Abstract

We describe an approach to calculating the optical performance of a wide range of nanofocusing X-ray optics using multislice scalar wave propagation with a complex X-ray refractive index. This approach produces results indistinguishable from methods such as coupled wave theory, and it allows one to reproduce other X-ray optical phenomena such as grazing incidence reflectivity where the direction of energy flow is changed significantly. Just as finite element analysis methods allow engineers to compute the thermal and mechanical responses of arbitrary structures too complex to model by analytical approaches, multislice propagation can be used to understand the properties of the real-world optics of finite extent and with local imperfections, allowing one to better understand the limits to nanoscale X-ray imaging.

© 2017 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
  26. J. M. Cowley and A. F. Moodie, “Fourier images. II. The out-of-focus patterns,” Proc. Phys. Soc. London B 70, 497–504 (1957).
    [Crossref]
  27. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nuc. Data Tab. 54, 181–342 (1993).
    [Crossref]
  28. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
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    [Crossref]
  30. L. Yu, M. C. Huang, M. Z. Chen, W. Z. Chen, W. D. Huang, and Z. Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23, 409–411 (1998).
    [Crossref]
  31. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004).
    [Crossref]
  32. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Ant. Prop. 14, 302–307 (1966).
    [Crossref]
  33. A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Elec. Compat. EMC-22, 191–202 (1980).
    [Crossref]
  34. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Di fference Time-Domain Method (Artech House, 2000), 2 ed.
  35. B. L. Henke, “Ultrasoft-x-ray reflection, refraction, and production of photoelectrons (100-1000-eV region),” Phys. Rev. A 6, 94–104 (1972).
    [Crossref]
  36. E. Spiller, “Low-loss reflection coatings using absorbing matrials,” App. Phys. Lett. 20, 365–367 (1972).
    [Crossref]
  37. M. J. Simpson and A. G. Michette, “The effects of manufacturing inaccuracies on the imaging propertiesyee of Fresnel zone plates,” Opt. Acta 30, 1455–1462 (1983).
    [Crossref]
  38. C. Pratsch, S. Rehbein, S. Werner, and G. Schneider, “Influence of random zone positioning errors on the resolving power of Fresnel zone plates,” Opt. Express 22, 30482–30491 (2014).
    [Crossref]
  39. K. Li, M. J. Wojcik, L. E. Ocola, R. Divan, and C. Jacobsen, “Multilayer on-chip stacked Fresnel zone plates: Hard x-ray fabrication and soft x-ray simulations,” J. Vac. Sci. Tech. B 33, 06FD04 (2015).
    [Crossref]

2015 (2)

K. Li, M. J. Wojcik, L. E. Ocola, R. Divan, and C. Jacobsen, “Multilayer on-chip stacked Fresnel zone plates: Hard x-ray fabrication and soft x-ray simulations,” J. Vac. Sci. Tech. B 33, 06FD04 (2015).
[Crossref]

K. Li and C. Jacobsen, “Rapid calculation of paraxial wave propagation for cylindrically symmetric optics,” J. Opt. Soc. Am. A 32, 2074–2081 (2015).
[Crossref]

2014 (3)

C. Pratsch, S. Rehbein, S. Werner, and G. Schneider, “Influence of random zone positioning errors on the resolving power of Fresnel zone plates,” Opt. Express 22, 30482–30491 (2014).
[Crossref]

M. Selin, E. Fogelqvist, A. Holmberg, P. Guttmann, U. Vogt, and H. M. Hertz, “3D simulation of the image formation in soft x-ray microscopes,” Opt. Exp. 22, 30756 (2014).
[Crossref]

C. Chang and A. Sakdinawat, “Ultra-high aspect ratio high-resolution nanofabrication for hard x-ray diffractive optics,” Nat. Comm. 5, 4243 (2014).
[Crossref]

2013 (1)

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. K. Robinson, and Y. S. Chu, “11 nm hard x-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

2012 (1)

W. Chao, P. Fischer, T. Tyliszczak, S. Rekawa, E. Anderson, and P. Naulleau, “Real space soft x-ray imaging at 10 nm spatial resolution,” Opt. Exp. 20, 9777–9783 (2012).
[Crossref]

2011 (1)

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
[Crossref] [PubMed]

2010 (2)

A. Sakdinawat and D. T. Attwood, “Nanoscale x-ray imaging,” Nat. Phot. 4, 840–848 (2010).
[Crossref]

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[Crossref]

2009 (1)

X. Huang, H. Miao, J. Steinbrener, J. Nelson, D. A. Shapiro, A. Stewart, J. Turner, and C. Jacobsen, “Signal-to-noise and radiation exposure considerations in conventional and diffraction x-ray microscopy,” Opt. Exp. 17, 13541–13553 (2009).
[Crossref]

2007 (2)

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

H. Yan, J. Maser, A. Macrander, Q. Shen, S. Vogt, G. B. Stephenson, and H. Kang, “Takagi-Taupin description of x-ray dynamical diffraction from diffractive optics with large numerical aperture,” Phys. Rev. B 76, 115438 (2007).
[Crossref]

2006 (1)

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Cryst. A 62, 248–261 (2006).
[Crossref]

2005 (1)

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

2004 (1)

2002 (1)

1998 (2)

L. Yu, M. C. Huang, M. Z. Chen, W. Z. Chen, W. D. Huang, and Z. Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. 23, 409–411 (1998).
[Crossref]

Y. Wang and C. Jacobsen, “A numerical study of resolution and contrast in soft x-ray contact microscopy,” J. Micros. 191, 159–169 (1998).
[Crossref]

1997 (2)

G. Schneider, “Zone plates with high efficiency in high orders of diffraction described by dynamical theory,” Appl. Phys. Lett. 71, 2242–2244 (1997).
[Crossref]

D. Smith, “The realization of atomic resolution with the electron microscope,” Rep. Prog. Phys. 60, 1513–1580 (1997).
[Crossref]

1995 (1)

Y. V. Kopylov, A. V. Popov, and A. V. Vinogradov, “Application of the parabolic wave equation to X-ray diffraction optics,” Opt. Comm. 118, 619–636 (1995).
[Crossref]

1994 (1)

A. R. Hare and G. R. Morrison, “Near-field soft X-ray diffraction modelled by the multislice method,” J. Mod. Opt. 41, 31–48 (1994).
[Crossref]

1993 (1)

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nuc. Data Tab. 54, 181–342 (1993).
[Crossref]

1992 (1)

J. Maser and G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Comm. 89, 355–362 (1992).
[Crossref]

1983 (1)

M. J. Simpson and A. G. Michette, “The effects of manufacturing inaccuracies on the imaging propertiesyee of Fresnel zone plates,” Opt. Acta 30, 1455–1462 (1983).
[Crossref]

1980 (1)

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Elec. Compat. EMC-22, 191–202 (1980).
[Crossref]

1974 (1)

1972 (2)

B. L. Henke, “Ultrasoft-x-ray reflection, refraction, and production of photoelectrons (100-1000-eV region),” Phys. Rev. A 6, 94–104 (1972).
[Crossref]

E. Spiller, “Low-loss reflection coatings using absorbing matrials,” App. Phys. Lett. 20, 365–367 (1972).
[Crossref]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Ant. Prop. 14, 302–307 (1966).
[Crossref]

1957 (2)

J. M. Cowley and A. F. Moodie, “Fourier images. II. The out-of-focus patterns,” Proc. Phys. Soc. London B 70, 497–504 (1957).
[Crossref]

J. M. Cowley and A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London B 70, 486–496 (1957).
[Crossref]

1923 (1)

A. H. Compton, “The total reflexion of x-rays,” Phil. Mag. (Ser. 6) 45, 1121–1131 (1923).
[Crossref]

Anderson, E.

W. Chao, P. Fischer, T. Tyliszczak, S. Rekawa, E. Anderson, and P. Naulleau, “Real space soft x-ray imaging at 10 nm spatial resolution,” Opt. Exp. 20, 9777–9783 (2012).
[Crossref]

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

Attwood, D.

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

Attwood, D. T.

A. Sakdinawat and D. T. Attwood, “Nanoscale x-ray imaging,” Nat. Phot. 4, 840–848 (2010).
[Crossref]

Bouet, N.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. K. Robinson, and Y. S. Chu, “11 nm hard x-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

Budai, J. D.

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
[Crossref] [PubMed]

Chang, C.

C. Chang and A. Sakdinawat, “Ultra-high aspect ratio high-resolution nanofabrication for hard x-ray diffractive optics,” Nat. Comm. 5, 4243 (2014).
[Crossref]

Chao, W.

W. Chao, P. Fischer, T. Tyliszczak, S. Rekawa, E. Anderson, and P. Naulleau, “Real space soft x-ray imaging at 10 nm spatial resolution,” Opt. Exp. 20, 9777–9783 (2012).
[Crossref]

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

Chen, M. Z.

Chen, W. Z.

Chu, Y. S.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. K. Robinson, and Y. S. Chu, “11 nm hard x-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

Compton, A. H.

A. H. Compton, “The total reflexion of x-rays,” Phil. Mag. (Ser. 6) 45, 1121–1131 (1923).
[Crossref]

Conley, R.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. K. Robinson, and Y. S. Chu, “11 nm hard x-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

J. Maser, G. Stephenson, S. Vogt, W. Yun, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, vol. 5539, A. Snigirev and D. Mancini, eds. (SPIE, 2004), vol. 5539, pp. 185–194.
[Crossref]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier images. I. The point source,” Proc. Phys. Soc. London B 70, 486–496 (1957).
[Crossref]

J. M. Cowley and A. F. Moodie, “Fourier images. II. The out-of-focus patterns,” Proc. Phys. Soc. London B 70, 497–504 (1957).
[Crossref]

David, C.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nuc. Data Tab. 54, 181–342 (1993).
[Crossref]

Divan, R.

K. Li, M. J. Wojcik, L. E. Ocola, R. Divan, and C. Jacobsen, “Multilayer on-chip stacked Fresnel zone plates: Hard x-ray fabrication and soft x-ray simulations,” J. Vac. Sci. Tech. B 33, 06FD04 (2015).
[Crossref]

Elser, V.

P. Thibault, V. Elser, C. Jacobsen, D. Shapiro, and D. Sayre, “Reconstruction of a yeast cell from x-ray diffraction data,” Acta Cryst. A 62, 248–261 (2006).
[Crossref]

Eom, D.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. K. Robinson, and Y. S. Chu, “11 nm hard x-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

Fischer, P.

W. Chao, P. Fischer, T. Tyliszczak, S. Rekawa, E. Anderson, and P. Naulleau, “Real space soft x-ray imaging at 10 nm spatial resolution,” Opt. Exp. 20, 9777–9783 (2012).
[Crossref]

Fogelqvist, E.

M. Selin, E. Fogelqvist, A. Holmberg, P. Guttmann, U. Vogt, and H. M. Hertz, “3D simulation of the image formation in soft x-ray microscopes,” Opt. Exp. 22, 30756 (2014).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Guizar-Sicairos, M.

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nuc. Data Tab. 54, 181–342 (1993).
[Crossref]

Gutiérrez-Vega, J. C.

Guttmann, P.

M. Selin, E. Fogelqvist, A. Holmberg, P. Guttmann, U. Vogt, and H. M. Hertz, “3D simulation of the image formation in soft x-ray microscopes,” Opt. Exp. 22, 30756 (2014).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Di fference Time-Domain Method (Artech House, 2000), 2 ed.

Handa, S.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).
[Crossref]

Harder, R.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. K. Robinson, and Y. S. Chu, “11 nm hard x-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

Hare, A. R.

A. R. Hare and G. R. Morrison, “Near-field soft X-ray diffraction modelled by the multislice method,” J. Mod. Opt. 41, 31–48 (1994).
[Crossref]

Harteneck, B.

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

Henke, B. L.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray interactions: Photoabsorption, scattering, transmission, and reflection at E=50–30,000 eV, Z=1–92,” At. Data Nuc. Data Tab. 54, 181–342 (1993).
[Crossref]

B. L. Henke, “Ultrasoft-x-ray reflection, refraction, and production of photoelectrons (100-1000-eV region),” Phys. Rev. A 6, 94–104 (1972).
[Crossref]

Hertz, H. M.

M. Selin, E. Fogelqvist, A. Holmberg, P. Guttmann, U. Vogt, and H. M. Hertz, “3D simulation of the image formation in soft x-ray microscopes,” Opt. Exp. 22, 30756 (2014).
[Crossref]

Holmberg, A.

M. Selin, E. Fogelqvist, A. Holmberg, P. Guttmann, U. Vogt, and H. M. Hertz, “3D simulation of the image formation in soft x-ray microscopes,” Opt. Exp. 22, 30756 (2014).
[Crossref]

Huang, M. C.

Huang, W. D.

Huang, X.

X. Huang, H. Yan, E. Nazaretski, R. Conley, N. Bouet, J. Zhou, K. Lauer, L. Li, D. Eom, D. Legnini, R. Harder, I. K. Robinson, and Y. S. Chu, “11 nm hard x-ray focus from a large-aperture multilayer Laue lens,” Sci. Rep. 3, 3562 (2013).
[PubMed]

X. Huang, H. Miao, J. Steinbrener, J. Nelson, D. A. Shapiro, A. Stewart, J. Turner, and C. Jacobsen, “Signal-to-noise and radiation exposure considerations in conventional and diffraction x-ray microscopy,” Opt. Exp. 17, 13541–13553 (2009).
[Crossref]

Ice, G. E.

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the method of multislice propagation, used to simulate a wavefield propagating though a non-homogeneous refractive medium [26]. Along the beam direction, the object is represented by a series of slices separated by distances Δz. At the entrance of a slice, the incident wavefield ψj is first modulated by the refractive effects of the slab of material, leading to a modified wavefield ψ j at the same plane. This wavefield is then propagated to the next slab entrance, yielding the next slice’s wavefield of ψj+1. Eventually one arrives at the exit wave leaving the medium, which can then be transferred to a far-field diffraction pattern using free-space propagation.

Fig. 2
Fig. 2

Calculating x-ray reflectivity using multislice propagation. The method of the calculation is shown in (a), where complications of diffraction from beam-defining apertures (as shown in c and d) are removed. A 10 keV x-ray plane wave is incident at the same grazing angle θ on gold reflective surfaces angled from both the bottom and the top. Within regions I1 and I3, a fraction r of the beam is reflected off of mirror surfaces, while the beam in region I3 is unaffected. The output region I4 contains the interference between rI1, I2, and rI3 which leads to local maxima and minima due to beam interference, but the total energy in I4 is unaffected by these local redistributions. As a result, we have r(I1 + I3) + I2 = I4, or r = (I4I2)/(I1 + I3) for the reflectivity of the optical surfaces. The resulting reflectivity curve r(θ) obtained by multislice propagation is shown in (b), plotted alongside the classical formula obtained by Henke [35] for unpolarized radiation (polarization effects are unnoticeably small in this case). As can be seen, the two curves are in excellent agreement, with only small differences of about 2–3% reflectivity at grazing incidence angles a bit below the critical angle of 2 δ = 7.73 mrad for Au at 10 keV. This shows that multislice propagation can reproduce even very strong x-ray beam effects due to refractive index variations. In (c) and (d), a 10 keV x-ray plane wave is incident on a 2 µm wide slit, followed by a gold reflective surface set at either 4 mrad (c) or 11 mrad (d) grazing angle of incidence. Dashed lines indicate the mirror surfaces. Multislice shows multiple optical effects in one calculation: diffraction from the slit, the angular dependence of x-ray reflectivity, standing wave phenomena above the surface of the mirror (magnified in the inset sub-images), and x-ray beam penetration over a limited distance into the mirror surface.

Fig. 3
Fig. 3

Simulation of the effects of surface roughness on x-ray reflectivity. Because the multislice method can deal with any arbitrary optical object, in (a) we show three simulated surface profiles with Gaussian-distributed departures from a flat with standard deviations of σ = 0, 0.5, 1.0, and 2.0 nm RMS. They come from an array of Gaussian-distributed random numbers followed by doing rolling average over 5 points and rescaling to the desired standard deviations. Then two more times of rolling average over 250–1000 points are carried out for better local smoothness and then the arrays are rescaled again. These four different surface profiles were then used in a multislice calculation (Δx = 0.1 nm and Δz = 10 nm) for grazing incidence reflection of 10 keV x-rays from a gold surface using the approach of Fig. 2(a), yielding the multislice reflectivity curves shown here in (b); shown on the same plot is the theoretical reflectivity [35] multiplied by the roughness correction factors of Eq. (5) for each value of σ. The exit wave was then propagated to the far-field, yielding an angular distribution of light set first by the finite mirror width giving a sin(θ)/θ distribution for σ = 0.0 nm, and then further redistribution of light into larger angles as the roughness is increased to σ = 0.5, 1.0, and 2.0 nm.

Fig. 4
Fig. 4

Comparison of multislice and CWT calculations of diffraction from gratings and zone plates. The CWT result was calculated using the approach described by Schneider et al. [13] for conditions in which simple scalar diffraction theory gives inaccurate results. In (a) we show that multislice can reproduce the results of an example CWT calculation [13, Fig. 8.10] showing non-scalar soft x-ray diffraction efficiency from a constant period grating slanted at the Bragg angle that would be used in a zone plate with an imaging magnification of M = 1000×. In (b) we show the diffraction efficiency for a t = 6 µm thick Au grating oriented along the beam direction (that is, without the grating tilted to meet the Bragg condition) as a function of grating bar width, and with an increasing number N of depth slices used for the multislice calculation. As can be seen, increasing the number of slices to N = 64 (with a slice thickness of Δz = 94 nm, as expected from Eq. (6) allows one to nicely reproduce the CWT results. In (c) we show the comparison for CWT for the 10 keV x-ray exit wave from an infinite grating of t = 6 µm Au with 29 nm:29 nm line:space ratio, against the end effects of a finite width grating (a situation that CWT cannot address). In (d) we show the effects of propagating this exit wave to the far field where one can measure both total transmittance and first-order diffraction efficiency for the infinite grating using CWT and the finite grating using increasing number of slices N in the multislice approach. Again, multislice with N = 64 nicely reproduces the CWT result (and differs from the simple scalar diffraction result), with a slight loss in transmittance due to the edge effects shown in (c).

Fig. 5
Fig. 5

Comparison of the multislice approach with increasing number of slices N against CWT for a case where scalar diffraction theory gives inaccurate results. This calculation is done for a t = 6 µm thick Au grating oriented along the beam direction (that is, without the grating tilted to meet the Bragg condition) with 10 keV x-ray illumination; at this thickness, scalar diffraction theory gives incorrect results for small grating periods. In (a), we show the calculated exit wave (the wavefield leaving the downstream side of the grating) as a function of increasing the number N of depth slices (or, equivalently, decreasing the multislice slab thickness Δz of Eq. (6) along with the CWT result. In (b) we show the RMS error in intensity and phase of the exit wave compared with the CWT result for several different grating thicknesses t. The phase error shows the most systematic trend of error versus thickness, with N = 64 slices already achieving a RMS phase error of less than 50 mrad in all cases. The diffraction efficiency calculation as a function of zone width drn for 1:1 line:space ratio, along with the scalar diffraction efficiency result [7], was shown in Fig. 4(b), which also showed that N = 64 slices gave accurate results for these parameters.

Fig. 6
Fig. 6

Multislice propagation allows one to simulate various flaws in the nanofabrication process, and see what their effects are on Fresnel zone plate diffraction efficiency. In (a), we show “normal” zones with the desired profile, surrounded on either side by two different examples of zone collapse: “collapsed plating” where the as-fabricated zones collapse on each other after electroplating, or “collapsed mold” where the plating mold collapses prior to electroplating of zone structures. In (b), we show the results of an increasing fraction of zones suffering either type of collapse, leading to a reduction of diffraction efficiency and an increase in the focal spot size when the zone plate is illuminated by a plane wave. The focus size is determined by the position of the first minimum from the axis. This calculation was carried out for Au regular zone plate of 2 µm thick, 40 µm diameter and 20 nm outermost zone width, illuminated by 10 keV x-rays.

Fig. 7
Fig. 7

Multislice propagation simulations from a regular Fresnel zone plate, and from a zone-doubled zone plate [9]. This is representative of a calculation for which CWT cannot be used, since the zone width (and, in the case of zone-doubled zone plates) the projected material composition within a zone changes from zone to zone. In this example two zone plates with t = 2 µm thickness and drN = 20 nm outermost zone width for use with 10 keV x-rays were compared: a standard Fresnel zone plate fabricated in Au, and a zone-doubled zone plate fabricated in Si with 20 nm of Ir deposited using Atomic Layer Deposition (ALD). In (a) we show the exit waves from these two zone plates calculated using multislice propagation with N = 64 slices, revealing very different characteristics. In (b) we show the radial intensity profile, as well as the radially-integrated energy distribution, indicating that zone-doubling produces a somewhat smaller focus spot due to lower diffraction efficiency from the coarser, inner-most zones with a line:space ratio that is far from the desired 1:1 value. In practice, the zone-doubling method allows one to produce higher spatial resolution zone plates because the electron beam fabricated structures can have lower grating density with fewer limitations due to electron beam sidescattering in photoresists and structure collapse during the nanofabrication process. The same radial intensity profile is shown as a 2D image in (c).

Fig. 8
Fig. 8

Partial filing of voxels through which the interface between two media passes. The fraction c of the voxel filled with material a is calculated, whether or not the interface is parallel (i), or perpendicular (ii), or in some other direction relative to the voxel faces. This approach was used in generating Fig. 2. As a test, a 1µm-thick gold grating structure illuminated by 10 keV x-rays was shifted by 0.0, 0.5, and 1.0 pixels relative to the voxel boundaries with Δx = 2 nm pixel size and Δz = 2 nm step size in the propagation direction. The grating exit waves (b) and far-field diffraction intensities (c) show the same characteristics no matter where the actual refractive index boundaries lie, as expected.

Tables (1)

Tables Icon

Table 1 Critical angles in mrad for grazing incidence x-ray reflection as calculated first using the theoretical result of 2 δ, and then as calculated at the maximum slope in the multislice reflectivity curve generated as shown in Fig. 2.

Equations (8)

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n ( x , y , z ) = 1 δ ( x , y , z ) i β ( x , y , z ) ,
ψ j ( x , y ) = ψ j ( x , y ) exp [ i 2 π Δ z λ ( δ ( x , y , z j ) + i β ( x , y , z j ) ) ]
ψ j + 1 ( x , y ) = 1 { { ψ j ( x , y ) } exp [ i 2 π Δ z λ 1 λ 2 ( u x 2 + u y 2 ) ] }
ψ j + 1 ( u x λ z , u y λ z ) = [ ψ j ( x , y ) exp ( i π x 2 + y 2 λ z ) ] i λ z exp [ i π λ z ( u x 2 + u y 2 ) ] .
η σ = exp [ ( 2 π 2 σ sin θ λ ) 2 ] = exp [ ( 4 π σ sin θ / λ ) 2 ]
Δ z = ϵ 2 ϵ 1 2 Δ x 2 λ .
ψ = ψ exp [ i 2 π c Δ z λ ( δ a + i β a ) ] exp [ i 2 π ( 1 c ) Δ z λ ( δ b + i β b ) ]
ψ = ψ { c exp [ i 2 π Δ z λ ( δ a + i β a ) ] + ( 1 c ) exp [ i 2 π Δ z λ ( δ b + i β b ) ] } .

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