Abstract

We have developed a sparse phase-stepping (SPS) method for x-ray Talbot–Lau interferometry, which first constructs a SPS intensity pattern of fewer images than the conventional phase-stepping (PS) method and then fills the data gap with neighboring pixels for phase retrieval. The SPS method is highly beneficial in practice since the fundamental difference in spatial resolution between the SPS and PS methods becomes negligible due to the blur caused by an interferometer. The concept of the SPS method has been proved by the experiment using a small effective source size. Furthermore, the experiment using a large effective source size has verified that in practical situations the SPS method can reduce the required number of images for phase retrieval and still offer the retrieved images with as high a spatial resolution as the PS method.

© 2014 Optical Society of America

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  1. U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
    [CrossRef]
  2. V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2317 (1995).
    [CrossRef]
  3. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
    [CrossRef]
  4. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
    [CrossRef]
  5. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
    [CrossRef]
  6. 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, 1175–1181 (2007).
    [CrossRef]
  7. I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett. 105, 248102 (2010).
    [CrossRef]
  8. H. Wen, E. Bennett, R. Kopace, A. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Opt. Lett. 35, 1932–1934 (2010).
    [CrossRef]
  9. H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based x-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express 19, 3339–3346 (2011).
    [CrossRef]
  10. G. Sato, T. Kondoh, H. Itoh, S. Handa, K. Yamaguchi, T. Nakamura, K. Nagai, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional gratings-based phase-contrast imaging using a conventional x-ray tube,” Opt. Lett. 36, 3551–3553 (2011).
    [CrossRef]
  11. W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
    [CrossRef]
  12. I. Zanette, M. Bech, F. Pfeiffer, and T. Weitkamp, “Interlaced phase stepping in phase-contrast x-ray tomography,” Appl. Phys. Lett. 98, 094101 (2011).
    [CrossRef]
  13. K. Nagai, H. Itoh, G. Sato, T. Nakamura, K. Yamaguchi, T. Kondoh, S. Handa, and T. Den, “New phase retrieval method for single-shot x-ray Talbot imaging using windowed Fourier transform,” Proc. SPIE 8127, 812706 (2011).
    [CrossRef]

2012 (1)

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

2011 (4)

I. Zanette, M. Bech, F. Pfeiffer, and T. Weitkamp, “Interlaced phase stepping in phase-contrast x-ray tomography,” Appl. Phys. Lett. 98, 094101 (2011).
[CrossRef]

K. Nagai, H. Itoh, G. Sato, T. Nakamura, K. Yamaguchi, T. Kondoh, S. Handa, and T. Den, “New phase retrieval method for single-shot x-ray Talbot imaging using windowed Fourier transform,” Proc. SPIE 8127, 812706 (2011).
[CrossRef]

H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based x-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express 19, 3339–3346 (2011).
[CrossRef]

G. Sato, T. Kondoh, H. Itoh, S. Handa, K. Yamaguchi, T. Nakamura, K. Nagai, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional gratings-based phase-contrast imaging using a conventional x-ray tube,” Opt. Lett. 36, 3551–3553 (2011).
[CrossRef]

2010 (2)

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

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

2007 (1)

2006 (2)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

1995 (2)

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

1965 (1)

U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[CrossRef]

Bech, M.

I. Zanette, M. Bech, F. Pfeiffer, and T. Weitkamp, “Interlaced phase stepping in phase-contrast x-ray tomography,” Appl. Phys. Lett. 98, 094101 (2011).
[CrossRef]

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

Bennett, E.

Bonse, U.

U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[CrossRef]

Bunk, O.

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, 1175–1181 (2007).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

David, C.

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

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, 1175–1181 (2007).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Den, T.

Donath, T.

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

Handa, S.

K. Nagai, H. Itoh, G. Sato, T. Nakamura, K. Yamaguchi, T. Kondoh, S. Handa, and T. Den, “New phase retrieval method for single-shot x-ray Talbot imaging using windowed Fourier transform,” Proc. SPIE 8127, 812706 (2011).
[CrossRef]

G. Sato, T. Kondoh, H. Itoh, S. Handa, K. Yamaguchi, T. Nakamura, K. Nagai, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional gratings-based phase-contrast imaging using a conventional x-ray tube,” Opt. Lett. 36, 3551–3553 (2011).
[CrossRef]

Hart, M.

U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[CrossRef]

Hattori, T.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Ingal, V. N.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

Itoh, H.

Jian, C.

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Kondoh, T.

Kopace, R.

Kottler, C.

Kun, G.

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Momose, A.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Nagai, K.

Nakamura, T.

Ouchi, C.

Pai, V.

Pei-Ping, Z.

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

Pfeiffer, F.

I. Zanette, M. Bech, F. Pfeiffer, and T. Weitkamp, “Interlaced phase stepping in phase-contrast x-ray tomography,” Appl. Phys. Lett. 98, 094101 (2011).
[CrossRef]

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, 1175–1181 (2007).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Rutishauser, S.

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

Sato, G.

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Setomoto, Y.

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

Stein, A.

Suzuki, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Takeda, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Teshima, T.

Weitkamp, T.

I. Zanette, M. Bech, F. Pfeiffer, and T. Weitkamp, “Interlaced phase stepping in phase-contrast x-ray tomography,” Appl. Phys. Lett. 98, 094101 (2011).
[CrossRef]

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

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Wen, H.

Xin, G.

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

Yamaguchi, K.

Yang-Chao, T.

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

Yashiro, W.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Zanette, I.

I. Zanette, M. Bech, F. Pfeiffer, and T. Weitkamp, “Interlaced phase stepping in phase-contrast x-ray tomography,” Appl. Phys. Lett. 98, 094101 (2011).
[CrossRef]

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

Zhi-Li, W.

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

Zi-Yu, W.

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

Appl. Phys. Lett. (2)

U. Bonse and M. Hart, “An x-ray interferometer,” Appl. Phys. Lett. 6, 155–156 (1965).
[CrossRef]

I. Zanette, M. Bech, F. Pfeiffer, and T. Weitkamp, “Interlaced phase stepping in phase-contrast x-ray tomography,” Appl. Phys. Lett. 98, 094101 (2011).
[CrossRef]

Chin. Phys. B (1)

W. Zhi-Li, G. Kun, C. Jian, G. Xin, Z. Pei-Ping, T. Yang-Chao, and W. Zi-Yu, “A new method for information retrieval in two-dimensional grating-based x-ray phase contrast imaging,” Chin. Phys. B 21, 118703 (2012).
[CrossRef]

J. Phys. D (1)

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D 28, 2314–2317 (1995).
[CrossRef]

Jpn. J. Appl. Phys. (1)

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, “Phase tomography by x-ray Talbot interferometry for biological imaging,” Jpn. J. Appl. Phys. 45, 5254–5262 (2006).
[CrossRef]

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2, 258–261 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Proc. SPIE (1)

K. Nagai, H. Itoh, G. Sato, T. Nakamura, K. Yamaguchi, T. Kondoh, S. Handa, and T. Den, “New phase retrieval method for single-shot x-ray Talbot imaging using windowed Fourier transform,” Proc. SPIE 8127, 812706 (2011).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66, 5486–5492 (1995).
[CrossRef]

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Figures (16)

Fig. 1.
Fig. 1.

(a) Intensity patterns obtained by the PS method, (b) FT method, and (c) SPS method. Images are recorded only at the filled circles. Images at the open circles are interpolated based on neighboring images.

Fig. 2.
Fig. 2.

Photograph of nylon mesh with the diameter of 500 μm.

Fig. 3.
Fig. 3.

(a) Intensity distribution of the pinhole image with the effective source size of 100 μm. (b) Contour map of 2D Fourier spectrum of the pinhole image. (c) Line profile of 2D Fourier spectrum of the (1,0) direction. (d) (0,1) direction, (e) (1,1) direction, and (f) (1,1) direction.

Fig. 4.
Fig. 4.

Differential phase map in the (1,0) direction retrieved by the (a) PS, (b) SPS, (c) FT method, global map of the (d) SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 100 μm.

Fig. 5.
Fig. 5.

Differential phase map in the (0,1) direction retrieved by the (a) PS, (b) SPS, and (c) FT method. (d) Global map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 100 μm.

Fig. 6.
Fig. 6.

Scattering map in the (1,0) direction retrieved by the (a) PS, (b) SPS, and (c) FT method. Global map of the (d) SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 100 μm.

Fig. 7.
Fig. 7.

Scattering map in the (0,1) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) global map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 100 μm.

Fig. 8.
Fig. 8.

Scattering map in the (1,1) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) global map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 100 μm.

Fig. 9.
Fig. 9.

Scattering map in the (1,0) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) global map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS, the source size is 100 μm.

Fig. 10.
Fig. 10.

(a) Intensity distribution of a pinhole image with the effective source size of 300 μm. (b) Contour map of 2D Fourier spectrum of the pinhole image. (c) Line profile of 2D Fourier spectrum of the (1,0) direction, (d) (0,1) direction, (e) (1,1) direction, and (f) (1,1) direction.

Fig. 11.
Fig. 11.

Differential phase map in the (1,0) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) whole map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 300 μm.

Fig. 12.
Fig. 12.

Differential phase map in the (0,1) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) whole map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 300 μm.

Fig. 13.
Fig. 13.

Scattering map in the (1,0) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) whole map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 300 μm.

Fig. 14.
Fig. 14.

Scattering map in the (0,1) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) whole map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 300 μm.

Fig. 15.
Fig. 15.

Scattering map in the (1,1) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) whole map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS, the source size is 300 μm.

Fig. 16.
Fig. 16.

Scattering map in the (1,1) direction retrieved by the (a) PS, (b) SPS, and (c) FT method, (d) whole map of the SPS, (e) line profiles of square area in (d), (f) scaled Fourier spectra and the ratio (SPS/PS), the source size is 300 μm.

Equations (5)

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fPS(x,y,xg,yg)kg,lg=11bkg,lg(x,y)exp[2πi(kgxgMx+lgygMy)+iφkg,lg(x,y)]=kg,lg=11ckg,lg(x,y)exp[2πi(kgxgMx+lgygMy)],ckg,lg(x,y)=bkg,lg(x,y)exp[iφkg,lg(x,y)],
A(x,y)=|c0,0s(x,y)/c0,0r(x,y)|,Pi,j(x,y)=d2πz0arg[ci,js(x,y)/ci,jr(x,y)],Vi,j(x,y)=|ci,js(x,y)/ci,jr(x,y)|/A(x,y),
fFT(x,y)kp,lp=11bkp,lp(x,y)exp[2πi(kpxPx+lpyPy)+iφkp,lp(x,y)],
fSPS(x,y,xg,yg)kg,lg,kp,lp=11bkg,lg,kp,lp(x,y)exp[2πi(kgxgMx+lgygMy)+2πi(kpxPx+lpyPy)+iφkp,lp(x,y)].
fSPS(x,y,xg1,yg)=fSPS(x1,y,xg,yg),fSPS(x,y,xg,yg1)=fSPS(x,y1,xg,yg),fSPS(x,y,xg1,yg1)=fSPS(x1,y1,xg,yg),

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