. Tomie, “The birth of the X-ray refractive lens,” Spectrochim. Acta, B At. Spectrosc. 65(3), 192–198 (2010).
[CrossRef]

2009 (2)

2008 (1)

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

2007 (2)

2006 (1)

2005 (2)

2001 (3)

T. Moreno and M. Idir, “SPOTX a ray tracing software for X-ray optics,” J. Phys. (France) 11, 527–531 (2001).

T. Yamada, N. Kawahara, M. Doi, T. Shoji, N. Tsuruoka, and H. Iwasaki, “A new ray-tracing program RIGTRACE for X-ray optical systems,” J. Synchrotron Radiat. 8(3), 1047–1050 (2001).
[CrossRef] [PubMed]

W. Leitenberger, S. M. Kuznetsov, and A. Snigirev, “Interferometric measurements with hard X-rays using a double slit,” Opt. Commun. 191(1–2), 91–96 (2001).
[CrossRef]

1998 (1)

1996 (2)

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[CrossRef]

T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. Tisch, D. D. Meyerhofer, and M. H. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996).
[CrossRef] [PubMed]

1986 (1)

B. Lai and F. Cerrina, “SHADOW: A synchrotron radiation ray tracing program,” Nucl. Instrum. Methods Phys. Res. A 246(1-3), 337–341 (1986).
[CrossRef]

Anderson, E. H.

Assoufid, L.

Attwood, D. T.

Boye, P.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Burghammer, M.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Cerrina, F.

B. Lai and F. Cerrina, “SHADOW: A synchrotron radiation ray tracing program,” Nucl. Instrum. Methods Phys. Res. A 246(1-3), 337–341 (1986).
[CrossRef]

Chao, W.

Cywiak, M.

Ditmire, T.

T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. Tisch, D. D. Meyerhofer, and M. H. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996).
[CrossRef] [PubMed]

Doi, M.

T. Yamada, N. Kawahara, M. Doi, T. Shoji, N. Tsuruoka, and H. Iwasaki, “A new ray-tracing program RIGTRACE for X-ray optical systems,” J. Synchrotron Radiat. 8(3), 1047–1050 (2001).
[CrossRef] [PubMed]

Endo, K.

Feldkamp, J. M.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Fischer, P.

Flores, J. M.

Gómez-Medina, R.

Gulden, J.

I. A. Vartanyants, A. P. Mancuso, A. Singer, O. M. Yefanov, and J. Gulden, “Coherence measurements and coherent diffractive imaging at FLASH,” J. Phys. At. Mol. Opt. Phys. 43(19), 194016 (2010).
[CrossRef]

Gumbrell, E. T.

T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. Tisch, D. D. Meyerhofer, and M. H. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996).
[CrossRef] [PubMed]

Guttmann, P.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Harteneck, B. D.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[CrossRef] [PubMed]

Heim, S.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Heymann, J. B.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Hutchinson, M. H.

T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. Tisch, D. D. Meyerhofer, and M. H. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996).
[CrossRef] [PubMed]

Idir, M.

T. Moreno and M. Idir, “SPOTX a ray tracing software for X-ray optics,” J. Phys. (France) 11, 527–531 (2001).

Ishikawa, T.

Iwasaki, H.

T. Yamada, N. Kawahara, M. Doi, T. Shoji, N. Tsuruoka, and H. Iwasaki, “A new ray-tracing program RIGTRACE for X-ray optical systems,” J. Synchrotron Radiat. 8(3), 1047–1050 (2001).
[CrossRef] [PubMed]

Kawahara, N.

T. Yamada, N. Kawahara, M. Doi, T. Shoji, N. Tsuruoka, and H. Iwasaki, “A new ray-tracing program RIGTRACE for X-ray optical systems,” J. Synchrotron Radiat. 8(3), 1047–1050 (2001).
[CrossRef] [PubMed]

Kewish, C. M.

Kim, J.

Kohn, V.

A. Snigirev, V. Kohn, I. Snigireva, A. Souvorov, and B. Lengeler, “Focusing high-energy x rays by compound refractive lenses,” Appl. Opt. 37(4), 653–662 (1998).
[CrossRef] [PubMed]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[CrossRef]

Kuznetsov, S. M.

W. Leitenberger, S. M. Kuznetsov, and A. Snigirev, “Interferometric measurements with hard X-rays using a double slit,” Opt. Commun. 191(1–2), 91–96 (2001).
[CrossRef]

Lai, B.

B. Lai and F. Cerrina, “SHADOW: A synchrotron radiation ray tracing program,” Nucl. Instrum. Methods Phys. Res. A 246(1-3), 337–341 (1986).
[CrossRef]

Larotonda, M. A.

Leitenberger, W.

W. Leitenberger, S. M. Kuznetsov, and A. Snigirev, “Interferometric measurements with hard X-rays using a double slit,” Opt. Commun. 191(1–2), 91–96 (2001).
[CrossRef]

Lengeler, B.

A. Snigirev, V. Kohn, I. Snigireva, A. Souvorov, and B. Lengeler, “Focusing high-energy x rays by compound refractive lenses,” Appl. Opt. 37(4), 653–662 (1998).
[CrossRef] [PubMed]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[CrossRef]

Liddle, J. A.

W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood, “Soft X-ray microscopy at a spatial resolution better than 15 nm,” Nature 435(7046), 1210–1213 (2005).
[CrossRef] [PubMed]

Liu, Y.

Livet, F.

F. Livet, “Diffraction with a coherent X-ray beam: dynamics and imaging,” Acta Crystallogr. A 63(2), 87–107 (2007).
[CrossRef] [PubMed]

Luther, B. M.

Macrander, A. T.

Mancuso, A. P.

I. A. Vartanyants, A. P. Mancuso, A. Singer, O. M. Yefanov, and J. Gulden, “Coherence measurements and coherent diffractive imaging at FLASH,” J. Phys. At. Mol. Opt. Phys. 43(19), 194016 (2010).
[CrossRef]

McNally, J. G.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Mendoza-Santoyo, F.

M. Cywiak, M. Servín, and F. Mendoza-Santoyo, “Wave-front propagation by Gaussian superposition,” Opt. Comm.195(5–6), 351–359, (2001).
[CrossRef]

Meyerhofer, D. D.

T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. Tisch, D. D. Meyerhofer, and M. H. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996).
[CrossRef] [PubMed]

Mimura, H.

Morales, A.

Moreno, T.

T. Moreno and M. Idir, “SPOTX a ray tracing software for X-ray optics,” J. Phys. (France) 11, 527–531 (2001).

Mori, Y.

Mueller, F.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Müller, W. G.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Nagashima, K.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Patommel, J.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Qian, J.

Rehbein, S.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Rekawa, S.

Riekel, C.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Rocca, J. J.

Saito, A.

Sano, Y.

Schneider, G.

G. Schneider, P. Guttmann, S. Heim, S. Rehbein, F. Mueller, K. Nagashima, J. B. Heymann, W. G. Müller, and J. G. McNally, “Three-dimensional cellular ultrastructure resolved by X-ray microscopy,” Nat. Methods 7(12), 985–987 (2010).
[CrossRef] [PubMed]

Schöder, S.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Schroer, C. G.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Schropp, A.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Schwab, A.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Servín, M.

Shoji, T.

T. Yamada, N. Kawahara, M. Doi, T. Shoji, N. Tsuruoka, and H. Iwasaki, “A new ray-tracing program RIGTRACE for X-ray optical systems,” J. Synchrotron Radiat. 8(3), 1047–1050 (2001).
[CrossRef] [PubMed]

Singer, A.

I. A. Vartanyants, A. P. Mancuso, A. Singer, O. M. Yefanov, and J. Gulden, “Coherence measurements and coherent diffractive imaging at FLASH,” J. Phys. At. Mol. Opt. Phys. 43(19), 194016 (2010).
[CrossRef]

Smith, R. A.

T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. Tisch, D. D. Meyerhofer, and M. H. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996).
[CrossRef] [PubMed]

Snigirev, A.

W. Leitenberger, S. M. Kuznetsov, and A. Snigirev, “Interferometric measurements with hard X-rays using a double slit,” Opt. Commun. 191(1–2), 91–96 (2001).
[CrossRef]

A. Snigirev, V. Kohn, I. Snigireva, A. Souvorov, and B. Lengeler, “Focusing high-energy x rays by compound refractive lenses,” Appl. Opt. 37(4), 653–662 (1998).
[CrossRef] [PubMed]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[CrossRef]

Snigireva, I.

A. Snigirev, V. Kohn, I. Snigireva, A. Souvorov, and B. Lengeler, “Focusing high-energy x rays by compound refractive lenses,” Appl. Opt. 37(4), 653–662 (1998).
[CrossRef] [PubMed]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[CrossRef]

Souvorov, A.

Stephan, S.

C. G. Schroer, P. Boye, J. M. Feldkamp, J. Patommel, A. Schropp, A. Schwab, S. Stephan, M. Burghammer, S. Schöder, and C. Riekel, “Coherent x-ray diffraction imaging with nanofocused illumination,” Phys. Rev. Lett. 101(9), 090801 (2008).
[CrossRef] [PubMed]

Tamasaku, K.

Tisch, J. W.

T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. Tisch, D. D. Meyerhofer, and M. H. Hutchinson, “Spatial coherence measurement of soft X-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77(23), 4756–4759 (1996).
[CrossRef] [PubMed]

Tomie, T.

T. Tomie, “The birth of the X-ray refractive lens,” Spectrochim. Acta, B At. Spectrosc. 65(3), 192–198 (2010).
[CrossRef]

Tsuruoka, N.

T. Yamada, N. Kawahara, M. Doi, T. Shoji, N. Tsuruoka, and H. Iwasaki, “A new ray-tracing program RIGTRACE for X-ray optical systems,” J. Synchrotron Radiat. 8(3), 1047–1050 (2001).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

20 microns slit diffraction profile versus the distance z between the slit and the detector position for λ = 0.1 nm.

Fig. 2
Fig. 2

Double slit diffraction geometry (slit size 5 microns, b = 30 microns and L vary from 1 cm to 300 cm)

Fig. 3
Fig. 3

Double Slit diffraction profile versus the distance z (from 1 mm up to 3 m) between the slit and the detector position for λ = 0.1 nm.

Fig. 4
Fig. 4

Double Slit diffraction geometry using two points sources.

Fig. 5
Fig. 5

Double Slit diffraction profile versus the distance z (from 1 mm up to 3 m) between the slit and the detector position for λ = 0.1 nm.

Fig. 6
Fig. 6

Intensity distribution at the 1st and second focal plane (p = 1 and p = 2) for a ZP with duty cycle of one.

Fig. 7
Fig. 7

Intensity distributions at the first and second focal plane (p = 1 and p = 2) for ZP with a duty cycle of 1/2.

Fig. 8
Fig. 8

Parabolic-Spherical Refractive lenses geometry. N = 1, f = 45.1875 m

Fig. 9
Fig. 9

Calculated diffraction pattern for a Be parabolic refractive lens (N = 1 (f = 2.2594 m), R = 100 microns).

Equations (11)

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Ψ 0 ( x , y ) = m P m exp ( ( x x m ) 2 + ( y y m ) 2 σ 2 ) exp ( i α m ( x 2 + y 2 ) ) exp ( i β m ( x + y ) ) .
Ψ P ( ξ , η ) = 1 i λ z Ψ 0 ( x , y ) exp ( i π λ z [ ( x ξ ) 2 + ( y η ) 2 ] ) d x d y .
Ψ P R O P ( ξ , η ) = i m = 0 M P m ( λ z + i a ) π σ 2 d exp ( x m 2 + y m 2 σ 2 + λ z p m + a q m σ 2 λ z + d ( g x m 2 + g y m 2 ) ( σ λ z ) 2 ) × exp ( i π λ z [ d π σ 2 a d ( ξ 2 + η 2 ) 2 ( f x m ξ + f y m η ) + λ z q m a p m π σ 2 ] ) × exp ( π 2 σ 2 d [ ( ξ d g x m π σ 2 λ z ) 2 + ( η d g y m π σ 2 λ z ) 2 ] ) .
a m = σ 2 ( α m λ z + π ) , d m = λ 2 z 2 + a m 2 , f x m = λ z 2 d m ( 2 x m λ z β m σ 2 a m ) ,     g x m = λ z 2 d m ( β m σ 2 λ z + 2 x m a m ) , f y m = λ z 2 d m ( 2 y m λ z β m σ 2 a m ) ,     g y m = λ z 2 d m ( β m σ 2 λ z + 2 y m a m ) ,
p m = f x 2 m + f y m 2 g x m 2 g y m 2 , q m = 2 ( f x m g x m + f y m g y m ) .
Ψ n ( x ) = P n exp ( i α n x ) exp ( i β n x 2 ) exp ( ( x A n ) 2 r n 2 ) exp ( i γ n [ x B n ] 2 ) .
Ψ n + 1 ( ξ , z ) = P n + 1 exp ( i α n + 1 ξ ) exp ( i β n + 1 ξ 2 ) exp ( ( ξ A n + 1 ) 2 r n + 1 2 ) exp ( i γ n + 1 [ ξ B n + 1 ] 2 ) .
P n + 1 = P n exp ( i 2 π z / λ ) i λ z π     r n 2 λ z λ z i r n 2 ( β n λ z + γ n λ z + π )     ×                          exp ( i γ n B n ) 2 ) exp ( i λ z ( A n ) 2 r n 4 ( β n λ z + γ n λ z + π ) ) , α n + 1 = 0 ,     β n + 1 = π λ z ,     γ n + 1 = π 2 r n 4 D n λ z ( β n λ z + γ n λ z + π ) , r n + 1 = D n π r n , A n + 1 = A n + α n λ z 2 π γ n λ z B n π + ( β n + γ n ) λ z A n π , B n + 1 = α n λ z 2 π γ n λ z B n π λ 2 z 2 β n λ z + γ n λ z + π A n π r n 4 , D n = λ 2 z 2 + r n 4 ( β n λ z + γ n λ z + π ) 2 .
α n = 2 π λ n tan ( θ n )     ,
α n + 1 = 2 π λ n + 1 tan ( θ n ) + 2 γ n + 1 B n + 1 2 ( β n + 1 + γ n + 1 ) A n + 1 , P n + 1 = P n + 1 exp ( i α n + 1 A n + 1 ) .
Z P ( x ) = A 2 + n 0 sin ( n π A / 2 ) n π exp ( i 2 π n λ [ p 2 + x 2 p ] ) .

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