Abstract

Computer simulations of nanofocusing by elliptical mirrors are presented wherein the diffraction and propagation of coherent hard x rays are predicted using wave-optical calculations. Surface height data acquired via microstitching interferometry were used to calculate the complex pupil function of a mirror, taking into account the Fresnel reflectivity and treating the surface topography as an aberration to a perfect elliptical mirror. The reflected wave-field amplitude and phase downstream of the mirror were obtained by numerically evaluating the Fresnel–Kirchhoff diffraction integral. Simulated intensity profiles and contours (isophotes) around the focal plane are presented for coherent illumination by a 15  keV point source, which indicate nearly diffraction-limited focusing at the 40  nm level. The effect of high spatial frequency microroughness on nanofocusing was investigated by low-pass filtering the Fourier spectrum of the residual height profile. Simulations using the filtered metrology data confirmed that roughness length scales shorter than 0.1  mm have a minor effect on the focal spot size and intensity.

© 2007 Optical Society of America

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2006 (2)

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, "Nanometer linear focusing of hard x rays by a multilayer Laue lens," Phys. Rev. Lett. 96, 127401 (2006).
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H. Schomerus and M. Hentschel, "Correcting ray optics at curved dielectric microresonator interfaces: Phase-space unification of Fresnel filtering and the Goos-Hänchen shift," Phys. Rev. Lett. 96, 243903 (2006).
[CrossRef] [PubMed]

2005 (5)

O. Hignette, P. Cloetens, G. Rostaing, P. Bernard, and C. Morawe, "Efficient sub 100 nm focusing of hard x rays," Rev. Sci. Instrum. 76, 063709 (2005).
[CrossRef]

H. Yumoto, H. Mimura, S. Matsuyama, H. Hara, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, "Fabrication of elliptically figured mirror for focusing hard x rays to size less than 50 nm," Rev. Sci. Instrum. 76, 063708 (2005).
[CrossRef]

W. Liu, G. E. Ice, J. Z. Tischler, A. Khounsary, C. Liu, L. Assoufid, and A. T. Macrander, "Short focal length Kirkpatrick-Baez mirrors for a hard x-ray nanoprobe," Rev. Sci. Instrum. 76, 113701 (2005).
[CrossRef]

H. Mimura, S. Matsuyama, H. Yumoto, H. Hara, K. Yamamura, Y. Sano, M. Shibahara, K. Endo, Y. Mori, Y. Nishino, K. Tamasaku, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Hard x-ray diffraction-limited nanofocusing with Kirkpatrick-Baez mirrors," Jpn. J. Appl. Phys. 44, L539-L542 (2005).
[CrossRef]

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, "Wave-optical evaluation of interference fringes and wavefront phase in a hard-x-ray beam totally reflected by mirror optics," Appl. Opt. 44, 6927-6932 (2005).
[CrossRef] [PubMed]

2004 (2)

T. M. Elfouhaily and C.-A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Wave Random Media 14, R1-R40 (2004).
[CrossRef]

M. Yasa, Y. Li, C. B. Mammen, J. Als-Nielsen, J. Hoszowska, C. Mocuta, and A. Freund, "Double focusing of hard x rays using combined multilayer and Bragg-Fresnel optics," Appl. Phys. Lett. 84, 4744-4746 (2004).
[CrossRef]

2003 (5)

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, "Microstitching interferometry for x-ray reflective optics," Rev. Sci. Instrum. 74, 2894-2898 (2003).
[CrossRef]

A. Freund, "Recent progress in x-ray optics at the ESRF," J. Phys. IV 104, 165-170 (2003).

D. H. Bilderback, "Review of capillary x-ray optics from the 2nd International Capillary Optics Meeting," X-Ray Spectrom. 32, 195-207 (2003).
[CrossRef]

K. Yamamura, K. Yamauchi, H. Mimura, Y. Sano, A. Saito, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, "Fabrication of elliptical mirror at nanometer-level accuracy for hard x-ray focusing by numerically controlled plasma chemical vaporization machining," Rev. Sci. Instrum. 74, 4549-4553 (2003).
[CrossRef]

O. Hignette, P. Cloetens, W.-K. Lee, W. Ludwig, and G. Rostaing, "Hard x-ray microscopy with reflecting mirrors status and perspectives of the ESRF technology," J. Phys. IV 104, 231-234 (2003).

2002 (2)

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, M. Kanaoka, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, "Wave-optical analysis of sub-micron focusing of hard x-ray beams by reflective optics," in X-ray Mirrors, Crystals, and Multilayers II, A. K. Freund, A. T. Macrander, T. Ishikawa, and J. L. Wood, eds., Proc. SPIE 4782, 271-276 (2002).
[CrossRef]

A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, Y. Mori, K. Yamauchi, K. Yamamura, and A. Saito, "Deterministic retrieval of surface waviness by means of topography with coherent x-rays," J. Synchrotron Radiat. 9, 223-228 (2002).
[CrossRef] [PubMed]

2001 (1)

K. F. Warnick and W. C. Chew, "Numerical simulation methods for rough surface scattering," Wave Random Media 11, R1-R30 (2001).
[CrossRef]

2000 (1)

G. E. Ice, J.-S. Chung, J. Z. Tischler, A. Lunt, and L. Assoufid, "Elliptical x-ray microprobe mirrors by differential deposition," Rev. Sci. Instrum. 71, 2635-2639 (2000).
[CrossRef]

1999 (1)

W. Yun, B. Lai, Z. Cai, J. Maser, D. Legnini, E. Gluskin, Z. Chen, A. A. Krasnoperova, Y. Vladimirsky, F. Cerrina, E. di Fabrizio, and M. Gentili, "Nanometer focusing of hard x rays by phase zone plates," Rev. Sci. Instrum. 70, 2238-2241 (1999).
[CrossRef]

1998 (1)

D. G. Stearns, D. P. Gaines, D. W. Sweeney, and E. M. Gullikson, "Nonspecular x-ray scattering in a multilayer-coated imaging system," J. Appl. Phys. 84, 1003-1028 (1998).
[CrossRef]

1997 (2)

C. Liu and J. A. Golovchenko, "Surface trapped x rays: whispering-gallery modes at λ = 0.7 Å," Phys. Rev. Lett. 79, 788-791 (1997).
[CrossRef]

G. E. Ice, "Microbeam-forming methods for synchrotron radiation," X-Ray Spectrom. 26, 315-326 (1997).
[CrossRef]

1996 (1)

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

1995 (4)

A. Snigirev, "The recent development of Bragg-Fresnel crystal optics. Experiments and applications at the ESRF (invited)," Rev. Sci. Instrum. 66, 2053-2058 (1995).
[CrossRef]

J. E. Harvey, K. L. Lewotsky, and A. Kotha, "Effects of surface scatter on the optical performance of x-ray synchrotron beam-line mirrors," Appl. Opt. 34, 3024-3031 (1995).
[CrossRef] [PubMed]

E. L. Church and P. Z. Takacs, "Specification of glancing- and normal-incidence x-ray mirrors," Opt. Eng. 34, 353-360 (1995).
[CrossRef]

J. Susini, "Design parameter for hard x-ray mirrors: the European Synchrotron Radiation Facility case," Opt. Eng. 34, 361-376 (1995).
[CrossRef]

1994 (1)

C. Welnak, G.-J. Chen, and F. Cerrina, "shadow: a synchrotron radiation and x-ray optics simulation tool," Nucl. Instrum. Methods Phys. Res. A 347, 344-347 (1994).
[CrossRef]

1993 (1)

1992 (1)

1989 (1)

E. L. Church and P. Z. Takacs, "Prediction of mirror performance from laboratory measurements," in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. SPIE 1160, 323-336 (1989).

1987 (1)

P. Z. Takacs, S. N. Qian, and J. Colbert, "Design of a long trace surface profiler," in Proc. SPIE 749, 59-64 (1987).

1986 (1)

B. Lai and F. Cerrina, "shadow: a synchrotron radiation ray tracing program," Nucl. Instrum. Methods Phys. Res. A 246, 337-341 (1986).
[CrossRef]

1977 (1)

1968 (1)

1948 (1)

1947 (1)

F. Goos and H. Hänchen, "Ein neuer und fundamentaler Versuch zur Totalreflexion," Ann. Phys. (Leipzig) 1, 333-346 (1947) (in German).
[CrossRef]

Als-Nielsen, J.

M. Yasa, Y. Li, C. B. Mammen, J. Als-Nielsen, J. Hoszowska, C. Mocuta, and A. Freund, "Double focusing of hard x rays using combined multilayer and Bragg-Fresnel optics," Appl. Phys. Lett. 84, 4744-4746 (2004).
[CrossRef]

Assoufid, L.

W. Liu, G. E. Ice, J. Z. Tischler, A. Khounsary, C. Liu, L. Assoufid, and A. T. Macrander, "Short focal length Kirkpatrick-Baez mirrors for a hard x-ray nanoprobe," Rev. Sci. Instrum. 76, 113701 (2005).
[CrossRef]

G. E. Ice, J.-S. Chung, J. Z. Tischler, A. Lunt, and L. Assoufid, "Elliptical x-ray microprobe mirrors by differential deposition," Rev. Sci. Instrum. 71, 2635-2639 (2000).
[CrossRef]

Baez, A. V.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Artech House, 1987).

Bernard, P.

O. Hignette, P. Cloetens, G. Rostaing, P. Bernard, and C. Morawe, "Efficient sub 100 nm focusing of hard x rays," Rev. Sci. Instrum. 76, 063709 (2005).
[CrossRef]

Bilderback, D. H.

D. H. Bilderback, "Review of capillary x-ray optics from the 2nd International Capillary Optics Meeting," X-Ray Spectrom. 32, 195-207 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, 2000).

Braud, J. P.

Cai, Z.

W. Yun, B. Lai, Z. Cai, J. Maser, D. Legnini, E. Gluskin, Z. Chen, A. A. Krasnoperova, Y. Vladimirsky, F. Cerrina, E. di Fabrizio, and M. Gentili, "Nanometer focusing of hard x rays by phase zone plates," Rev. Sci. Instrum. 70, 2238-2241 (1999).
[CrossRef]

Carniglia, C. K.

Cerrina, F.

W. Yun, B. Lai, Z. Cai, J. Maser, D. Legnini, E. Gluskin, Z. Chen, A. A. Krasnoperova, Y. Vladimirsky, F. Cerrina, E. di Fabrizio, and M. Gentili, "Nanometer focusing of hard x rays by phase zone plates," Rev. Sci. Instrum. 70, 2238-2241 (1999).
[CrossRef]

C. Welnak, G.-J. Chen, and F. Cerrina, "shadow: a synchrotron radiation and x-ray optics simulation tool," Nucl. Instrum. Methods Phys. Res. A 347, 344-347 (1994).
[CrossRef]

B. Lai and F. Cerrina, "shadow: a synchrotron radiation ray tracing program," Nucl. Instrum. Methods Phys. Res. A 246, 337-341 (1986).
[CrossRef]

Chen, G.-J.

C. Welnak, G.-J. Chen, and F. Cerrina, "shadow: a synchrotron radiation and x-ray optics simulation tool," Nucl. Instrum. Methods Phys. Res. A 347, 344-347 (1994).
[CrossRef]

Chen, Z.

W. Yun, B. Lai, Z. Cai, J. Maser, D. Legnini, E. Gluskin, Z. Chen, A. A. Krasnoperova, Y. Vladimirsky, F. Cerrina, E. di Fabrizio, and M. Gentili, "Nanometer focusing of hard x rays by phase zone plates," Rev. Sci. Instrum. 70, 2238-2241 (1999).
[CrossRef]

Chew, W. C.

K. F. Warnick and W. C. Chew, "Numerical simulation methods for rough surface scattering," Wave Random Media 11, R1-R30 (2001).
[CrossRef]

Chung, J.-S.

G. E. Ice, J.-S. Chung, J. Z. Tischler, A. Lunt, and L. Assoufid, "Elliptical x-ray microprobe mirrors by differential deposition," Rev. Sci. Instrum. 71, 2635-2639 (2000).
[CrossRef]

Church, E. L.

E. L. Church and P. Z. Takacs, "Specification of glancing- and normal-incidence x-ray mirrors," Opt. Eng. 34, 353-360 (1995).
[CrossRef]

E. L. Church and P. Z. Takacs, "Specification of surface figure and finish in terms of system performance," Appl. Opt. 32, 3344-3353 (1993).
[CrossRef] [PubMed]

E. L. Church and P. Z. Takacs, "Prediction of mirror performance from laboratory measurements," in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. SPIE 1160, 323-336 (1989).

Cloetens, P.

O. Hignette, P. Cloetens, G. Rostaing, P. Bernard, and C. Morawe, "Efficient sub 100 nm focusing of hard x rays," Rev. Sci. Instrum. 76, 063709 (2005).
[CrossRef]

O. Hignette, P. Cloetens, W.-K. Lee, W. Ludwig, and G. Rostaing, "Hard x-ray microscopy with reflecting mirrors status and perspectives of the ESRF technology," J. Phys. IV 104, 231-234 (2003).

Colbert, J.

P. Z. Takacs, S. N. Qian, and J. Colbert, "Design of a long trace surface profiler," in Proc. SPIE 749, 59-64 (1987).

Conley, R.

H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley, A. T. Macrander, and S. Vogt, "Nanometer linear focusing of hard x rays by a multilayer Laue lens," Phys. Rev. Lett. 96, 127401 (2006).
[CrossRef] [PubMed]

di Fabrizio, E.

W. Yun, B. Lai, Z. Cai, J. Maser, D. Legnini, E. Gluskin, Z. Chen, A. A. Krasnoperova, Y. Vladimirsky, F. Cerrina, E. di Fabrizio, and M. Gentili, "Nanometer focusing of hard x rays by phase zone plates," Rev. Sci. Instrum. 70, 2238-2241 (1999).
[CrossRef]

Elfouhaily, T. M.

T. M. Elfouhaily and C.-A. Guérin, "A critical survey of approximate scattering wave theories from random rough surfaces," Wave Random Media 14, R1-R40 (2004).
[CrossRef]

Endo, K.

H. Yumoto, H. Mimura, S. Matsuyama, H. Hara, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, "Fabrication of elliptically figured mirror for focusing hard x rays to size less than 50 nm," Rev. Sci. Instrum. 76, 063708 (2005).
[CrossRef]

H. Mimura, S. Matsuyama, H. Yumoto, H. Hara, K. Yamamura, Y. Sano, M. Shibahara, K. Endo, Y. Mori, Y. Nishino, K. Tamasaku, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Hard x-ray diffraction-limited nanofocusing with Kirkpatrick-Baez mirrors," Jpn. J. Appl. Phys. 44, L539-L542 (2005).
[CrossRef]

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, "Wave-optical evaluation of interference fringes and wavefront phase in a hard-x-ray beam totally reflected by mirror optics," Appl. Opt. 44, 6927-6932 (2005).
[CrossRef] [PubMed]

K. Yamauchi, K. Yamamura, H. Mimura, Y. Sano, A. Saito, K. Ueno, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, "Microstitching interferometry for x-ray reflective optics," Rev. Sci. Instrum. 74, 2894-2898 (2003).
[CrossRef]

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K. Yamamura, K. Yamauchi, H. Mimura, Y. Sano, A. Saito, K. Endo, A. Souvorov, M. Yabashi, K. Tamasaku, T. Ishikawa, and Y. Mori, "Fabrication of elliptical mirror at nanometer-level accuracy for hard x-ray focusing by numerically controlled plasma chemical vaporization machining," Rev. Sci. Instrum. 74, 4549-4553 (2003).
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H. Yumoto, H. Mimura, S. Matsuyama, H. Hara, K. Yamamura, Y. Sano, K. Ueno, K. Endo, Y. Mori, M. Yabashi, Y. Nishino, K. Tamasaku, T. Ishikawa, and K. Yamauchi, "Fabrication of elliptically figured mirror for focusing hard x rays to size less than 50 nm," Rev. Sci. Instrum. 76, 063708 (2005).
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Figures (9)

Fig. 1
Fig. 1

Schematic of elliptical mirror geometry. The semimajor axis a, semiminor axis b, and the eccentricity e are defined in terms of the source distance s 1 = SO , focal length s 2 = OD , and the grazing incidence angle θ (described in the text). The distance between the foci of the ellipse is 2F and the center of the mirror length L is located at the coordinates ( z 0 , x 0 ) .

Fig. 2
Fig. 2

Phase change for σ-polarized x rays on total reflection, for grazing angles of incidence θ z below the critical angle θ c .

Fig. 3
Fig. 3

Simulation geometry for grazing incidence reflection by an ideally smooth elliptical mirror (figure not to scale). The coordinate z m denotes a point on the mirror surface, and the points ( x s , z s ) and ( x d , z d ) , respectively, represent the locations of a secondary source and the detection point, separated by a distance d d (as described in the text). The wave field E 1 is calculated over the effective aperture dimension | x s   min | + | x s   max | = L   sin   θ , defined by the width of the pupil function. The outgoing wave segment converges onto the focal point ( x f , z f ) , and the wave field E OUT is calculated by evaluating the Fresnel–Kirchhoff integral.

Fig. 4
Fig. 4

Path-length differences introduced by surface height fluctuations h ( z m ) along the mirror length L. The path length between the foci of the ellipse, f 1 and f 2 , is changed from the nominal value s 1 + s 2 due to the surface topography by the amount W ( z m ) = 2 h ( z m ) sin   θ z .

Fig. 5
Fig. 5

Elliptical height profile for the central 39   mm × 12   mm of the SN-2 mirror surface, reconstructed by stitching together 33 overlapping 5   mm × 5   mm subapertures measured with a microscope interferometer model MicroXAM RTS. The interferometer has been customized for the APS by the manufacturer ADE Phase Shift.

Fig. 6
Fig. 6

Residual height profile for the SN-2 mirror, calculated from height data obtained using microstitching interferometry. The profile is one pixel wide ( 10 μ m ) , taken from the center of the surface map. The rms residual height error is 2.16   nm .

Fig. 7
Fig. 7

(a) Intensity profiles in the optimal focal plane for the SN-2 mirror; the solid curve is the simulated intensity profile having a FWHM of 40.68   nm based on microstitched residual height data. The dashed curve represents the intensity profile in the focal plane of a perfectly smooth ellipse with the same design parameters, giving a diffraction-limited FWHM of 38.82   nm . (b) Filled contours representing areas of equal intensity (isophotes) for the SN-2 mirror based on microstitched residual height data; contours normalized to the maximum intensity. (c) Normalized isophotes for the perfectly smooth elliptical mirror, showing the diffraction-limited intensity pattern centered at the focal point. The vertical dashed lines at Z = + 70 μ m in (b) and Z = 0 μ m in (c) show the optimum focal planes.

Fig. 8
Fig. 8

Intensity profiles in the optimal focal plane for the SN-2 mirror using filtered microstitched residual height data sets. The FFT filter cutoff (indicated in units of spatial wavelength) increases from the data resolution up to a maximum of 50   mm . Each profile is normalized to the peak intensity of the lowest curve and is vertically offset by 0.2 units for clarity. The lowest profile is simulated using the original data and has a FWHM of 40.68   nm , and the top intensity profile corresponds to a cutoff wavelength longer than the mirror surface that reproduces the diffraction-limited FWHM of 38.82   nm .

Fig. 9
Fig. 9

(a) Variation of focal plane intensity FWHM as a function of FFT filter cutoff. The low-pass FFT filter spatial frequency cutoff is shown on the top axis, with the corresponding spatial wavelength displayed on the lower x axis. (b) Peak intensity in the focal plane simulated with filtered data, relative to the simulated peak intensity using the original microstitched data. It is evident that the calculated peak intensity and FWHM are influenced primarily by components of the Fourier spectrum corresponding to spatial wavelengths greater than 0.1   mm , as there are obvious deviations in the calculated values when the residual height profile is filtered. A small jump in the calculated FWHM near the filter cutoff wavelength of 5   mm results from the removal of several peaks appearing in the Fourier spectrum of the microstitched data in this region, related to the specific surface topography.

Tables (1)

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Table 1 Summary of Numerical Results from Simulations using Filtered Microstitched Data

Equations (14)

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E OUT ( x , y ) = S A { E 0 ( x , y ) } .
S A { B E 0 ( 1 ) ( x , y ) + C E 0 ( 2 ) ( x , y ) } = B S A { E 0 ( 1 ) ( x , y ) } + C S A { E 0 ( 2 ) ( x , y ) } ,
E 0 ( x , y ) = E 0 ( x 1 , y 1 ) δ ( x x 1 , y y 1 ) d x 1 d y 1 .
E OUT ( x 2 , y 2 ) = E 0 ( x 1 , y 1 ) G ( x 2 , y 2 ; x 1 , y 1 ) d x 1 d y 1 ,
E OUT ( x 2 , y 2 ) = E 0 ( x 1 , y 1 ) G ( x 2 x 1 , y 2 y 1 ) d x 1 d y 1 ,
E OUT = E 0 G .
E ˜ OUT ( f x , f y ) = E ˜ 0 ( f x , f y ) G ˜ ( f x , f y ) ,
r ( z m ) = sin   θ z sin 2 θ z 2 δ sin   θ z + sin 2 θ z 2 δ ,
ϕ ( z m ) = 2   arctan ( 2 δ sin 2 θ z sin   θ z )
P ( x s ) = { E 0 r ( x s ) , | x s |     L   sin   θ 0 , | x s | >     L   sin   θ .
E 1 ( x s , z s ) = P ( x s ) exp [ i k d f ( x s , z s ) ] d f ( x s , z s ) ,
P ^ ( x s ) = P ( x s ) exp [ i k W ( x s ) ] .
E 1 ( x s , z s ) = P ( x s ) exp { i k [ d f ( x s , z s ) + W ( x s ) ] } d f ( x s , z s ) .
E OUT ( x d , z d ) = i 2 λ E 1 ( x s , z s ) × exp [ i k d d ( x d x s , z d z s ) ] d d ( x d x s , z d z s ) × ( cos   γ 1 cos   γ 2 ) d x s ,

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