Abstract

We assesses the efficiency of x-ray Talbot interferometry (XTI), a technique based on the Talbot effect for measuring a wavefront gradient, in terms of how quickly it can capture a high-quality phase image with a large signal-to-noise ratio for a given incident photon number. Photon statistics cause errors in the phase of the moiré fringes and impose a detection limit on the wavefront gradient. The relation between the incident photon number and the detection limit is determined, and a figure of merit of XTI for a monochromatic cone beam is then defined. The dependence of the figure of merit on optical system parameters, such as grating pitch and position, is then discussed. The effects of varying the pattern height and linewidth of the second grating are shown for rectangular and trapezoidal teeth. Finally, we show how to design a practical cone-beam Talbot interferometer for certain boundary conditions.

© 2008 Optical Society of America

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  3. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
    [CrossRef]
  4. T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 54101-54103 (2005).
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  5. A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jpn. J. Appl. Phys., Part 1 45, 314-316 (2006).
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  6. T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
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  7. A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352-360 (2004).
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  8. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “Quantitative x-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296-6304 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  29. The condition M⩾3 is required for Eq. and the condition M⩾4 is required for Eq. . For this reason the result Eq. is correct for M⩾4. If M=3, ΔΨ depends on φxs because the term of q−1 is not negligible in Eq. .
  30. We assumed that the distribution of intensity from each groove is roughly represented as a Gaussian function of a standard deviation of σs. Then the full width at half-maximum (FWHM) of it is approximately given by 2.35σs.
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    [CrossRef]

2007 (3)

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[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] [PubMed]

2006 (5)

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., Part 1 45, 5254-5262 (2006).
[CrossRef]

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (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]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S (2006).
[CrossRef]

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jpn. J. Appl. Phys., Part 1 45, 314-316 (2006).
[CrossRef]

2005 (5)

A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys., Part 1 44, 6355-6367 (2005).
[CrossRef]

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 54101-54103 (2005).
[CrossRef]

Y. Utsumi, T. Kishimoto, T. Hattori, and H. Hara, “Large-area x-ray lithography system for LIGA process operating in wide energy range of synchrotron radiation,” Jpn. J. Appl. Phys., Part 1 44, 5500-5504 (2005).
[CrossRef]

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

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

2004 (2)

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352-360 (2004).
[CrossRef]

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
[CrossRef]

2000 (1)

R. Fitzgerald, “Phase-sensitive x-ray imaging,” Phys. Today 53, 23-26 (2000).
[CrossRef]

1999 (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

1997 (1)

1995 (1)

1990 (2)

M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: An overview,” Ind. Metrology 1, 79-99 (1990).
[CrossRef]

O. Kafri and I. Glatt, The Physics of Moiré Metrology (Wiley-Interscience, 1990).

1989 (1)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics XXVII (Elsevier, 1989).
[CrossRef]

1986 (1)

E. W. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plasticmoulding (LIGA process),” Microelectronics 4, 35-56 (1986).
[CrossRef]

1978 (1)

J. H. Bruning, Optical Shop Testing, D.Malacara, ed. (Wiley-Interscience, 1978).

1974 (1)

1971 (1)

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic object, with application to structure determination of phase object,” Opt. Acta 18, 677-682 (1971).
[CrossRef]

1965 (1)

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).
[CrossRef]

Arrizón, V.

Baruchel, J.

Baumann, J.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Becker, E. W.

E. W. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plasticmoulding (LIGA process),” Microelectronics 4, 35-56 (1986).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Brangaccio, D. J.

Bruning, J. H.

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] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S (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]

Cloetens, P.

David, C.

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (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, 1175-1181 (2007).
[CrossRef] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (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]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S (2006).
[CrossRef]

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 54101-54103 (2005).
[CrossRef]

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

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

De Martino, C.

Diaz, A.

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

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 54101-54103 (2005).
[CrossRef]

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

Ehrfeld, W.

E. W. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plasticmoulding (LIGA process),” Microelectronics 4, 35-56 (1986).
[CrossRef]

Engelhardt, M.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Fitzgerald, R.

R. Fitzgerald, “Phase-sensitive x-ray imaging,” Phys. Today 53, 23-26 (2000).
[CrossRef]

Gallagher, J. E.

Glatt, I.

O. Kafri and I. Glatt, The Physics of Moiré Metrology (Wiley-Interscience, 1990).

Guigay, J. P.

P. Cloetens, J. P. Guigay, C. De Martino, and J. Baruchel, “Fractional Talbot imaging of phase gratings with hard X rays,” Opt. Lett. 22, 1059-1061 (1997).
[CrossRef] [PubMed]

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic object, with application to structure determination of phase object,” Opt. Acta 18, 677-682 (1971).
[CrossRef]

Hagmann, P.

E. W. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plasticmoulding (LIGA process),” Microelectronics 4, 35-56 (1986).
[CrossRef]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
[CrossRef]

Hara, H.

Y. Utsumi, T. Kishimoto, T. Hattori, and H. Hara, “Large-area x-ray lithography system for LIGA process operating in wide energy range of synchrotron radiation,” Jpn. J. Appl. Phys., Part 1 44, 5500-5504 (2005).
[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., Part 1 45, 5254-5262 (2006).
[CrossRef]

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

Y. Utsumi, T. Kishimoto, T. Hattori, and H. Hara, “Large-area x-ray lithography system for LIGA process operating in wide energy range of synchrotron radiation,” Jpn. J. Appl. Phys., Part 1 44, 5500-5504 (2005).
[CrossRef]

Herriott, D. R.

Hunabiki, Y.

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

Kafri, O.

O. Kafri and I. Glatt, The Physics of Moiré Metrology (Wiley-Interscience, 1990).

Kawamoto, S.

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jpn. J. Appl. Phys., Part 1 45, 314-316 (2006).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352-360 (2004).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
[CrossRef]

Kishimoto, T.

Y. Utsumi, T. Kishimoto, T. Hattori, and H. Hara, “Large-area x-ray lithography system for LIGA process operating in wide energy range of synchrotron radiation,” Jpn. J. Appl. Phys., Part 1 44, 5500-5504 (2005).
[CrossRef]

Kottler, C.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (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, 1175-1181 (2007).
[CrossRef] [PubMed]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S (2006).
[CrossRef]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352-360 (2004).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
[CrossRef]

López-Olazagasti, E.

Maner, A.

E. W. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plasticmoulding (LIGA process),” Microelectronics 4, 35-56 (1986).
[CrossRef]

Matsumoto, M.

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

Momose, A.

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jpn. J. Appl. Phys., Part 1 45, 314-316 (2006).
[CrossRef]

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., Part 1 45, 5254-5262 (2006).
[CrossRef]

A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys., Part 1 44, 6355-6367 (2005).
[CrossRef]

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352-360 (2004).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
[CrossRef]

Moritake, M.

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

Münchmeyer, D.

E. W. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plasticmoulding (LIGA process),” Microelectronics 4, 35-56 (1986).
[CrossRef]

Nöhammer, B.

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 54101-54103 (2005).
[CrossRef]

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics XXVII (Elsevier, 1989).
[CrossRef]

Pfeiffer, F.

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] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S (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]

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

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

Rohbeck, T.

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

Rosenfeld, D. P.

Schuster, M.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Stampanoni, M.

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

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

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., Part 1 45, 5254-5262 (2006).
[CrossRef]

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352-360 (2004).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
[CrossRef]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
[CrossRef]

Takeda, H.

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

Takeda, M.

M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: An overview,” Ind. Metrology 1, 79-99 (1990).
[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., Part 1 45, 5254-5262 (2006).
[CrossRef]

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

Takeuchi, A.

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

Takiguchi, K.

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).
[CrossRef]

Tanaka, M.

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

Uesugi, K.

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

Utsumi, Y.

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

Y. Utsumi, T. Kishimoto, T. Hattori, and H. Hara, “Large-area x-ray lithography system for LIGA process operating in wide energy range of synchrotron radiation,” Jpn. J. Appl. Phys., Part 1 44, 5500-5504 (2005).
[CrossRef]

Weitkamp, T.

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S (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]

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

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 54101-54103 (2005).
[CrossRef]

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

White, A. D.

Winthrop, J. T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Worthington, C. R.

Yashiro, W.

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

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., Part 1 45, 5254-5262 (2006).
[CrossRef]

Ziegler, E.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

T. Weitkamp, B. Nöhammer, A. Diaz, and C. David, “X-ray wavefront analysis and optics characterization with a grating interferometer,” Appl. Phys. Lett. 86, 54101-54103 (2005).
[CrossRef]

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, “High-resolution differential phase contrast imaging using a magnifying projection geometry with a microfocus x-ray source,” Appl. Phys. Lett. 90, 224101 (2007).
[CrossRef]

Ind. Metrology (1)

M. Takeda, “Spatial-carrier fringe-pattern analysis and its applications to precision interferometry and profilometry: An overview,” Ind. Metrology 1, 79-99 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys., Part 1 (5)

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., Part 1 45, 5254-5262 (2006).
[CrossRef]

Y. Utsumi, T. Kishimoto, T. Hattori, and H. Hara, “Large-area x-ray lithography system for LIGA process operating in wide energy range of synchrotron radiation,” Jpn. J. Appl. Phys., Part 1 44, 5500-5504 (2005).
[CrossRef]

A. Momose and S. Kawamoto, “X-ray Talbot interferometry with capillary plates,” Jpn. J. Appl. Phys., Part 1 45, 314-316 (2006).
[CrossRef]

A. Momose, “Recent advances in x-ray phase imaging,” Jpn. J. Appl. Phys., Part 1 44, 6355-6367 (2005).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jpn. J. Appl. Phys., Part 1 42, L866-L868 (2003).
[CrossRef]

Microelectronics (1)

E. W. Becker, W. Ehrfeld, P. Hagmann, A. Maner, and D. Münchmeyer, “Fabrication of microstructures with high aspect ratios and great structural heights by synchrotron radiation lithography, galvanoforming, and plasticmoulding (LIGA process),” Microelectronics 4, 35-56 (1986).
[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. Acta (1)

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic object, with application to structure determination of phase object,” Opt. Acta 18, 677-682 (1971).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Philos. Mag. (1)

H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401-407 (1836).
[CrossRef]

Phys. Rev. Lett. (1)

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[CrossRef] [PubMed]

Phys. Today (1)

R. Fitzgerald, “Phase-sensitive x-ray imaging,” Phys. Today 53, 23-26 (2000).
[CrossRef]

Proc. SPIE (4)

T. Weitkamp, A. Diaz, B. Nöhammer, F. Pfeiffer, T. Rohbeck, P. Cloetens, M. Stampanoni, and C. David, “Hard x-ray phase imaging and tomography with a grating interferometer,” Proc. SPIE 5535, 137-142 (2004).
[CrossRef]

A. Momose, S. Kawamoto, I. Koyama, and Y. Suzuki, “Phase tomography using an x-ray Talbot interferometer,” Proc. SPIE 5535, 352-360 (2004).
[CrossRef]

A. Momose, W. Yashiro, M. Moritake, Y. Takeda, K. Uesugi, A. Takeuchi, Y. Suzuki, M. Tanaka, and T. Hattori, “Biomedical imaging by Talbot-type x-ray phase tomography,” Proc. SPIE 6318, 63180T (2006).
[CrossRef]

T. Weitkamp, C. David, C. Kottler, O. Bunk, and F. Pfeiffer, “Tomography with grating interferometers at low-brilliance sources,” Proc. SPIE 6318, 63180S (2006).
[CrossRef]

Other (7)

M. Matsumoto, K. Takiguchi, M. Tanaka, Y. Hunabiki, H. Takeda, A. Momose, Y. Utsumi, and T. Hattori, “Fabrication of diffraction grating for x-ray Talbot interferometer,” in High Aspect Ratio Micro Structure Technology Workshop 2005 (Harmst, 2005), p. 22.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics XXVII (Elsevier, 1989).
[CrossRef]

O. Kafri and I. Glatt, The Physics of Moiré Metrology (Wiley-Interscience, 1990).

J. H. Bruning, Optical Shop Testing, D.Malacara, ed. (Wiley-Interscience, 1978).

The condition M⩾3 is required for Eq. and the condition M⩾4 is required for Eq. . For this reason the result Eq. is correct for M⩾4. If M=3, ΔΨ depends on φxs because the term of q−1 is not negligible in Eq. .

We assumed that the distribution of intensity from each groove is roughly represented as a Gaussian function of a standard deviation of σs. Then the full width at half-maximum (FWHM) of it is approximately given by 2.35σs.

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Figures (6)

Fig. 1
Fig. 1

Experimental setup of the cone-beam x-ray Talbot interferometer, where R s , R 1 , and R 2 are the distances of the x-ray source to the sample, the first grating; and the second grating, z 12 is the distance between the first and the second gratings; and d 1 and d 2 are the pitches of the first and second gratings.

Fig. 2
Fig. 2

Examples of Monte Carlo simulation of the cone-beam XTI. The upper figures are simulated moiré images that would be captured by an area detector placed immediately behind the second grating [corresponding to m = 0 in Eq. (23) and ε = 0 in Eq. (26)]. The left and right figures correspond to the cases of I 0 = 10 14 and 10 12 photons/sr (corresponding to 9.1 × 10 3 and 9.1 × 10 1 photons/pixel, respectively). The samples in the figures are polystyrene spheres with diameters of 0.25 mm (upper left), 0.50 mm (upper right), 0.75 mm (lower left), and 1.00 mm (lower right), which are put just in front of the first grating ( R s = R 1 ) . The lower figures are Ψ ( x , y ) Ψ 0 ( x , y ) images, each of which corresponds to the case of the upper one. Here a Ψ ( x , y ) image was obtained from five moiré images ( M = 5 ) by using the fringe scanning technique [see Eq. (20)].

Fig. 3
Fig. 3

Dependence of Δ Ψ on the incident photon number. The abscissa corresponds to 1 I total . Crosses are results corresponding to Fig. 2. The solid black line was calculated with Eq. (57). Other symbols are results of the simulation in the cases where the linewidth of the second grating ( l 2 ) or distances of the two gratings from the source are changed: triangles, l 2 = d 2 4 and R 2 = 2.62 m ; open circles, l 2 = 3 d 2 4 and R 2 = 2.62 m ; gray filled circles, l 2 = d 2 2 and R 2 = 2 × 2.62 m ; gray filled squares, l 2 = d 2 2 and R 2 = 0.5 × 2.62 m . Broken black, dotted, solid gray, and broken gray lines were calculated with Eq. (57) for the above four cases, respectively.

Fig. 4
Fig. 4

Curve ( p k ) exp [ ( ( p k ) 2 2 ) ] plotted as a function of k p .

Fig. 5
Fig. 5

Optimum distances of the first and second gratings from the source, R 1 , op and R 2 , op , and the optimal pitch of the first grating d 1 that maximize ( p d 2 λ ) ( μ 1 R 2 ) are plotted as a function of the pitch of the second grating d 2 when λ = 0.3 Å : (a) p = 1 2 and (b) p = 3 2 . The plot of d 1 is, in fact, independent of λ and p.

Fig. 6
Fig. 6

(a) Dependence of c 1 c 0 on the pattern height h 2 and linewidth l 2 (normalized by d 2 ) of the rectangular second grating. (b) Plot of c 1 c 0 for a trapezoidal model as functions of the averaged linewidth l 2 and the width of slope w 2 (normalized by d 2 ), where the pattern height h 2 was fixed at 100 μ m .

Equations (146)

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I G 1 ( x , y , z 12 ) I 0 Δ S t n μ n ¯ b n ¯ ( z 12 ) exp ( i 2 π n d 2 x ) R 2 2 ,
μ n = μ ( n p d 1 ) ,
b n ¯ ( z 12 ) = n a n + n a n * exp [ i π ( ( n + n ) 2 n 2 ) λ d 1 2 R 1 z 12 R 1 + z 12 ] ,
R 2 = R 1 + z 12 ,
d 2 = R 2 R 1 d 1 .
μ ( L ) = exp [ ( L L coh ) 2 2 ] ,
L coh λ R 1 ( 2 π σ s ) .
z 12 = p d 1 2 λ R 1 R 1 p ( d 1 2 λ ) .
b n ¯ ( z 12 ) = n a n + n a n * exp [ i π p ( ( n + n ) 2 n 2 ) ] .
R 2 = R 1 2 R 1 p ( d 1 2 λ ) .
κ 2 , 1 = R 1 R 1 p ( d 1 2 λ )
= R 2 + p ( d 2 2 λ ) R 2 .
z 12 = p λ d 1 d 2 .
I ( x , y , p ) I 0 Δ S t 1 R 2 2 n μ n b n ( z 12 ) c n exp [ i 2 π n ( y θ + χ d 2 + ζ s φ x s ( x s , y s ) ) ] ,
ζ s = { p d s λ ( R s R 1 ) z s d 2 ( R s R 1 ) .
φ x s ( x s , y s ) = λ 2 π Φ ( x s , y s ) x s ,
Φ ( x s , y s ) = 2 π λ δ ( x s , y s , z ) d z .
Φ ( x s n p d s , y s ) Φ ( x s ) n p d s Φ ( x s , y s ) x s ,
Ψ ( x , y ) 2 π ( y θ d 2 + ζ s φ x s ( x s , y s ) )
Ψ ( x , y ) arg [ m = 0 M 1 I m ( x , y , p ) exp ( 2 π i m M ) ] ,
φ x s ( x s , y s ) λ 2 π Δ x s ( Ψ ( x , y ) Ψ 0 ( x , y ) ) ,
Δ Ψ = 2 π Δ x s Δ φ x s λ .
N mes , m ( x , y , p ) = I m ( x , y , p ) + N m
= Δ S I 0 t n q n exp ( i 2 π n ( ζ s φ x s ( x s , y s ) + m M ) ) + N m .
q n μ n b n ( z 12 ) c n R 2 2 ,
N m = N st , m + ε ,
Δ Ψ = ε q 1 M Δ S t I 0 1 2 .
N st , m = 0 ,
N st , m N st , m = { I m ( x , y , p ) ( m = m ) 0 ( m m ) ,
arg [ m = 0 M 1 N mes , m ( x , y , p ) exp ( 2 π i m M ) ] = arg [ m = 0 M 1 ( I m ( x , y , p ) + N st , m ) exp ( 2 π i m M ) ]
= arg [ I + N st ] ,
I ( x , y , p ) m = 0 M 1 I m ( x , y , p ) exp ( 2 π i m M ) ,
N st m = 0 M 1 N st , m exp ( 2 π i m M ) .
arg [ I + N st ] arg [ I ] + arg [ 1 + N st I ]
Ψ ( x , y ) + Δ Ψ st ,
Δ Ψ st Img [ N st I ] .
Δ Ψ st 1 2 i [ N st I N st * I * ]
= 0 .
arg [ m = 0 M 1 N mes , m ( x , y , p ) exp ( 2 π i m M ) ] Ψ ( x , y ) .
Δ Ψ st 2 1 4 [ N st 2 I 2 2 N st 2 I 2 + N st * 2 I * 2 ] .
N st 2 m = 0 M 1 m = 0 M 1 N st , m N st , m exp ( 2 π i ( m m ) M )
= m = 0 M 1 I m ( x , y , p )
= M Δ S I 0 t l q l M exp ( i 2 π l M ζ s φ x s ( x s , y s ) ) ,
N st 2 m = 0 M 1 m = 0 M 1 N st , m N st , m exp ( 2 π i ( m + m ) M )
= m = 0 M 1 I m ( x , y , p ) exp ( 2 π i 2 m M )
= Δ S I 0 t n q n exp ( i 2 π n ( ζ s φ x s ( x s , y s ) ) ) m = 0 M 1 exp ( 2 π i ( n 2 ) m M )
= M Δ S I 0 t l q l M + 2 exp ( i 2 π ( l M + 2 ) ζ s φ x s ( x s , y s ) ) ,
N st 2 M Δ S I 0 t q 0
N st 2 0 .
Δ Ψ st 2 1 2 [ M Δ S I 0 t q 0 I 2 ]
1 2 M Δ S I 0 t q 0 ( M Δ S I 0 t q 1 ) 2
= 1 2 q 0 M Δ S I 0 t q 1 2 ,
I = Δ S I 0 t n q n exp ( i 2 π n ( ζ s φ x s ( x s , y s ) ) ) m = 0 M 1 exp ( 2 π i ( n 1 ) m M )
M Δ S I 0 t q 1 .
Δ Ψ = Δ Ψ st 2
q 0 2 M Δ S I 0 t 1 q 1
= q 0 q 1 1 2 1 I total Δ S ,
Δ Ψ q 0 I total Δ S + ε 2 q 1 I total Δ S 1 2 .
η Ψ = 2 q 1 q 0
= μ 1 R 2 b 1 b 0 c 1 c 0 2 .
Δ x s Δ φ x s = λ 2 π η Ψ I total Δ S .
Δ Det , s = κ 2 , s 1 Δ Det ,
Δ φ x s = λ 2 π Δ x s η Ψ I total Δ S .
Δ Det , s Δ φ x s = λ Δ Det , s 2 π Δ x s η Ψ I total Δ S
= λ Δ Det 2 π p d 2 η Ψ I total Δ S .
η = p d 2 η Ψ λ ,
= p d 2 λ μ 1 R 2 b 1 b 0 c 1 c 0 2 .
Δ Det , s Δ φ x s = Δ Det 2 π η I total Δ S .
δ ( x s , y s , z ) = r e λ 2 2 π ρ ( x s , y s , z ) ,
Δ φ x s = r e λ 2 2 π ( D ( x s , y s ) x s ) DL ,
D ( x s , y s ) = ρ ( x s , y s , z ) d z .
Δ Det , s ( D ( x s , y s ) x s ) DL = Δ Det r e λ 2 η I total Δ S .
( D ) DL Δ Det , s ( D ( x s , y s ) x s ) DL .
( D ) DL = Δ Det r e λ 2 η I total Δ S ,
( D ) DL = Δ Det η I total Δ S ,
η = r e λ 2 η
= r e λ p d 2 η Ψ .
p d 2 λ μ 1 R 2 = p d 2 λ R 2 μ ( p d 1 ) ,
= 1 2 π σ s p k exp [ ( ( p k ) 2 2 ) ] ,
k = L coh d 1
= λ R 1 2 π σ s d 1
= λ R 2 2 π σ s d 2 .
k = p .
R 1 , op = 2 π σ s p d 1 λ .
R 2 , op = 2 π σ s p d 2 λ .
R 1 , op = p λ 1 1 ( 2 π σ s d 2 ) + 1 ( 2 π σ s ) 2 .
d 1 = d 2 1 + d 2 2 π σ s .
( p d 2 λ μ 1 R 2 ) max = exp ( 1 2 ) 2 π σ s ,
Δ Det , s Δ φ x s exp ( 1 2 ) σ s Δ Det I total Δ S .
tan π ( 1 l 2 d 2 ) = 2 π [ ( 1 l 2 d 2 ) + T ( h 2 ) ( l 2 d 2 ) ] 1 T ( h 2 ) ,
tan π ( 1 l 2 , op d 2 ) = 2 π ( 1 l 2 , op d 2 ) .
Δ Det , s Δ φ x s 3.8 σ s I total Δ Det Δ S 1 A air ( R 2 ) A sub 2 A sample ϵ .
( D ) DL 3.8 1 r e λ 2 σ s I total Δ Det Δ S 1 A air ( R 2 ) A sub 2 A sample ϵ .
Δ Det , s = { ( 1 κ 2 , s 1 ) σ s } 2 + ( Δ Det κ 2 , s 1 ) 2 .
R 1 d 0 = z 12 d 2 .
z 12 = p d 1 ( α d 2 ) λ .
1 d 0 = p λ α d 1 R 1
= p λ α 2 d 2 R 2 .
1 d 0 = α d 1 1 d 2 .
Δ Det , s = { ( 1 κ 2 , s 1 ) Σ s } 2 + ( Δ Det κ 2 , s 1 ) 2 ,
Δ φ x s = κ 2 , s 2 π η ( 2.35 σ s d 0 ) I total Δ S .
σ s d 0 = α p 2 π k .
η σ s d 0 ( α p k ) 1 2 exp [ ( ( α p k ) 2 2 ) ] 2 π d 0 ,
R 1 , op = 2 2 π σ s α p d 1 λ ,
R 2 , op = 2 2 π σ s α 2 p d 2 λ .
σ s d 0 = 1 2 π 2 .
Δ φ x s exp ( 1 4 ) 2.35 × 2 π ( 1 2 ) 1 2 d 0 κ 2 , s I total Δ S .
Δ Ψ = 1 η poly , Ψ 1 I 0 M t Δ S ,
η poly , Ψ 2 q 1 ( λ , σ s ) I ¯ 0 ( λ , σ s ) A ( λ ) ϵ ( λ ) d λ q 0 ( λ ) I ¯ 0 ( λ , σ s ) A ( λ ) ϵ ( λ ) d λ ,
Δ Ψ = q 0 ( λ ) I total Δ S I ¯ 0 ( λ , σ s ) A ( λ ) ϵ ( λ ) d λ + ε 2 q 1 ( λ , σ s ) I total Δ S I ¯ 0 ( λ , σ s ) A ( λ ) ϵ ( λ ) d λ 1 2 .
φ x s λ = r e λ ¯ π ρ ( x s , y s , z ) x s d z
= 2 λ φ x s .
Δ Ψ λ = 2 Δ λ λ ¯ Ψ .
Δ Det η poly I total Δ S ( D ) DL π 4 Δ Det p d 2 r e Δ λ ,
η poly p d 2 r e λ ¯ η poly , Ψ .
C s 1 η poly , Ψ I total Δ S ( D ) DL C s π 4 λ ¯ Δ λ ,
C s Δ Det p d 2 r e λ ¯ .
α d 2 d 1 = R 2 R 1 .
I ( x , y , p ) I 0 Δ S t 1 R 2 2 n μ α n b α n ( z 12 ) c α n exp [ i 2 π n ( y θ + χ d 2 + ζ s φ x s ( x , y ) ) ] ,
ζ s = { α p d s λ ( R s R 1 ) z s d 2 ( R s R 1 ) ,
Δ Det , s Δ φ x s = λ Δ Det , s 2 π α p d s η Ψ I total Δ S ,
= λ Δ Det 2 π α p ( α d 2 ) η Ψ I total Δ S .
η = α 2 p d 2 λ 2 q α q 0
= ( α 2 p d 2 λ μ α R 2 ) b α b 0 c α c 0 2 ,
α 2 p d 2 λ μ α R 2 = 1 2 π σ s α p k exp [ ( ( α p k ) 2 2 ) ] ,
k = λ R 2 2 π σ s α d 2 .
k = α p ,
R 1 , op = α 2 p λ 1 1 ( 2 π σ s d 2 ) + 1 ( 2 π σ s ) 2 ,
R 2 , op = α 2 2 π σ s d 2 p λ ,
( α 2 p d 2 λ μ α R 2 ) max = exp ( 1 2 ) 2 π σ s ,
d 1 = α d 2 1 + d 2 2 π σ s .
d 2 d 1 = C C p d 1 2 .
d 2 d 1 = C C d 1 2 .
d 2 d 1 = 1 1 d 1 2 π σ s .
c 0 = 1 d 2 d 2 2 d 2 2 T ( x ) d x ,
c 1 = 1 d 2 d 2 2 d 2 2 T ( x ) cos 2 π x d 2 d x .
d 2 2 d 2 2 T ( x ) sin 2 π x d 2 d x = 0 .
T rect ( x ) = { 1 ( c 0 2 d 2 x c 0 2 d 2 ) 0 ( d 2 2 d 2 < x < c 0 2 d 2 , c 0 2 d 2 < x < d 2 2 d 2 ) .
c 1 , rect = 1 d 2 ( c 0 2 ) d 2 ( c 0 2 ) d 2 cos 2 π x d 2 d x .
c 1 , rect c 1 = 1 d 2 [ ( c 0 2 ) d 2 ( c 0 2 ) d 2 cos 2 π x d 2 d x d 2 2 d 2 2 T ( x ) cos 2 π x d 2 d x ]
= 1 d 2 [ 0 ( c 0 2 ) d 2 ( 1 T ( x ) ) cos 2 π x d 2 d x ( c 0 2 ) d 2 d 2 2 T ( x ) cos 2 π x d 2 d x ] + 1 d 2 [ ( c 0 2 ) d 2 0 ( 1 T ( x ) ) cos 2 π x d 2 d x d 2 2 ( c 0 2 ) d 2 T ( x ) cos 2 π x d 2 d x ] .
0 ( c 0 2 ) d 2 ( 1 T ( x ) ) cos 2 π x d 2 d x ( c 0 2 ) d 2 d 2 2 T ( x ) cos 2 π x d 2 d x cos 2 π c 0 2 0 ( c 0 2 ) d 2 ( 1 T ( x ) ) d x cos 2 π c 0 2 ( c 0 2 ) d 2 d 2 2 T ( x ) d x = cos 2 π c 0 2 ( c 0 2 c 0 , + ) d 2 ,
c 0 , + , + 1 d 2 0 d 2 2 T ( x ) d x .
( c 0 2 ) d 2 0 ( 1 T ( x ) ) cos 2 π x d 2 d x d 2 2 ( c 0 2 ) d 2 T ( x ) cos 2 π x d 2 d x cos 2 π c 0 2 ( c 0 2 ) d 2 0 ( 1 T ( x ) ) d x cos 2 π c 0 2 d 2 2 ( c 0 2 ) d 2 T ( x ) d x = cos 2 π c 0 2 ( c 0 2 c 0 , ) d 2 ,
c 0 , 1 d 2 d 2 2 0 T ( x ) d x .
c 1 , rect c 1 0 .

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