Abstract

The current advances in new generation X-ray sources are calling for the development and improvement of high-performance optics. Techniques for high-sensitivity phase sensing and wavefront characterisation, preferably performed at-wavelength, are increasingly required for quality control, optimisation and development of such devices. We here show that the recently proposed unified modulated pattern analysis (UMPA) can be used for these purposes. We characterised two polymer X-ray refractive lenses and quantified the effect of beam damage and shape errors on their refractive properties. Measurements were performed with two different setups for UMPA and validated with conventional X-ray grating interferometry. Due to its adaptability to different setups, the ease of implementation and cost-effectiveness, we expect UMPA to find applications for high-throughput quantitative optics characterisation and wavefront sensing.

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2017 (2)

Y. Kayser, C. David, U. Flechsig, J. Krempasky, V. Schlott, and R. Abela, “X-ray grating interferometer for in situ and at-wavelength wavefront metrology,” J. of Synchrotron Radiat. 24(1), 150–162 (2017).
[Crossref]

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, “X-ray phase-contrast imaging and metrology through unified modulated pattern analysis,” Phys. Rev. Lett. 118(20), 203903 (2017).
[Crossref] [PubMed]

2016 (3)

Y. Kashyap, H. Wang, and K. Sawhney, “Development of a speckle-based portable device for in situ metrology of synchrotron x-ray mirrors,” J. Synchrotron Radiat. 23(Pt 5), 1131–1136 (2016).
[Crossref] [PubMed]

Y. Kashyap, H. Wang, and K. Sawhney, “Speckle-based at-wavelength metrology of x-ray mirrors with super accuracy,” Rev. Sci. Instrum. 87(5), 052001 (2016).
[Crossref] [PubMed]

F. J. Koch, C. Detlefs, T. J. Schröter, D. Kunka, A. Last, and J. Mohr, “Quantitative characterization of x-ray lenses from two fabrication techniques with grating interferometry,” Opt. Express 24(9), 9168–9177 (2016).
[Crossref] [PubMed]

2015 (4)

H. Wang, J. Sutter, and K. Sawhney, “Advanced in situ metrology for x-ray beam shaping with super precision,” Opt. Express 23(2), 1605–1614 (2015).
[Crossref] [PubMed]

H. Wang, Y. Kashyap, and K. Sawhney, “Speckle based x-ray wavefront sensing with nanoradian angular sensitivity,” Opt. Express 23(18), 23310–23317 (2015).
[Crossref] [PubMed]

S. Berujon, E. Ziegler, and P. Cloetens, “X-ray pulse wavefront metrology using speckle tracking,” J. Synchrotron Radiat. 22(4), 886–894 (2015).
[Crossref] [PubMed]

H. Wang, Y. Kashyap, D. Laundy, and K. Sawhney, “Two-dimensional in situ metrology of x-ray mirrors using the speckle scanning technique,” J. Synchrotron Radiat. 22(4), 925–929 (2015).
[Crossref] [PubMed]

2014 (2)

S. Berujon, H. Wang, S. Alcock, and K. Sawhney, “At-wavelength metrology of hard x-ray mirror using near field speckle,” Opt. Express 22(6), 6438–6446 (2014).
[Crossref] [PubMed]

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise x-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

2013 (3)

H. Wang, S. Berujon, and K. Sawhney, “Development of at-wavelength metrology using grating-based shearing interferometry at Diamond Light Source,” J. Phys. Conf. Ser. 425(5), 052021 (2013).
[Crossref]

S. Berujon, H. Wang, and K. Sawhney, “At-wavelength metrology using the x-ray speckle tracking technique: case study of a x-ray compound refractive lens,” J. Phys. Conf. Ser. 425(5), 052020 (2013).
[Crossref]

K. S. Morgan, P. Modregger, S. C. Irvine, S. Rutishauser, V. A. Guzenko, M. Stampanoni, and C. David, “A sensitive x-ray phase contrast technique for rapid imaging using a single phase grid analyzer,” Opt. Lett. 38(22), 4605–4608 (2013).
[Crossref] [PubMed]

2012 (4)

S. Berujon and E. Ziegler, “Grating-based at-wavelength metrology of hard x-ray reflective optics,” Opt. Lett. 37(21), 4464–4466 (2012).
[Crossref] [PubMed]

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

S. Berujon, H. Wang, and K. Sawhney, “X-ray multimodal imaging using a random-phase object,” Phys. Rev. A 86(6), 063813 (2012).
[Crossref]

H. Wang, S. Berujon, and K. Sawhney, “Characterization of a one dimensional focusing compound refractive lens using the rotating shearing interferometer technique,” AIP Conf. Proc. 1466(1), 223–228 (2012).
[Crossref]

2011 (5)

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett. 99(22), 221104 (2011).
[Crossref]

B. Flöter, P. Juranić, P. Großmann, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedkte, “Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements,” Nucl. Instr. Meth. Phys. Res. A 635(1, Supplement), S108–S112 (2011).
[Crossref]

C. Rau, U. Wagner, Z. Pesic, and A. De Fanis, “Coherent imaging at the Diamond beamline I13,” Phys. Status Solidi A 208(11), 2522–2525 (2011).
[Crossref]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref] [PubMed]

J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive x-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19(22), 21333–21344 (2011).
[Crossref] [PubMed]

2010 (6)

H. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Opt. Lett. 35(12), 1932–1934 (2010).
[Crossref] [PubMed]

C. M. Kewish, M. Guizar-Sicairos, C. Liu, J. Qian, B. Shi, C. Benson, A. M. Khounsary, J. Vila-Comamala, O. Bunk, J. R. Fienup, A. T. Macrander, and L. Assoufid, “Reconstruction of an astigmatic hard x-ray beam and alignment of K-B mirrors from ptychographic coherent diffraction data,” Opt. Express 18(22), 23420–23427 (2010).
[Crossref] [PubMed]

B. Flöter, P. Juranić, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedtke, “EUV Hartmann sensor for wavefront measurements at the free-electron LASer in Hamburg,” New J. Phys. 12(8), 083015 (2010).
[Crossref]

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. 17(3), 299–307 (2010).
[Crossref] [PubMed]

M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instr. Meth. Phys. Res. A 616(2), 162–171 (2010).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105(24), 248102 (2010).
[Crossref]

2009 (1)

E. Reznikova, T. Weitkamp, V. Nazmov, M. Simon, A. Last, and V. Saile, “Transmission hard x-ray microscope with increased view field using planar refractive objectives and condensers made of SU-8 polymer,” J. Phys. Conf. Ser. 186(1), 012070 (2009).
[Crossref]

2008 (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

2007 (3)

E. Reznikova, T. Weitkamp, V. Nazmov, A. Last, M. Simon, and V. Saile, “Investigation of phase contrast hard x-ray microscopy using planar sets of refractive crossed linear parabolic lenses made from SU-8 polymer,” Phys. Status Solidi A 204(8), 2811–2816 (2007).
[Crossref]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 78(9), 093707 (2007).
[Crossref] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating-based differential phase contrast-imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
[Crossref] [PubMed]

2005 (2)

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86(5), 054101 (2005).
[Crossref]

2004 (5)

F. Siewert, T. Noll, T. Schlegel, T. Zeschke, and H. Lammert, “The nanometer optical component measuring machine: a new sub-nm topography measuring device for x-ray optics at BESSY,” AIP Conf. Proc. 705(1), 847–850 (2004).
[Crossref]

V. Nazmov, E. Reznikova, A. Somogyi, J. Mohr, and V. Saile, “Planar sets of cross x-ray refractive lenses from SU-8 polymer,” Proc. SPIE 5539, 235–243 (2004).
[Crossref]

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
[Crossref]

S. C. Mayo and B. Sexton, “Refractive microlens array for wave-front analysis in the medium to hard x-ray range,” Opt. Lett. 29(8), 866–868 (2004).
[Crossref] [PubMed]

2003 (1)

A. Snigirev, I. Snigireva, M. Drakopoulos, V. Nazmov, E. Reznikova, S. Kuznetsov, M. Grigoriev, J. Mohr, and V. Saile, “Focusing properties of x-ray polymer refractive lenses from SU-8 resist layer,” Proc. SPIE 5195, 21–31 (2003).
[Crossref]

2002 (1)

B. Lengeler, C. Schroer, B. Benner, A. Gerhardus, T. F. Günzler, M. Kuhlmann, J. Meyer, and C. Zimprich, “Parabolic refractive x-ray lenses,” J. Synchrotron Radiat. 9(3), 119–124 (2002).
[Crossref] [PubMed]

1999 (1)

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

1997 (2)

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]

1987 (1)

P. Z. Takacs, S.-N. Qian, and J. Colbert, “Design of a long trace surface profiler,” Proc. SPIE 0749, 59–64 (1987).
[Crossref]

1982 (1)

E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
[Crossref]

1971 (1)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

1900 (1)

J. Hartmann, “Bemerkungen über den Bau und die Justierung von Spektrographen,” Z. Instrumentenkd. 20, 47 (1900).

Abela, R.

Y. Kayser, C. David, U. Flechsig, J. Krempasky, V. Schlott, and R. Abela, “X-ray grating interferometer for in situ and at-wavelength wavefront metrology,” J. of Synchrotron Radiat. 24(1), 150–162 (2017).
[Crossref]

Achenbach, S.

V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
[Crossref]

Alcock, S.

Assoufid, L.

Backer, E. W.

E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
[Crossref]

Baruchel, J.

Bauer, G.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. 17(3), 299–307 (2010).
[Crossref] [PubMed]

Baumann, J.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 78(9), 093707 (2007).
[Crossref] [PubMed]

Bech, M.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Benner, B.

B. Lengeler, C. Schroer, B. Benner, A. Gerhardus, T. F. Günzler, M. Kuhlmann, J. Meyer, and C. Zimprich, “Parabolic refractive x-ray lenses,” J. Synchrotron Radiat. 9(3), 119–124 (2002).
[Crossref] [PubMed]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

Bennett, E. E.

Benson, C.

Berujon, S.

S. Berujon, E. Ziegler, and P. Cloetens, “X-ray pulse wavefront metrology using speckle tracking,” J. Synchrotron Radiat. 22(4), 886–894 (2015).
[Crossref] [PubMed]

S. Berujon, H. Wang, S. Alcock, and K. Sawhney, “At-wavelength metrology of hard x-ray mirror using near field speckle,” Opt. Express 22(6), 6438–6446 (2014).
[Crossref] [PubMed]

S. Berujon, H. Wang, and K. Sawhney, “At-wavelength metrology using the x-ray speckle tracking technique: case study of a x-ray compound refractive lens,” J. Phys. Conf. Ser. 425(5), 052020 (2013).
[Crossref]

H. Wang, S. Berujon, and K. Sawhney, “Development of at-wavelength metrology using grating-based shearing interferometry at Diamond Light Source,” J. Phys. Conf. Ser. 425(5), 052021 (2013).
[Crossref]

H. Wang, S. Berujon, and K. Sawhney, “Characterization of a one dimensional focusing compound refractive lens using the rotating shearing interferometer technique,” AIP Conf. Proc. 1466(1), 223–228 (2012).
[Crossref]

S. Berujon, H. Wang, and K. Sawhney, “X-ray multimodal imaging using a random-phase object,” Phys. Rev. A 86(6), 063813 (2012).
[Crossref]

S. Berujon and E. Ziegler, “Grating-based at-wavelength metrology of hard x-ray reflective optics,” Opt. Lett. 37(21), 4464–4466 (2012).
[Crossref] [PubMed]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref] [PubMed]

Betz, H.

E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
[Crossref]

Boerner, M.

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

Brönnimann, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Buchheim, J.

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise x-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Bucourt, S.

M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instr. Meth. Phys. Res. A 616(2), 162–171 (2010).
[Crossref]

Bunk, O.

Cloetens, P.

Colbert, J.

P. Z. Takacs, S.-N. Qian, and J. Colbert, “Design of a long trace surface profiler,” Proc. SPIE 0749, 59–64 (1987).
[Crossref]

David, C.

Y. Kayser, C. David, U. Flechsig, J. Krempasky, V. Schlott, and R. Abela, “X-ray grating interferometer for in situ and at-wavelength wavefront metrology,” J. of Synchrotron Radiat. 24(1), 150–162 (2017).
[Crossref]

K. S. Morgan, P. Modregger, S. C. Irvine, S. Rutishauser, V. A. Guzenko, M. Stampanoni, and C. David, “A sensitive x-ray phase contrast technique for rapid imaging using a single phase grid analyzer,” Opt. Lett. 38(22), 4605–4608 (2013).
[Crossref] [PubMed]

J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive x-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19(22), 21333–21344 (2011).
[Crossref] [PubMed]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref] [PubMed]

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett. 99(22), 221104 (2011).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105(24), 248102 (2010).
[Crossref]

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. 17(3), 299–307 (2010).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating-based differential phase contrast-imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
[Crossref] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 78(9), 093707 (2007).
[Crossref] [PubMed]

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86(5), 054101 (2005).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

De Fanis, A.

C. Rau, U. Wagner, Z. Pesic, and A. De Fanis, “Coherent imaging at the Diamond beamline I13,” Phys. Status Solidi A 208(11), 2522–2525 (2011).
[Crossref]

Detlefs, C.

Di Michiel, M.

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

Diaz, A.

J. Vila-Comamala, A. Diaz, M. Guizar-Sicairos, A. Mantion, C. M. Kewish, A. Menzel, O. Bunk, and C. David, “Characterization of high-resolution diffractive x-ray optics by ptychographic coherent diffractive imaging,” Opt. Express 19(22), 21333–21344 (2011).
[Crossref] [PubMed]

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. 17(3), 299–307 (2010).
[Crossref] [PubMed]

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86(5), 054101 (2005).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

Donath, T.

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett. 99(22), 221104 (2011).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105(24), 248102 (2010).
[Crossref]

Dovillaire, G.

M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instr. Meth. Phys. Res. A 616(2), 162–171 (2010).
[Crossref]

Drakopoulos, M.

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

A. Snigirev, I. Snigireva, M. Drakopoulos, V. Nazmov, E. Reznikova, S. Kuznetsov, M. Grigoriev, J. Mohr, and V. Saile, “Focusing properties of x-ray polymer refractive lenses from SU-8 resist layer,” Proc. SPIE 5195, 21–31 (2003).
[Crossref]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

Ehrfeld, W.

E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
[Crossref]

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Engelhardt, M.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 78(9), 093707 (2007).
[Crossref] [PubMed]

Escolano, L.

M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instr. Meth. Phys. Res. A 616(2), 162–171 (2010).
[Crossref]

Falkenberg, G.

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise x-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Fienup, J. R.

Flechsig, U.

Y. Kayser, C. David, U. Flechsig, J. Krempasky, V. Schlott, and R. Abela, “X-ray grating interferometer for in situ and at-wavelength wavefront metrology,” J. of Synchrotron Radiat. 24(1), 150–162 (2017).
[Crossref]

Flöter, B.

B. Flöter, P. Juranić, P. Großmann, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedkte, “Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements,” Nucl. Instr. Meth. Phys. Res. A 635(1, Supplement), S108–S112 (2011).
[Crossref]

B. Flöter, P. Juranić, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedtke, “EUV Hartmann sensor for wavefront measurements at the free-electron LASer in Hamburg,” New J. Phys. 12(8), 083015 (2010).
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Georgi, S.

A. Last, O. Márkus, S. Georgi, and J. Mohr, “Röntgenoptische Messung des Seitenwandwinkels direktlithografischer refraktiver Röntgenlinsen,” in Proceedings of MEMS, Mikroelektronik, Systeme, Mikrosystemtechnik-Kongress 6 (VDE, 2015), pp. 508–510.

Gerhardus, A.

B. Lengeler, C. Schroer, B. Benner, A. Gerhardus, T. F. Günzler, M. Kuhlmann, J. Meyer, and C. Zimprich, “Parabolic refractive x-ray lenses,” J. Synchrotron Radiat. 9(3), 119–124 (2002).
[Crossref] [PubMed]

Glashauser, W.

E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2004), 3rd ed.

Grigoriev, M.

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

A. Snigirev, I. Snigireva, M. Drakopoulos, V. Nazmov, E. Reznikova, S. Kuznetsov, M. Grigoriev, J. Mohr, and V. Saile, “Focusing properties of x-ray polymer refractive lenses from SU-8 resist layer,” Proc. SPIE 5195, 21–31 (2003).
[Crossref]

Großmann, P.

B. Flöter, P. Juranić, P. Großmann, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedkte, “Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements,” Nucl. Instr. Meth. Phys. Res. A 635(1, Supplement), S108–S112 (2011).
[Crossref]

Grünzweig, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Guigay, J. P.

Guizar-Sicairos, M.

Günzler, T. F.

B. Lengeler, C. Schroer, B. Benner, A. Gerhardus, T. F. Günzler, M. Kuhlmann, J. Meyer, and C. Zimprich, “Parabolic refractive x-ray lenses,” J. Synchrotron Radiat. 9(3), 119–124 (2002).
[Crossref] [PubMed]

Guzenko, V. A.

Hariharan, P.

P. Hariharan, “Interferometric testing of optical surfaces: absolute measurements of flatness,” Opt. Eng. 36(9), 2478–2481 (1997).
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Hartmann, J.

J. Hartmann, “Bemerkungen über den Bau und die Justierung von Spektrographen,” Z. Instrumentenkd. 20, 47 (1900).

Heuberger, A.

E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
[Crossref]

Idir, M.

M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instr. Meth. Phys. Res. A 616(2), 162–171 (2010).
[Crossref]

Irvine, S. C.

Juranic, P.

B. Flöter, P. Juranić, P. Großmann, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedkte, “Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements,” Nucl. Instr. Meth. Phys. Res. A 635(1, Supplement), S108–S112 (2011).
[Crossref]

B. Flöter, P. Juranić, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedtke, “EUV Hartmann sensor for wavefront measurements at the free-electron LASer in Hamburg,” New J. Phys. 12(8), 083015 (2010).
[Crossref]

Kapitzki, S.

B. Flöter, P. Juranić, P. Großmann, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedkte, “Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements,” Nucl. Instr. Meth. Phys. Res. A 635(1, Supplement), S108–S112 (2011).
[Crossref]

B. Flöter, P. Juranić, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedtke, “EUV Hartmann sensor for wavefront measurements at the free-electron LASer in Hamburg,” New J. Phys. 12(8), 083015 (2010).
[Crossref]

Kashyap, Y.

Y. Kashyap, H. Wang, and K. Sawhney, “Development of a speckle-based portable device for in situ metrology of synchrotron x-ray mirrors,” J. Synchrotron Radiat. 23(Pt 5), 1131–1136 (2016).
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Y. Kashyap, H. Wang, and K. Sawhney, “Speckle-based at-wavelength metrology of x-ray mirrors with super accuracy,” Rev. Sci. Instrum. 87(5), 052001 (2016).
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H. Wang, Y. Kashyap, D. Laundy, and K. Sawhney, “Two-dimensional in situ metrology of x-ray mirrors using the speckle scanning technique,” J. Synchrotron Radiat. 22(4), 925–929 (2015).
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H. Wang, Y. Kashyap, and K. Sawhney, “Speckle based x-ray wavefront sensing with nanoradian angular sensitivity,” Opt. Express 23(18), 23310–23317 (2015).
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Kayser, Y.

Y. Kayser, C. David, U. Flechsig, J. Krempasky, V. Schlott, and R. Abela, “X-ray grating interferometer for in situ and at-wavelength wavefront metrology,” J. of Synchrotron Radiat. 24(1), 150–162 (2017).
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Keitel, B.

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C. Kottler, C. David, F. Pfeiffer, and O. Bunk, “A two-directional approach for grating-based differential phase contrast-imaging using hard x-rays,” Opt. Express 15(3), 1175–1181 (2007).
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H. Wang, Y. Kashyap, D. Laundy, and K. Sawhney, “Two-dimensional in situ metrology of x-ray mirrors using the speckle scanning technique,” J. Synchrotron Radiat. 22(4), 925–929 (2015).
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M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instr. Meth. Phys. Res. A 616(2), 162–171 (2010).
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Márkus, O.

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M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instr. Meth. Phys. Res. A 616(2), 162–171 (2010).
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Mohr, J.

F. J. Koch, C. Detlefs, T. J. Schröter, D. Kunka, A. Last, and J. Mohr, “Quantitative characterization of x-ray lenses from two fabrication techniques with grating interferometry,” Opt. Express 24(9), 9168–9177 (2016).
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V. Nazmov, E. Reznikova, A. Somogyi, J. Mohr, and V. Saile, “Planar sets of cross x-ray refractive lenses from SU-8 polymer,” Proc. SPIE 5539, 235–243 (2004).
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V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
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A. Last, O. Márkus, S. Georgi, and J. Mohr, “Röntgenoptische Messung des Seitenwandwinkels direktlithografischer refraktiver Röntgenlinsen,” in Proceedings of MEMS, Mikroelektronik, Systeme, Mikrosystemtechnik-Kongress 6 (VDE, 2015), pp. 508–510.

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Münchmeyer, D.

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E. Reznikova, T. Weitkamp, V. Nazmov, A. Last, M. Simon, and V. Saile, “Investigation of phase contrast hard x-ray microscopy using planar sets of refractive crossed linear parabolic lenses made from SU-8 polymer,” Phys. Status Solidi A 204(8), 2811–2816 (2007).
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V. Nazmov, E. Reznikova, A. Somogyi, J. Mohr, and V. Saile, “Planar sets of cross x-ray refractive lenses from SU-8 polymer,” Proc. SPIE 5539, 235–243 (2004).
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V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
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E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
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C. Rau, U. Wagner, Z. Pesic, and A. De Fanis, “Coherent imaging at the Diamond beamline I13,” Phys. Status Solidi A 208(11), 2522–2525 (2011).
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E. Reznikova, T. Weitkamp, V. Nazmov, M. Simon, A. Last, and V. Saile, “Transmission hard x-ray microscope with increased view field using planar refractive objectives and condensers made of SU-8 polymer,” J. Phys. Conf. Ser. 186(1), 012070 (2009).
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V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
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V. Nazmov, E. Reznikova, A. Somogyi, J. Mohr, and V. Saile, “Planar sets of cross x-ray refractive lenses from SU-8 polymer,” Proc. SPIE 5539, 235–243 (2004).
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V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
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V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
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V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
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A. Snigirev, I. Snigireva, M. Drakopoulos, V. Nazmov, E. Reznikova, S. Kuznetsov, M. Grigoriev, J. Mohr, and V. Saile, “Focusing properties of x-ray polymer refractive lenses from SU-8 resist layer,” Proc. SPIE 5195, 21–31 (2003).
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H. Wang, J. Sutter, and K. Sawhney, “Advanced in situ metrology for x-ray beam shaping with super precision,” Opt. Express 23(2), 1605–1614 (2015).
[Crossref] [PubMed]

H. Wang, Y. Kashyap, and K. Sawhney, “Speckle based x-ray wavefront sensing with nanoradian angular sensitivity,” Opt. Express 23(18), 23310–23317 (2015).
[Crossref] [PubMed]

S. Berujon, H. Wang, S. Alcock, and K. Sawhney, “At-wavelength metrology of hard x-ray mirror using near field speckle,” Opt. Express 22(6), 6438–6446 (2014).
[Crossref] [PubMed]

S. Berujon, H. Wang, and K. Sawhney, “At-wavelength metrology using the x-ray speckle tracking technique: case study of a x-ray compound refractive lens,” J. Phys. Conf. Ser. 425(5), 052020 (2013).
[Crossref]

H. Wang, S. Berujon, and K. Sawhney, “Development of at-wavelength metrology using grating-based shearing interferometry at Diamond Light Source,” J. Phys. Conf. Ser. 425(5), 052021 (2013).
[Crossref]

H. Wang, S. Berujon, and K. Sawhney, “Characterization of a one dimensional focusing compound refractive lens using the rotating shearing interferometer technique,” AIP Conf. Proc. 1466(1), 223–228 (2012).
[Crossref]

S. Berujon, H. Wang, and K. Sawhney, “X-ray multimodal imaging using a random-phase object,” Phys. Rev. A 86(6), 063813 (2012).
[Crossref]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref] [PubMed]

Schäfer, B.

B. Flöter, P. Juranić, P. Großmann, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedkte, “Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements,” Nucl. Instr. Meth. Phys. Res. A 635(1, Supplement), S108–S112 (2011).
[Crossref]

B. Flöter, P. Juranić, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedtke, “EUV Hartmann sensor for wavefront measurements at the free-electron LASer in Hamburg,” New J. Phys. 12(8), 083015 (2010).
[Crossref]

Schlegel, T.

F. Siewert, T. Noll, T. Schlegel, T. Zeschke, and H. Lammert, “The nanometer optical component measuring machine: a new sub-nm topography measuring device for x-ray optics at BESSY,” AIP Conf. Proc. 705(1), 847–850 (2004).
[Crossref]

Schlenker, M.

Schlott, V.

Y. Kayser, C. David, U. Flechsig, J. Krempasky, V. Schlott, and R. Abela, “X-ray grating interferometer for in situ and at-wavelength wavefront metrology,” J. of Synchrotron Radiat. 24(1), 150–162 (2017).
[Crossref]

Schroer, C.

B. Lengeler, C. Schroer, B. Benner, A. Gerhardus, T. F. Günzler, M. Kuhlmann, J. Meyer, and C. Zimprich, “Parabolic refractive x-ray lenses,” J. Synchrotron Radiat. 9(3), 119–124 (2002).
[Crossref] [PubMed]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

Schröter, T. J.

Schuster, M.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “Inspection of refractive x-ray lenses using high-resolution differential phase contrast imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 78(9), 093707 (2007).
[Crossref] [PubMed]

Sexton, B.

Shack, R. V.

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

Shi, B.

Siemens, R. v.

E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
[Crossref]

Siewert, F.

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise x-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

F. Siewert, T. Noll, T. Schlegel, T. Zeschke, and H. Lammert, “The nanometer optical component measuring machine: a new sub-nm topography measuring device for x-ray optics at BESSY,” AIP Conf. Proc. 705(1), 847–850 (2004).
[Crossref]

Simon, M.

E. Reznikova, T. Weitkamp, V. Nazmov, M. Simon, A. Last, and V. Saile, “Transmission hard x-ray microscope with increased view field using planar refractive objectives and condensers made of SU-8 polymer,” J. Phys. Conf. Ser. 186(1), 012070 (2009).
[Crossref]

E. Reznikova, T. Weitkamp, V. Nazmov, A. Last, M. Simon, and V. Saile, “Investigation of phase contrast hard x-ray microscopy using planar sets of refractive crossed linear parabolic lenses made from SU-8 polymer,” Phys. Status Solidi A 204(8), 2811–2816 (2007).
[Crossref]

Simon, R.

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

Siu, K. K. W.

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

Snigirev, A.

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
[Crossref]

A. Snigirev, I. Snigireva, M. Drakopoulos, V. Nazmov, E. Reznikova, S. Kuznetsov, M. Grigoriev, J. Mohr, and V. Saile, “Focusing properties of x-ray polymer refractive lenses from SU-8 resist layer,” Proc. SPIE 5195, 21–31 (2003).
[Crossref]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

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]

Snigireva, I.

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
[Crossref]

A. Snigirev, I. Snigireva, M. Drakopoulos, V. Nazmov, E. Reznikova, S. Kuznetsov, M. Grigoriev, J. Mohr, and V. Saile, “Focusing properties of x-ray polymer refractive lenses from SU-8 resist layer,” Proc. SPIE 5195, 21–31 (2003).
[Crossref]

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

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]

Somogyi, A.

V. Nazmov, E. Reznikova, A. Somogyi, J. Mohr, and V. Saile, “Planar sets of cross x-ray refractive lenses from SU-8 polymer,” Proc. SPIE 5539, 235–243 (2004).
[Crossref]

Stampanoni, M.

Stangl, J.

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. 17(3), 299–307 (2010).
[Crossref] [PubMed]

Stein, A. F.

Störmer, M.

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise x-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Sutter, J.

Tabata, O.

V. Saile, U. Wallradbe, O. Tabata, and J. G. Korvink, LIGA and its Applications (Wiley-VCH, 2009), Vol. 7 of Advanced Micro & Nanosystems.

Takacs, P. Z.

P. Z. Takacs, S.-N. Qian, and J. Colbert, “Design of a long trace surface profiler,” Proc. SPIE 0749, 59–64 (1987).
[Crossref]

Thibault, P.

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, “X-ray phase-contrast imaging and metrology through unified modulated pattern analysis,” Phys. Rev. Lett. 118(20), 203903 (2017).
[Crossref] [PubMed]

Tiedkte, K.

B. Flöter, P. Juranić, P. Großmann, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedkte, “Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements,” Nucl. Instr. Meth. Phys. Res. A 635(1, Supplement), S108–S112 (2011).
[Crossref]

Tiedtke, K.

B. Flöter, P. Juranić, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedtke, “EUV Hartmann sensor for wavefront measurements at the free-electron LASer in Hamburg,” New J. Phys. 12(8), 083015 (2010).
[Crossref]

Tümmler, J.

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

Vila-Comamala, J.

Wagner, U.

C. Rau, U. Wagner, Z. Pesic, and A. De Fanis, “Coherent imaging at the Diamond beamline I13,” Phys. Status Solidi A 208(11), 2522–2525 (2011).
[Crossref]

Wallradbe, U.

V. Saile, U. Wallradbe, O. Tabata, and J. G. Korvink, LIGA and its Applications (Wiley-VCH, 2009), Vol. 7 of Advanced Micro & Nanosystems.

Wang, H.

Y. Kashyap, H. Wang, and K. Sawhney, “Speckle-based at-wavelength metrology of x-ray mirrors with super accuracy,” Rev. Sci. Instrum. 87(5), 052001 (2016).
[Crossref] [PubMed]

Y. Kashyap, H. Wang, and K. Sawhney, “Development of a speckle-based portable device for in situ metrology of synchrotron x-ray mirrors,” J. Synchrotron Radiat. 23(Pt 5), 1131–1136 (2016).
[Crossref] [PubMed]

H. Wang, Y. Kashyap, D. Laundy, and K. Sawhney, “Two-dimensional in situ metrology of x-ray mirrors using the speckle scanning technique,” J. Synchrotron Radiat. 22(4), 925–929 (2015).
[Crossref] [PubMed]

H. Wang, J. Sutter, and K. Sawhney, “Advanced in situ metrology for x-ray beam shaping with super precision,” Opt. Express 23(2), 1605–1614 (2015).
[Crossref] [PubMed]

H. Wang, Y. Kashyap, and K. Sawhney, “Speckle based x-ray wavefront sensing with nanoradian angular sensitivity,” Opt. Express 23(18), 23310–23317 (2015).
[Crossref] [PubMed]

S. Berujon, H. Wang, S. Alcock, and K. Sawhney, “At-wavelength metrology of hard x-ray mirror using near field speckle,” Opt. Express 22(6), 6438–6446 (2014).
[Crossref] [PubMed]

S. Berujon, H. Wang, and K. Sawhney, “At-wavelength metrology using the x-ray speckle tracking technique: case study of a x-ray compound refractive lens,” J. Phys. Conf. Ser. 425(5), 052020 (2013).
[Crossref]

H. Wang, S. Berujon, and K. Sawhney, “Development of at-wavelength metrology using grating-based shearing interferometry at Diamond Light Source,” J. Phys. Conf. Ser. 425(5), 052021 (2013).
[Crossref]

H. Wang, S. Berujon, and K. Sawhney, “Characterization of a one dimensional focusing compound refractive lens using the rotating shearing interferometer technique,” AIP Conf. Proc. 1466(1), 223–228 (2012).
[Crossref]

S. Berujon, H. Wang, and K. Sawhney, “X-ray multimodal imaging using a random-phase object,” Phys. Rev. A 86(6), 063813 (2012).
[Crossref]

H. Wang, K. Sawhney, S. Berujon, E. Ziegler, S. Rutishauser, and C. David, “X-ray wavefront characterization using a rotating shearing interferometer technique,” Opt. Express 19(17), 16550–16559 (2011).
[Crossref] [PubMed]

Weitkamp, T.

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett. 99(22), 221104 (2011).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105(24), 248102 (2010).
[Crossref]

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. 17(3), 299–307 (2010).
[Crossref] [PubMed]

E. Reznikova, T. Weitkamp, V. Nazmov, M. Simon, A. Last, and V. Saile, “Transmission hard x-ray microscope with increased view field using planar refractive objectives and condensers made of SU-8 polymer,” J. Phys. Conf. Ser. 186(1), 012070 (2009).
[Crossref]

E. Reznikova, T. Weitkamp, V. Nazmov, A. Last, M. Simon, and V. Saile, “Investigation of phase contrast hard x-ray microscopy using planar sets of refractive crossed linear parabolic lenses made from SU-8 polymer,” Phys. Status Solidi A 204(8), 2811–2816 (2007).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005).
[Crossref] [PubMed]

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86(5), 054101 (2005).
[Crossref]

Wen, H. H.

Zanette, I.

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, “X-ray phase-contrast imaging and metrology through unified modulated pattern analysis,” Phys. Rev. Lett. 118(20), 203903 (2017).
[Crossref] [PubMed]

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett. 99(22), 221104 (2011).
[Crossref]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105(24), 248102 (2010).
[Crossref]

Zdora, M.-C.

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, “X-ray phase-contrast imaging and metrology through unified modulated pattern analysis,” Phys. Rev. Lett. 118(20), 203903 (2017).
[Crossref] [PubMed]

Zeschke, T.

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise x-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

F. Siewert, T. Noll, T. Schlegel, T. Zeschke, and H. Lammert, “The nanometer optical component measuring machine: a new sub-nm topography measuring device for x-ray optics at BESSY,” AIP Conf. Proc. 705(1), 847–850 (2004).
[Crossref]

Zhou, T.

M.-C. Zdora, P. Thibault, T. Zhou, F. J. Koch, J. Romell, S. Sala, A. Last, C. Rau, and I. Zanette, “X-ray phase-contrast imaging and metrology through unified modulated pattern analysis,” Phys. Rev. Lett. 118(20), 203903 (2017).
[Crossref] [PubMed]

Ziegler, E.

Zimprich, C.

B. Lengeler, C. Schroer, B. Benner, A. Gerhardus, T. F. Günzler, M. Kuhlmann, J. Meyer, and C. Zimprich, “Parabolic refractive x-ray lenses,” J. Synchrotron Radiat. 9(3), 119–124 (2002).
[Crossref] [PubMed]

AIP Conf. Proc. (3)

F. Siewert, T. Noll, T. Schlegel, T. Zeschke, and H. Lammert, “The nanometer optical component measuring machine: a new sub-nm topography measuring device for x-ray optics at BESSY,” AIP Conf. Proc. 705(1), 847–850 (2004).
[Crossref]

H. Wang, S. Berujon, and K. Sawhney, “Characterization of a one dimensional focusing compound refractive lens using the rotating shearing interferometer technique,” AIP Conf. Proc. 1466(1), 223–228 (2012).
[Crossref]

V. Nazmov, E. Reznikova, M. Boerner, J. Mohr, V. Saile, A. Snigirev, I. Snigireva, M. Di Michiel, M. Drakopoulos, R. Simon, and M. Grigoriev, “Refractive lenses fabricated by deep SR lithography and LIGA technology for x-ray energies from 1 keV to 1 MeV,” AIP Conf. Proc. 705(1), 752–755 (2004).
[Crossref]

Appl. Phys. Lett. (3)

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett. 99(22), 221104 (2011).
[Crossref]

K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer,” Appl. Phys. Lett. 100(12), 124102 (2012).
[Crossref]

T. Weitkamp, B. Nöhammer, A. Diaz, C. David, and E. Ziegler, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86(5), 054101 (2005).
[Crossref]

J. of Synchrotron Radiat. (1)

Y. Kayser, C. David, U. Flechsig, J. Krempasky, V. Schlott, and R. Abela, “X-ray grating interferometer for in situ and at-wavelength wavefront metrology,” J. of Synchrotron Radiat. 24(1), 150–162 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

R. V. Shack and B. C. Platt, “Production and use of a lenticular Hartmann screen,” J. Opt. Soc. Am. 61, 656 (1971).

J. Phys. Conf. Ser. (3)

H. Wang, S. Berujon, and K. Sawhney, “Development of at-wavelength metrology using grating-based shearing interferometry at Diamond Light Source,” J. Phys. Conf. Ser. 425(5), 052021 (2013).
[Crossref]

S. Berujon, H. Wang, and K. Sawhney, “At-wavelength metrology using the x-ray speckle tracking technique: case study of a x-ray compound refractive lens,” J. Phys. Conf. Ser. 425(5), 052020 (2013).
[Crossref]

E. Reznikova, T. Weitkamp, V. Nazmov, M. Simon, A. Last, and V. Saile, “Transmission hard x-ray microscope with increased view field using planar refractive objectives and condensers made of SU-8 polymer,” J. Phys. Conf. Ser. 186(1), 012070 (2009).
[Crossref]

J. Synchrotron Radiat. (7)

B. Lengeler, C. Schroer, J. Tümmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard x-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

B. Lengeler, C. Schroer, B. Benner, A. Gerhardus, T. F. Günzler, M. Kuhlmann, J. Meyer, and C. Zimprich, “Parabolic refractive x-ray lenses,” J. Synchrotron Radiat. 9(3), 119–124 (2002).
[Crossref] [PubMed]

Y. Kashyap, H. Wang, and K. Sawhney, “Development of a speckle-based portable device for in situ metrology of synchrotron x-ray mirrors,” J. Synchrotron Radiat. 23(Pt 5), 1131–1136 (2016).
[Crossref] [PubMed]

S. Berujon, E. Ziegler, and P. Cloetens, “X-ray pulse wavefront metrology using speckle tracking,” J. Synchrotron Radiat. 22(4), 886–894 (2015).
[Crossref] [PubMed]

H. Wang, Y. Kashyap, D. Laundy, and K. Sawhney, “Two-dimensional in situ metrology of x-ray mirrors using the speckle scanning technique,” J. Synchrotron Radiat. 22(4), 925–929 (2015).
[Crossref] [PubMed]

A. Diaz, C. Mocuta, J. Stangl, M. Keplinger, T. Weitkamp, F. Pfeiffer, C. David, T. H. Metzger, and G. Bauer, “Coherence and wavefront characterization of Si-111 monochromators using double-grating interferometry,” J. Synchrotron Radiat. 17(3), 299–307 (2010).
[Crossref] [PubMed]

F. Siewert, J. Buchheim, T. Zeschke, M. Störmer, G. Falkenberg, and R. Sankari, “On the characterization of ultra-precise x-ray optical components: advances and challenges in ex situ metrology,” J. Synchrotron Radiat. 21(5), 968–975 (2014).
[Crossref] [PubMed]

Microsys. Technol. (1)

V. Nazmov, E. Reznikova, J. Mohr, A. Snigirev, I. Snigireva, S. Achenbach, and V. Saile, “Fabrication and preliminary testing of x-ray lenses in thick SU-8 resist layers,” Microsys. Technol. 10(10), 716–721 (2004).
[Crossref]

Nat. Mater. (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Nature (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]

Naturwissenschaften (1)

E. W. Backer, W. Ehrfeld, D. Münchmeyer, H. Betz, A. Heuberger, S. Pongratz, W. Glashauser, H. J. Michel, and R. v. Siemens, “Production of separation-nozzle systems for uranium enrichment by a combination of x-ray lithography and galvanoplastics,” Naturwissenschaften 69(11), 520–523 (1982).
[Crossref]

New J. Phys. (1)

B. Flöter, P. Juranić, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedtke, “EUV Hartmann sensor for wavefront measurements at the free-electron LASer in Hamburg,” New J. Phys. 12(8), 083015 (2010).
[Crossref]

Nucl. Instr. Meth. Phys. Res. A (2)

B. Flöter, P. Juranić, P. Großmann, S. Kapitzki, B. Keitel, K. Mann, E. Plönjes, B. Schäfer, and K. Tiedkte, “Beam parameters of FLASH beamline BL1 from Hartmann wavefront measurements,” Nucl. Instr. Meth. Phys. Res. A 635(1, Supplement), S108–S112 (2011).
[Crossref]

M. Idir, P. Mercere, M. H. Modi, G. Dovillaire, X. Levecq, S. Bucourt, L. Escolano, and P. Sauvageot, “X-ray active mirror coupled with a Hartmann wavefront sensor,” Nucl. Instr. Meth. Phys. Res. A 616(2), 162–171 (2010).
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Opt. Eng. (1)

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Figures (7)

Fig. 1
Fig. 1 Experimental arrangements for the characterisation of the refractive lenses. Setups using the UMPA method with (a) a 1D phase grating rotated around the optical axis in 46 steps and (b) a piece of sandpaper, which was translated perpendicular to the beam directions on a spiral pattern in 24 steps. (c) X-ray grating interferometry setup for validation, consisting of a 1D phase grating and a 1D absorption grating placed in front of the detector. Phase-stepping was performed with five steps over one grating period.
Fig. 2
Fig. 2 Refraction angle αx in the horizontal direction of the line-focus lens A measured with UMPA using (a) a piece of sandpaper, (b) a periodic phase grating as a phase modulator, and (c) total phase shift integrated from (b). Line profiles along the mid-line of the lens for (d) the horizontal refraction angles and (e) the integrated phase shift signals. Support elements can be observed on both sides of the lens aperture.
Fig. 3
Fig. 3 Refraction angle signals α and integrated phase shifts Φ of the point-focus lens B. Refraction angle αx in the horizontal direction measured with UMPA using (a) sandpaper, (b) a periodic phase grating and (c) conventional two-grating XGI. (d) Absolute phase shift determined by 1D integration of (c). Refraction angle αy in the vertical direction measured with UMPA using (e) sandpaper, (f) a periodic phase grating and (g), (h) absolute phase shift signals integrated from (a) and (e) or (b) and (f), respectively. (i) 3D surface plot of the beam phase profile after lens B in panel (h) measured in the detector plane with the UMPA grating setup. (j), (k) Profile plots through the horizontal and vertical refraction angle signals in (a)–(c) and (e)–(f), respectively.
Fig. 4
Fig. 4 Residuals from a linear fit to the refraction angle of lens B (wavefront gradient error) in (a), (d) the horizontal, (b), (e) the vertical direction, and (c), (f) absolute deviation, measured with the UMPA sandpaper and grating setups, respectively. Residuals from (g) the horizontal refraction angle and (h) absolute deviation in the horizontal, obtained from the two-grating interferometer. (i), (j) Absolute deviation from the horizontal refraction angle signal of lens A for the UMPA sandpaper and grating setups, respectively.
Fig. 5
Fig. 5 Variation of the measured focal length along the height of lens A (line-focus lens). The decrease of the focal length closer to the substrate is due to shape errors more pronounced in this area.
Fig. 6
Fig. 6 Cuts through the propagated wave fields after the point-focus lens B (beam damage) and line-focus lens A (shape errors), calculated with an angular spectrum propagator from the phase signal measured with the UMPA grating setup in the detector plane. (a), (c) Longitudinal cuts through the centre of lens B in the xz- and yz-planes, respectively. The dashed lines indicate the distance from the detector where the focal length was found for the horizontal (orange line) and the vertical (cyan line) focussing direction (numbers give the distance from the detector (bottom) and from the lens (top)). (b), (d) Transverse slices along the vertical y through the wave field of lens B at the focal distances in the horizontal and vertical directions, respectively. (e) Longitudinal cut through the propagated wave field of lens A in the top part of the lens (pixel row 39 from the top) and distance of the focal line from the detector (magenta line) and (f) corresponding transverse slice at this z-position.
Fig. 7
Fig. 7 (a), (c), (e) Longitudinal cuts (xz-plane) through the propagated wave fields of lens A in the top, middle and bottom part of the lens. Magenta lines indicating the position of the calculated focal length measured from the detector plane (bottom value) or lens position (top value). (b), (d), (f) Transverse cuts at the focal distances at z = 3.72 m, z = 3.68 m, and z = 3.66 m from the lens.

Equations (8)

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I j ( m , n ) = T ( m , n ) [ I + D ( m , n ) ( I 0 j ( m + u x , n + u y ) I ) ]
( α x , α y ) = ( u x , u y ) p eff d ,
( Φ x , Φ y ) = 2 π λ ( α x , α y ) .
I r , s ( m , n ; x g ) = a 0 r , s ( m , n ) + a 1 r , s ( m , n ) cos ( 2 π p x g + ϕ 1 r , s ( m , n ) ) ,
Φ x = 2 π λ α x = p λ d T ( ϕ 1 , s ϕ 1 , r ) ,
1 / f x = α x x 1 / f y = α y y
f = R 2 δ .
= exp ( i k z 1 q 2 λ 2 ) ,

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