Abstract

We study extreme-ultraviolet wave propagation within optically thick nanostructures by means of high-resolution coherent diffractive imaging using high-harmonic radiation. Exit waves from different objects are reconstructed by phase retrieval algorithms, and are shown to be dominated by waveguiding within the sample. The experiments provide a direct visualization of extreme-ultraviolet guided modes, and demonstrate that multiple scattering is a generic feature in extruded nanoscale geometries. The observations are successfully reproduced in numerical and semi-analytical simulations.

© 2015 Optical Society of America

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2014 (1)

M. Zürch, J. Rothhardt, S. Hadrich, S. Demmler, M. Krebs, J. Limpert, A. Tunnermann, A. Guggenmos, U. Kleineberg, and C. Spielmann, “Real-time and sub-wavelength ultrafast coherent diffraction imaging in the extreme ultraviolet,” Sci. Rep. 4, 7356 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (3)

2011 (1)

2010 (2)

K. S. Raines, S. Salha, R. L. Sandberg, H. Jiang, J. A. Rodriguez, B. P. Fahimian, H. C. Kapteyn, J. Du, and J. Miao, “Three-dimensional structure determination from a single view,” Nature 463, 214–217 (2010).
[Crossref]

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

2009 (1)

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, “Single-shot diffractive imaging with a table-top femtosecond soft x-ray laser-harmonics source,” Phys. Rev. Lett. 103, 028104 (2009).
[Crossref] [PubMed]

2008 (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[Crossref] [PubMed]

C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80, 117–140 (2008).
[Crossref]

2007 (3)

A. Poonawala and P. Milanfar, “Mask design for optical microlithography mdash;an inverse imaging problem,” IEEE Trans. Image Processing 16, 774–788 (2007).
[Crossref]

R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hädrich, D. M. Gaudiosi, J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. Song, J. Miao, Y. Liu, and F. Salmassi, “Lensless diffractive imaging using tabletop coherent high-harmonic soft-x-ray beams,” Phys. Rev. Lett. 99, 098103 (2007).
[Crossref] [PubMed]

S. Marchesini, “Invited article: a [corrected] unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]

2006 (5)

C. Fuhse and T. Salditt, “Propagation of x-rays in ultra-narrow slits,” Optics Communications 265, 140–146 (2006).
[Crossref]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

C. Fuhse and T. Salditt, “Finite-difference field calculations for two-dimensionally confined x-ray waveguides,” Appl. Opt. 45, 4603–4608 (2006).
[Crossref] [PubMed]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys 2, 839–843 (2006).
[Crossref]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[Crossref] [PubMed]

2005 (3)

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. t. Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Problems 21, 37 (2005).
[Crossref]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. USA 102, 15343–15346 (2005).
[Crossref] [PubMed]

2004 (2)

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

J. Spence, U. Weierstall, and M. Howells, “Coherence and sampling requirements for diffractive imaging,” Ultramicroscopy 101, 149–152 (2004).
[Crossref] [PubMed]

2003 (2)

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[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]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1994 (1)

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

1989 (1)

X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompré, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751–5761 (1989).
[Crossref] [PubMed]

1987 (1)

1982 (1)

Abbey, B.

B. Chen, B. Abbey, R. Dilanian, E. Balaur, G. van Riessen, M. Junker, C. Q. Tran, M. W. M. Jones, A. G. Peele, I. McNulty, D. J. Vine, C. T. Putkunz, H. M. Quiney, and K. A. Nugent, “Diffraction imaging: The limits of partial coherence,” Phys. Rev. B 86, 235401 (2012).
[Crossref]

Adams, D. E.

Alkemade, P. F. A.

H. F. Schouten, N. Kuzmin, G. Dubois, T. D. Visser, G. Gbur, P. F. A. Alkemade, H. Blok, G. W. t. Hooft, D. Lenstra, and E. R. Eliel, “Plasmon-assisted two-slit transmission: Young’s experiment revisited,” Phys. Rev. Lett. 94, 053901 (2005).
[Crossref]

Anderson, E. H.

J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval of diffraction patterns from noncrystalline samples using the oversampling method,” Phys. Rev. B 67, 174104 (2003).
[Crossref]

Aquila, A.

Bajt, S.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys 2, 839–843 (2006).
[Crossref]

Balaur, E.

B. Chen, B. Abbey, R. Dilanian, E. Balaur, G. van Riessen, M. Junker, C. Q. Tran, M. W. M. Jones, A. G. Peele, I. McNulty, D. J. Vine, C. T. Putkunz, H. M. Quiney, and K. A. Nugent, “Diffraction imaging: The limits of partial coherence,” Phys. Rev. B 86, 235401 (2012).
[Crossref]

Barthelmess, M.

Barty, A.

A. V. Martin, F. Wang, N. D. Loh, T. Ekeberg, F. R. N. C. Maia, M. Hantke, G. van der Schot, C. Y. Hampton, R. G. Sierra, A. Aquila, S. Bajt, M. Barthelmess, C. Bostedt, J. D. Bozek, N. Coppola, S. W. Epp, B. Erk, H. Fleckenstein, L. Foucar, M. Frank, H. Graafsma, L. Gumprecht, A. Hartmann, R. Hartmann, G. Hauser, H. Hirsemann, P. Holl, S. Kassemeyer, N. Kimmel, M. Liang, L. Lomb, S. Marchesini, K. Nass, E. Pedersoli, C. Reich, D. Rolles, B. Rudek, A. Rudenko, J. Schulz, R. L. Shoeman, H. Soltau, D. Starodub, J. Steinbrener, F. Stellato, L. Strüder, J. Ullrich, G. Weidenspointner, T. A. White, C. B. Wunderer, A. Barty, I. Schlichting, M. J. Bogan, and H. N. Chapman, “Noise-robust coherent diffractive imaging with a single diffraction pattern,” Opt. Express 20, 16650–16661 (2012).
[Crossref]

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High-resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200 (2006).
[Crossref]

H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege, S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider, E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson, D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert, F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond diffractive imaging with a soft-x-ray free-electron laser,” Nat. Phys 2, 839–843 (2006).
[Crossref]

Beerlink, A.

K. Giewekemeyer, P. Thibault, S. Kalbfleisch, A. Beerlink, C. M. Kewish, M. Dierolf, F. Pfeiffer, and T. Salditt, “Quantitative biological imaging by ptychographic x-ray diffraction microscopy,” Proc. Nat. Acad. Sci. U.S.A. 107, 529–534 (2010).
[Crossref]

Beetz, T.

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Figures (8)

Fig. 1
Fig. 1 Schematic of the experimental setup displaying the high-harmonic generation chamber, the diffraction grating and the imaging chamber (CDI chamber). The inset illustrates the modification of the exit wave induced by propagating waveguide modes within the extended structure (three even modes denoted by mode indices).
Fig. 2
Fig. 2 (a) Coherent diffraction patterns (a), (d) on a logarithmic scale, reconstructed exit wave amplitudes (b), (e) in arb. units and phases (c), (f) in radians of objects A (top row) and B (bottom row). Insets in (b), (e) display scanning electron micrographs. Scale bars: 5 μm−1 (a), (d); 1 μm (b), (e). Identical colormaps are used for both objects in (a), (d) and (c), (f).
Fig. 3
Fig. 3 (a) Tilted view SEM image of object A. (b) Magnitude of the reconstructed exit field, indicating positions of cross-sections chosen for comparison with wave-guide simulations. (c) Finite element simulations of the propagation in slab waveguides of dimensions and materials corresponding to experimental conditions (see text). Left column: Evolution of field magnitudes along the propagation direction for three waveguide widths chosen according to the respective lineouts in (a) and (b). Right column: Comparison of simulated exit field magnitudes (dotted red) with experimental profiles (solid blue). Scale bars are 1 μm.
Fig. 4
Fig. 4 Real (top curves) and imaginary (bottom curves) parts of the propagation constant for the first three even TM (solid) and TE (dotted) modes, normalized to the vacuum wave number. Inset: Transmission coefficients through a 700 nm long gold waveguide of different widths.
Fig. 5
Fig. 5 (a) Schematic of the layered structure of object A (top) and B (bottom). (b) Scanning electron micrograph of object A, recorded from the back side (gold). Focused ion beam milling was carried out from the front side. Note that the apparent width of the waveguides depends on the SEM focus settings, which gives rise to an uncertainty in the measured widths of several tens of nanometers.
Fig. 6
Fig. 6 (a) Single acquisition (5 s exposure time) diffraction data; logarithmic color scale. (b) Reconstructed amplitude based on diffraction pattern in (a). (c) Average of 200 individual acquisitions and corresponding reconstruction (d). Scale bars: 1 μm.
Fig. 7
Fig. 7 Phase retrieval transfer function for the reconstructions shown in Fig. 2(b) and 2(c).
Fig. 8
Fig. 8 (a) Bitmap image of the apertures in object A. Insets show a close-up (dashed red box) and a lineout (blue solid). (b) Magnitude of two-dimensional Fourier transform of the binary object. The inner part corresponds to the range of scattering angles collected by the detector area. (c) Inverse Fourier transform after truncation to the detector area. Minor fluctuations at high spatial frequencies are found. The features are clearly distinct from the waveguiding observed in the experiments (e.g., Fig. 3).

Equations (2)

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k w h = 2 arctan 1 γ k w + m π
k w h = 2 tan 1 n w 2 γ n c 2 k w + m π

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