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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 12 — Jun. 6, 2011
  • pp: 11120–11127
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High efficiency nano-focusing kinoform optics for synchrotron radiation

L. Alianelli, K. J. S. Sawhney, R. Barrett, I. Pape, A. Malik, and M. C. Wilson  »View Author Affiliations


Optics Express, Vol. 19, Issue 12, pp. 11120-11127 (2011)
http://dx.doi.org/10.1364/OE.19.011120


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Abstract

Modern synchrotron sources have provided for decades intense beams of photons over a large energy spectrum. The availability of improved optics and detectors has opened up new opportunities for the study of matter at the micrometre and nanometre scale in many disciplines. Whilst exploitation of micro-focused beams occurs almost daily in many beamlines, the production of beams of 100 nm is achieved on few instruments which use specialised optics. Refractive lenses, zone plates, curved mirrors, multilayers, and multilayer Laue lenses, can all focus x-rays to less than 50 nm under strict beam stability conditions. Focusing the synchrotron radiation to beam sizes smaller than 10 nm is considered the ultimate goal for the current decade. Silicon micro-technology has so far provided some of the most advanced x-ray refractive lenses; we report on design and characterisation of a novel silicon kinoform lens that is capable of delivering nano-beams with high efficiency.

© 2011 OSA

1. Introduction

One of the key successes of synchrotron facilities is that they provide access to scientists from a wide range of disciplines in chemistry, life science and material science. Optics development has enabled in the past 20 years experiments with highly brilliant and coherent beams of hard x-rays. Hard x-ray nano-probes benefit studies of complex, inhomogeneous and disordered materials, with applications in microelectronics, material science, biology, geology, cultural heritage, environmental and earth science. The following (non-exhaustive) list of scientific applications of micro- and nano-focused beams justifies the fact that many large synchrotron facilities are investing in the development of focusing optics to be used in nano-probe beamlines where users will have access to hard x-ray beams of 10 – 30 nm in size: (i) nano-beams give local information, at the nano-scale rather than averaged over several nano-objects; therefore they are required in investigation of samples that are inhomogeneous or very small like whiskers and nano-wires; (ii) smaller and smaller beams are used in studies of adsorption onto particulate samples deposited on surfaces. In many cases some sample areas show well ordered and crystalline structures in small regions whilst other regions are disordered. It is particularly interesting to determine the structure of the ordered regions but also to be able to texture map the interface to observe the transition between the regions; (iii) imaging and diffraction with very hard x-rays are used for experiments in structural and process engineering. Stress, strain, deformation behaviour and phase transformations are analysed in samples of industrial interest using either highly collimated or highly focused beams; (iv) high resolution diffraction at medium x-ray energy for material science and magnetism can require sub-micron beams in studies of individual crystallites or domains. A technique that would benefit from an x-ray nano-probe is resonant diffraction applied to the study of anti-ferromagnetic domains, ferromagnetic domains and potentially orbital-ordering domains; (v) high resolution beams are required in diamond anvil cell experiments as minute sample areas (1-10 µm in dimension) are often used in extreme conditions studies; (vi) finally, focus quality evaluation would make kinoform optics, discussed in this paper, suitable for imaging and coherent diffraction experiments as they improve signal to noise ratio in experiments such as x-ray photon correlation spectroscopy and reflectometry [1

1. A. R. Sandy, S. Narayanan, M. Sprung, J.-D. Su, K. Evans-Lutterodt, A. F. Isakovic, and A. Stein, “Kinoform optics applied to X-ray photon correlation spectroscopy,” J. Synchrotron Radiat. 17(3), 314–320 (2010). [CrossRef] [PubMed]

,2

2. M. K. Tiwari, L. Alianelli, I. P. Dolbnya, and K. J. S. Sawhney, “Application of kinoform lens for X-ray reflectivity analysis,” J. Synchrotron Radiat. 17(2), 237–242 (2010). [CrossRef] [PubMed]

].

In this paper we present recent work on kinoform lenses development, and suggest that they have reached sufficient maturity to be used as proper high flux nano-focusing devices.

It is understood that reflective and refractive optics with very short focal lengths can introduce geometrical aberrations when the curvature is not optimized for the shape of the incident wave front. The nano-focusing lenses described in this paper, in addition to the phase profile that minimises absorption, are designed with a double refractive surface, which improves the collimation of the beam prior to nano-focusing. The lenses were tested at the ID06 undulator beamline at the ESRF. The smallest focal length was f = 75 mm, and the measured focused spot full width at half maximum (fwhm) was s = 225 nm. The focal spot quality is satisfactory, and a peak-to-background ratio larger than 100 was measured.

2. Double kinoform lens system for nano-focusing

Kinoform lenses are semi-transparent refractive objects, in which the profile is optimised for a specific radiation wavelength and the transmission is maximised due to the very small amount of lens material [3

3. K. Evans-Lutterodt, J. Ablett, A. Stein, C. C. Kao, D. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. Gammel, H. Huggins, G. Bogart, S. Ustin, and L. Ocola, “Single-element elliptical hard x-ray micro-optics,” Opt. Express 11(8), 919–926 (2003). [CrossRef] [PubMed]

]. The focal length of a lens with a single refractive surface, in the thin lens approximation, is f = R / δ where R is the radius of curvature at the apex of the surface, and δ is the refractive index real part decrement δ = Re (1 - n). The principal obstacles to the use of refractive optics to focus x-rays are the smallness of δ ~10−5 – 10−6, and the large absorption. Synchrotron focusing lenses have to be fabricated with very strong curvatures in materials with low Z number. A kinoform profile minimizes absorption. The profile is obtained by setting the length of the kinoform steps equal to multiples ofL=λ/δ, where λ is the x-ray wavelength. This ensures that each part of the kinoform lens which is illuminated by a collimated monochromatic wave, refracts the wave without introducing phase changes, apart from a factor multiple of 2π.

New generation kinoform focusing lenses were developed with improved design and fabrication [3

3. K. Evans-Lutterodt, J. Ablett, A. Stein, C. C. Kao, D. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. Gammel, H. Huggins, G. Bogart, S. Ustin, and L. Ocola, “Single-element elliptical hard x-ray micro-optics,” Opt. Express 11(8), 919–926 (2003). [CrossRef] [PubMed]

,4

4. K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99(13), 134801 (2007). [CrossRef] [PubMed]

] and micro-focusing of the hard x-ray beam was obtained with a lens with f = 150 mm [3

3. K. Evans-Lutterodt, J. Ablett, A. Stein, C. C. Kao, D. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. Gammel, H. Huggins, G. Bogart, S. Ustin, and L. Ocola, “Single-element elliptical hard x-ray micro-optics,” Opt. Express 11(8), 919–926 (2003). [CrossRef] [PubMed]

]. In this paper we show that a lens with the same focal length, and slightly different design, will provide nano-focusing of the x-rays with good flux transmission over a large aperture. By decreasing the radius of the nano-focusing lens to R = 0.5 µm, rather than using refractive arrays, we have obtained an optics with focal length of f = 75 mm only.

It has been correctly reported that when a single refractive lens is used, the ideal surface to focus a planar wave to a point is elliptical [3

3. K. Evans-Lutterodt, J. Ablett, A. Stein, C. C. Kao, D. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. Gammel, H. Huggins, G. Bogart, S. Ustin, and L. Ocola, “Single-element elliptical hard x-ray micro-optics,” Opt. Express 11(8), 919–926 (2003). [CrossRef] [PubMed]

,5

5. E. Hecht, Optics. 2nd ed. 1987, Reading, MA: Addison-Wesley. [PubMed]

]. However synchrotron radiation is not perfectly collimated, and the presence of thermal deformations of the crystal monochromators usually introduces further wave front distortions. For these reasons, strongly curved optical surfaces should follow a computer generated shape adjusted to the incoming wave front [6

6. M. Sánchez del Río, European Synchrotron Radiation Facility, Grenoble, France (personal communication 2010).

]. A simple solution to decrease aberrations and increase demagnification ratio by the nano-focusing optics, is illustrated in Fig. 1
Fig. 1 Simplified sketch of the nano-focusing system: lens A is designed for collimation, whilst lens B strongly demagnifies the beam. Geometrical aberrations are minimised by the use of two elliptical surfaces. The beam refracted by lens A and incident on lens B is highly collimated. Therefore it is highly parallel to the kinoform steps shown in the SEM in Fig. 2, and phase conservation across the optics is enhanced.
.

Optical design was accomplished using the Shadow ray-tracing code [7

7. C. Welnak, G. J. Chen, and F. Cerrina, “SHADOW: a synchrotron radiation and x-rayoptics simulation tool,” Nucl. Instrum. Methods A347, 344–347 (1994).

] which provides detailed simulation of geometrical focusing. The undulator beam divergence is αundulator = 22 µrad (fwhm); the collimator reduces the divergence of the beam incident on the nano-focusing refractive lens to α < 1 µrad as illustrated in Fig. 3
Fig. 3 Simulation of beam divergence incident on the nano-focusing lens. The solid line is the undulator beam divergence (αundulator = 22 µrad) incident on lens A in Fig. 1. The dotted line is the beam divergence after the collimator lens and incident on lens B (α < 1 µrad).
.

The influence of a phase error introduced by the beam divergence can be estimated by calculating the phase of the x-rays across one or more kinoform steps, as a function of α. The difference in phase between two parallel rays traveling in air and in the material is:

Δϕ=ϕairϕmaterial=2π1nλL=2π.
(1)

When the first ray travels at an angle α the difference in phase is:

Δϕ=ϕairϕmaterial=2πcosαnλL=2π(11cosαδ).
(2)

From Eq. (2) it follows that for very small values of α:

Δϕ~2π(1α22δ).
(3)

The deviation of Δφ from the ideal 2π value is in the range 10−1-10−2 for more divergent beams (bending magnet case) and 10−5-10−4 in the case of very collimated undulator beams. The pre-collimating lens removes these phase errors.

3. Nano-focusing tests and lens efficiency

The kinoform lenses developed at Diamond Light Source have in the past been characterised on bending magnet and undulator beamlines [8

8. L. Alianelli, K.J.S. Sawhney, D.W.K. Jenkins, I.M. Loader, R. Stevens, A. Snigirev and I. Snigireva, Development of Refractive X-ray Focusing Optics at Diamond Light Source” SPIE Proceedings 670507 (2007).

11

11. L. Alianelli, K. J. S. Sawhney, I. Snigireva, A. Snigirev, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “Focusing kinoform lenses: optical design and experimental validation,” AIP Conf. Proc. 1234, 633–636 (2010). [CrossRef]

]. The data in Figs. 4
Fig. 4 High resolution image of lens linear focus (f = 150 mm, E = 8 keV). The lens shadow is shown and part of the direct beam from the slits upstream the lens can be seen.
-7
Fig. 7 Measured transmission by the lenses with f = 150 mm and 75 mm at E = 8 keV. The data are the ratio between transmitted flux and incident flux versus lens aperture. The theoretical peak efficiency of the system is 75%.
summarise results of measurements of vertically focused beam on the ID06 undulator source at the ESRF. A liquid nitrogen cooled Si-111 double crystal, fixed exit monochromator was used to adjust the x-ray energy. The lenses were mounted at distance P = 55 m from the source. Alignment was optimised using an imaging detector with micrometer spatial resolution, placed in the focal plane of the lenses. An image of the line focus showing lens dimensions and part of the direct unfocused beam outside the lens area is illustrated in Fig. 4.

Beam size, depth of focus and lens transmission were measured scanning the edge of a metal wire in the focal plane. The plot in Fig. 5
Fig. 5 Transmitted flux from knife edge scan in the focal plane of lens with f = 75 mm, E = 8 keV. The derivative represents the focused spot profile. The data shown are the raw data, without any correction for wire scan shape etc.
is the signal from the wire, measured in transmission mode. The derivative representing that focus shape has a size s = 225 nm fwhm. Analysis of these data shows that there are two decades intensities between peak intensity and background. This is an indication of good quality of design and fabrication especially when considering that the measurement was made without additional slits or apertures between lens and diagnostic. The measured vertical source size on ID06 is Sundulator = 45 ± 5 µm fwhm [12

12. A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-ray nanointerferometer based on si refractive bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009). [CrossRef] [PubMed]

] and the calculated focused beam size would be less than 200 nm for an effective aperture A ~100 µm. The larger measured beam size is caused by a number of possible unwanted effects, namely: scattering from the scalloping of the lens walls, which could mask a more pronounced peak; vibrations of wire or equipment; lens could be non perfectly aligned.

The plot in Fig. 6
Fig. 6 Focal spot size s (fwhm) versus focal distance. The experimental points summarise data from several measurements at the same beamline ID06 at the ESRF, with x-ray energy in the range E = [812] keV. The solid lines are calculated by convoluting the size of the geometrically demagnified beam with the size of the diffraction limited beam, assuming a lens effective aperture of A = 100 µm (black line), 200 µm (red line), with no diffraction at all (blue line).
illustrates measured focal spot size s versus focusing distance. The solid lines represents the expected demagnified beam sizes, after convolution with the diffraction limited beam size, assuming lens effective apertures A = 100 µm, 200 µm or no diffraction limit at all. The plot indicates that the lenses developed up to now do not produce a diffraction limited spot size. It also shows that aperture is approximately 100 µm: whilst such acceptance value is not too competitive for micro-focusing optics, it would be rather attractive for shorter focal nano-focusing optics. For instance the reported apertures of current nano-focusing silicon planar CRL's and multilayer Laue lenses is in the range 30 - 50 µm [13

13. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. Van der Hart, and M. Kuchler, “Hard x-ray nanoprobe based on refractive x-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005). [CrossRef]

15

15. H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, “Focusing of hard x-rays to 16 nanometers with a multilayer Laue lens,” Appl. Phys. Lett. 92(22), 221114 (2008). [CrossRef]

].

Kinoform lens efficiency is higher than the efficiency expected from planar silicon refractive lenses due to reduced absorption. At 8 keV x-ray energy the theoretical peak efficiency of this kinoform lens system is 75%. The measured transmitted flux is shown in Fig. 7: this was measured by recording the focused flux (without tails, i.e. between 10% and 90% of the maximum intensity in the knife edge scan) versus lens aperture, normalized to the incident flux. Measured peak efficiency is well above 50%. The flux gain G is the ratio of focused flux density to incident flux density. Lens efficiency varies across the aperture, however using the transmission data we obtain G > 200 for A = 200 µm for the lens with f = 150 mm. Transmission, gain and aperture of the new proposed kinoform lenses are larger than previously reported values for similar optics [14

14. A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. D. Michiel, G. Vaughan, V. Kohn, and S. Kuznetsov, “High energy X-ray nanofocusing by silicon planar lenses,” J. Phys.: Conf. Ser. 186, 012072 (2009). [CrossRef]

].

4. Discussion

Both lens efficiency and apertures described in this paper are comparable to or higher than the theoretical efficiency and apertures of high aspect ratio zone plates. This is a striking result when one considers that the lenses were fabricated in an old generation etcher. Lenses with better aspect ratio and reduced roughness or scalloping can be fabricated in modern fast cycle etchers. This implies that increased flux in the focused beam and shorter tails will be obtained with kinoform lenses provided that advanced silicon etchers become available to the synchrotron optics community. The improvements witnessed in recent years in the field of mirror and zone plates fabrication will be matched with refractive lenses when the most advanced silicon technologies are used.

This work shows that tightly focused beams can be obtained using moderate focal lengths. Refractive in-line focusing optics are easily kept and used in vacuum hence they do not require any window between optics and sample. Moreover kinoform lenses described in this paper do not produce unwanted diffraction or higher order focusing [16

16. H. Yan, “X-ray nanofocusing by kinoform lenses: a comparative study using different modeling approaches,” Phys. Rev. B 81(7), 075402 (2010). [CrossRef]

] and can be used as an add-on on many synchrotron experimental end-stations with the only requirement of suitable alignment stages.

Acknowledgements

The kinoform lens development is funded by the STFC through grants ST/F001665 and ST/F001606. We acknowledge the European Synchrotron Radiation Facility for providing beam time on ID06. This work is supported by the NanoFOX JRA within the European Commission I3 Project ELISA.

References and links

1.

A. R. Sandy, S. Narayanan, M. Sprung, J.-D. Su, K. Evans-Lutterodt, A. F. Isakovic, and A. Stein, “Kinoform optics applied to X-ray photon correlation spectroscopy,” J. Synchrotron Radiat. 17(3), 314–320 (2010). [CrossRef] [PubMed]

2.

M. K. Tiwari, L. Alianelli, I. P. Dolbnya, and K. J. S. Sawhney, “Application of kinoform lens for X-ray reflectivity analysis,” J. Synchrotron Radiat. 17(2), 237–242 (2010). [CrossRef] [PubMed]

3.

K. Evans-Lutterodt, J. Ablett, A. Stein, C. C. Kao, D. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. Gammel, H. Huggins, G. Bogart, S. Ustin, and L. Ocola, “Single-element elliptical hard x-ray micro-optics,” Opt. Express 11(8), 919–926 (2003). [CrossRef] [PubMed]

4.

K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99(13), 134801 (2007). [CrossRef] [PubMed]

5.

E. Hecht, Optics. 2nd ed. 1987, Reading, MA: Addison-Wesley. [PubMed]

6.

M. Sánchez del Río, European Synchrotron Radiation Facility, Grenoble, France (personal communication 2010).

7.

C. Welnak, G. J. Chen, and F. Cerrina, “SHADOW: a synchrotron radiation and x-rayoptics simulation tool,” Nucl. Instrum. Methods A347, 344–347 (1994).

8.

L. Alianelli, K.J.S. Sawhney, D.W.K. Jenkins, I.M. Loader, R. Stevens, A. Snigirev and I. Snigireva, Development of Refractive X-ray Focusing Optics at Diamond Light Source” SPIE Proceedings 670507 (2007).

9.

L. Alianelli, K. J. S. Sawhney, M. K. Tiwari, I. P. Dolbnya, R. Stevens, D. W. K. Jenkins, I. M. Loader, M. C. Wilson, and A. Malik, “Germanium and silicon kinoform focusing lenses for hard x-rays,” J. Phys.: Conf. Ser. 186, 012062 (2009). [CrossRef]

10.

L. Alianelli, K. J. S. Sawhney, M. K. Tiwari, I. P. Dolbnya, R. Stevens, D. W. K. Jenkins, I. M. Loader, M. C. Wilson, and A. Malik, “Characterization of germanium linear kinoform lenses at Diamond Light Source,” J. Synchrotron Radiat. 16(Pt 3), 325–329 (2009). [CrossRef] [PubMed]

11.

L. Alianelli, K. J. S. Sawhney, I. Snigireva, A. Snigirev, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “Focusing kinoform lenses: optical design and experimental validation,” AIP Conf. Proc. 1234, 633–636 (2010). [CrossRef]

12.

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-ray nanointerferometer based on si refractive bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009). [CrossRef] [PubMed]

13.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. Van der Hart, and M. Kuchler, “Hard x-ray nanoprobe based on refractive x-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005). [CrossRef]

14.

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. D. Michiel, G. Vaughan, V. Kohn, and S. Kuznetsov, “High energy X-ray nanofocusing by silicon planar lenses,” J. Phys.: Conf. Ser. 186, 012072 (2009). [CrossRef]

15.

H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, “Focusing of hard x-rays to 16 nanometers with a multilayer Laue lens,” Appl. Phys. Lett. 92(22), 221114 (2008). [CrossRef]

16.

H. Yan, “X-ray nanofocusing by kinoform lenses: a comparative study using different modeling approaches,” Phys. Rev. B 81(7), 075402 (2010). [CrossRef]

OCIS Codes
(220.3630) Optical design and fabrication : Lenses
(340.0340) X-ray optics : X-ray optics

ToC Category:
X-ray Optics

History
Original Manuscript: April 18, 2011
Manuscript Accepted: May 6, 2011
Published: May 23, 2011

Citation
L. Alianelli, K. J. S. Sawhney, R. Barrett, I. Pape, A. Malik, and M. C. Wilson, "High efficiency nano-focusing kinoform optics for synchrotron radiation," Opt. Express 19, 11120-11127 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-12-11120


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References

  1. A. R. Sandy, S. Narayanan, M. Sprung, J.-D. Su, K. Evans-Lutterodt, A. F. Isakovic, and A. Stein, “Kinoform optics applied to X-ray photon correlation spectroscopy,” J. Synchrotron Radiat. 17(3), 314–320 (2010). [CrossRef] [PubMed]
  2. M. K. Tiwari, L. Alianelli, I. P. Dolbnya, and K. J. S. Sawhney, “Application of kinoform lens for X-ray reflectivity analysis,” J. Synchrotron Radiat. 17(2), 237–242 (2010). [CrossRef] [PubMed]
  3. K. Evans-Lutterodt, J. Ablett, A. Stein, C. C. Kao, D. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. Gammel, H. Huggins, G. Bogart, S. Ustin, and L. Ocola, “Single-element elliptical hard x-ray micro-optics,” Opt. Express 11(8), 919–926 (2003). [CrossRef] [PubMed]
  4. K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic, A. Taylor, and D. M. Tennant, “Using compound kinoform hard-x-ray lenses to exceed the critical angle limit,” Phys. Rev. Lett. 99(13), 134801 (2007). [CrossRef] [PubMed]
  5. E. Hecht, Optics. 2nd ed. 1987, Reading, MA: Addison-Wesley. [PubMed]
  6. M. Sánchez del Río, European Synchrotron Radiation Facility, Grenoble, France (personal communication 2010).
  7. C. Welnak, G. J. Chen, and F. Cerrina, “SHADOW: a synchrotron radiation and x-rayoptics simulation tool,” Nucl. Instrum. Methods A347, 344–347 (1994).
  8. L. Alianelli, K.J.S. Sawhney, D.W.K. Jenkins, I.M. Loader, R. Stevens, A. Snigirev and I. Snigireva, Development of Refractive X-ray Focusing Optics at Diamond Light Source” SPIE Proceedings 670507 (2007).
  9. L. Alianelli, K. J. S. Sawhney, M. K. Tiwari, I. P. Dolbnya, R. Stevens, D. W. K. Jenkins, I. M. Loader, M. C. Wilson, and A. Malik, “Germanium and silicon kinoform focusing lenses for hard x-rays,” J. Phys.: Conf. Ser. 186, 012062 (2009). [CrossRef]
  10. L. Alianelli, K. J. S. Sawhney, M. K. Tiwari, I. P. Dolbnya, R. Stevens, D. W. K. Jenkins, I. M. Loader, M. C. Wilson, and A. Malik, “Characterization of germanium linear kinoform lenses at Diamond Light Source,” J. Synchrotron Radiat. 16(Pt 3), 325–329 (2009). [CrossRef] [PubMed]
  11. L. Alianelli, K. J. S. Sawhney, I. Snigireva, A. Snigirev, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “Focusing kinoform lenses: optical design and experimental validation,” AIP Conf. Proc. 1234, 633–636 (2010). [CrossRef]
  12. A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-ray nanointerferometer based on si refractive bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009). [CrossRef] [PubMed]
  13. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. Van der Hart, and M. Kuchler, “Hard x-ray nanoprobe based on refractive x-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005). [CrossRef]
  14. A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. D. Michiel, G. Vaughan, V. Kohn, and S. Kuznetsov, “High energy X-ray nanofocusing by silicon planar lenses,” J. Phys.: Conf. Ser. 186, 012072 (2009). [CrossRef]
  15. H. C. Kang, H. Yan, R. P. Winarski, M. V. Holt, J. Maser, C. Liu, R. Conley, S. Vogt, A. T. Macrander, and G. B. Stephenson, “Focusing of hard x-rays to 16 nanometers with a multilayer Laue lens,” Appl. Phys. Lett. 92(22), 221114 (2008). [CrossRef]
  16. H. Yan, “X-ray nanofocusing by kinoform lenses: a comparative study using different modeling approaches,” Phys. Rev. B 81(7), 075402 (2010). [CrossRef]

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