Abstract

Modern synchrotron beamlines often take the form of critical illumination systems, where an incoherent source of limited spatial extent is re-imaged to an experimental plane of interest. Unique constraints of synchrotron sources and beamlines, however, may preclude the use of the simple Zernike approximation for calculating the object-image coherence relationship. Here, we perform a rigorous analysis of the object-image coherence relationship valid for synchrotron beamlines. The analysis shows that beamline aberrations have an effect on the coherence properties. Effects of various low-order aberrations on the coherence properties are explicitly studied.

© 2003 Optical Society of America

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References

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  1. F. Polack, D. Joyeux, J. Svatos, D. Phalippou, “Applications of wavefront division interferometers in soft X-rays,” Rev. Sci. Instrum. 66, 2180–2183 (1995).
    [CrossRef]
  2. S. Marchesini, R. Coïsson, “Two-dimensional coherence measurements with Fresnel mirrors,” Opt. Eng. 35, 3597–3601 (1996).
    [CrossRef]
  3. Y. Takayama, T. Hatano, T. Miyahara, W. Okamoto, “Relationship between spatial coherence of synchrotron radiation and emittance,” J. Synchrotron Radiat. 5, 1187–1194 (1998).
    [CrossRef]
  4. C. Chang, P. Naulleau, E. Anderson, K. Rosfjord, D. Attwood, “Diffractive optical elements based on Fourier optical techniques: A new class of optics for extreme ultraviolet and soft x-ray wavelengths,” Appl. Opt. 41, 7384–7390 (2002).
    [CrossRef] [PubMed]
  5. R. Coïsson, “Spatial coherence of synchrotron radiation,” Appl. Opt. 34, 904–908 (1995).
    [CrossRef] [PubMed]
  6. Joseph W. Goodman, Statistical Optics, (Wiley, New York, 1985), critical illumination: Sec. 7.2.1, pp. 306; Zernike approximation: Sec. 7.2.2, pp. 311; Van Cittert-Zernike theorem: Sec. 5.6, pp. 207–211; generalized Van Cittert-Zernike theorem: Sec. 5.6.4, pp. 218–222; Huygens-Fresnel principle: Sec. 5.4.1, pp. 196–198 and Sec. 7.1.4, pp. 296–300.
  7. D. T. Attwood, Soft X-rays and Extreme Ultraviolet Radiation: Principles and Applications, (Cambridge University, Cambridge, UK, 1999).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, (Cambridge University, Cambridge, UK, 1997), Zernike approximation: Sec. 10.5.2, pp. 522–524; Van Cittert-Zernike theorem: Sec. 10.4.2, pp. 508–512; displacement theorem: Sec. 9.1, pp. 460–464.
  9. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica V, 785–795 (1938).
    [CrossRef]
  10. D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
    [CrossRef]
  11. C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
    [CrossRef]
  12. Bernard R. A. Nijboer, The Diffraction Theory of Aberrations, (Batavia, Groningen, the Netherlands, 1942). Chap. 4.
  13. A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
    [CrossRef]
  14. P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
    [CrossRef]
  15. P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
    [CrossRef]

2002 (1)

2000 (1)

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

1999 (3)

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
[CrossRef]

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

1998 (2)

Y. Takayama, T. Hatano, T. Miyahara, W. Okamoto, “Relationship between spatial coherence of synchrotron radiation and emittance,” J. Synchrotron Radiat. 5, 1187–1194 (1998).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

1996 (1)

S. Marchesini, R. Coïsson, “Two-dimensional coherence measurements with Fresnel mirrors,” Opt. Eng. 35, 3597–3601 (1996).
[CrossRef]

1995 (2)

F. Polack, D. Joyeux, J. Svatos, D. Phalippou, “Applications of wavefront division interferometers in soft X-rays,” Rev. Sci. Instrum. 66, 2180–2183 (1995).
[CrossRef]

R. Coïsson, “Spatial coherence of synchrotron radiation,” Appl. Opt. 34, 904–908 (1995).
[CrossRef] [PubMed]

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica V, 785–795 (1938).
[CrossRef]

Anderson, E.

Attwood, D.

C. Chang, P. Naulleau, E. Anderson, K. Rosfjord, D. Attwood, “Diffractive optical elements based on Fourier optical techniques: A new class of optics for extreme ultraviolet and soft x-ray wavelengths,” Appl. Opt. 41, 7384–7390 (2002).
[CrossRef] [PubMed]

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Attwood, D. T.

D. T. Attwood, Soft X-rays and Extreme Ultraviolet Radiation: Principles and Applications, (Cambridge University, Cambridge, UK, 1999).
[CrossRef]

Batson, P.

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Beguiristain, R.

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

Bokor, J.

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, (Cambridge University, Cambridge, UK, 1997), Zernike approximation: Sec. 10.5.2, pp. 522–524; Van Cittert-Zernike theorem: Sec. 10.4.2, pp. 508–512; displacement theorem: Sec. 9.1, pp. 460–464.

Bresloff, C.

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Chang, C.

C. Chang, P. Naulleau, E. Anderson, K. Rosfjord, D. Attwood, “Diffractive optical elements based on Fourier optical techniques: A new class of optics for extreme ultraviolet and soft x-ray wavelengths,” Appl. Opt. 41, 7384–7390 (2002).
[CrossRef] [PubMed]

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Coïsson, R.

S. Marchesini, R. Coïsson, “Two-dimensional coherence measurements with Fresnel mirrors,” Opt. Eng. 35, 3597–3601 (1996).
[CrossRef]

R. Coïsson, “Spatial coherence of synchrotron radiation,” Appl. Opt. 34, 904–908 (1995).
[CrossRef] [PubMed]

Goldberg, K. A.

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Goodman, Joseph W.

Joseph W. Goodman, Statistical Optics, (Wiley, New York, 1985), critical illumination: Sec. 7.2.1, pp. 306; Zernike approximation: Sec. 7.2.2, pp. 311; Van Cittert-Zernike theorem: Sec. 5.6, pp. 207–211; generalized Van Cittert-Zernike theorem: Sec. 5.6.4, pp. 218–222; Huygens-Fresnel principle: Sec. 5.4.1, pp. 196–198 and Sec. 7.1.4, pp. 296–300.

Gullikson, E. M.

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

Hatano, T.

Y. Takayama, T. Hatano, T. Miyahara, W. Okamoto, “Relationship between spatial coherence of synchrotron radiation and emittance,” J. Synchrotron Radiat. 5, 1187–1194 (1998).
[CrossRef]

Holyst, R.

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
[CrossRef]

Joyeux, D.

F. Polack, D. Joyeux, J. Svatos, D. Phalippou, “Applications of wavefront division interferometers in soft X-rays,” Rev. Sci. Instrum. 66, 2180–2183 (1995).
[CrossRef]

Kevan, S. D.

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
[CrossRef]

Koike, M.

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

Lee, S. H.

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Marchesini, S.

S. Marchesini, R. Coïsson, “Two-dimensional coherence measurements with Fresnel mirrors,” Opt. Eng. 35, 3597–3601 (1996).
[CrossRef]

Medecki, H.

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

Miyahara, T.

Y. Takayama, T. Hatano, T. Miyahara, W. Okamoto, “Relationship between spatial coherence of synchrotron radiation and emittance,” J. Synchrotron Radiat. 5, 1187–1194 (1998).
[CrossRef]

Naulleau, P.

C. Chang, P. Naulleau, E. Anderson, K. Rosfjord, D. Attwood, “Diffractive optical elements based on Fourier optical techniques: A new class of optics for extreme ultraviolet and soft x-ray wavelengths,” Appl. Opt. 41, 7384–7390 (2002).
[CrossRef] [PubMed]

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, D. Attwood, J. Bokor, “Extreme-ultraviolet phase-shifting point-diffraction interferometer: a wave-front metrology tool with subangstrom reference-wave accuracy,” Appl. Opt. 38, 7252–7263 (1999).
[CrossRef]

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Nijboer, Bernard R. A.

Bernard R. A. Nijboer, The Diffraction Theory of Aberrations, (Batavia, Groningen, the Netherlands, 1942). Chap. 4.

Okamoto, W.

Y. Takayama, T. Hatano, T. Miyahara, W. Okamoto, “Relationship between spatial coherence of synchrotron radiation and emittance,” J. Synchrotron Radiat. 5, 1187–1194 (1998).
[CrossRef]

Phalippou, D.

F. Polack, D. Joyeux, J. Svatos, D. Phalippou, “Applications of wavefront division interferometers in soft X-rays,” Rev. Sci. Instrum. 66, 2180–2183 (1995).
[CrossRef]

Polack, F.

F. Polack, D. Joyeux, J. Svatos, D. Phalippou, “Applications of wavefront division interferometers in soft X-rays,” Rev. Sci. Instrum. 66, 2180–2183 (1995).
[CrossRef]

Poniewierski, A.

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
[CrossRef]

Price, A. C.

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
[CrossRef]

Rosfjord, K.

Sorensen, L. B.

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
[CrossRef]

Svatos, J.

F. Polack, D. Joyeux, J. Svatos, D. Phalippou, “Applications of wavefront division interferometers in soft X-rays,” Rev. Sci. Instrum. 66, 2180–2183 (1995).
[CrossRef]

Takayama, Y.

Y. Takayama, T. Hatano, T. Miyahara, W. Okamoto, “Relationship between spatial coherence of synchrotron radiation and emittance,” J. Synchrotron Radiat. 5, 1187–1194 (1998).
[CrossRef]

Tejnil, E.

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

Toner, J.

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
[CrossRef]

Underwood, J.

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, (Cambridge University, Cambridge, UK, 1997), Zernike approximation: Sec. 10.5.2, pp. 522–524; Van Cittert-Zernike theorem: Sec. 10.4.2, pp. 508–512; displacement theorem: Sec. 9.1, pp. 460–464.

Zernike, F.

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica V, 785–795 (1938).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

D. Attwood, P. Naulleau, K. A. Goldberg, E. Tejnil, C. Chang, R. Beguiristain, P. Batson, J. Bokor, E. M. Gullikson, M. Koike, H. Medecki, J. Underwood, “Tunable coherent radiation in the soft X-ray and extreme ultraviolet spectral regions,” IEEE J. Quantum Electron. 35, 709–720 (1999).
[CrossRef]

J. Synchrotron Radiat. (1)

Y. Takayama, T. Hatano, T. Miyahara, W. Okamoto, “Relationship between spatial coherence of synchrotron radiation and emittance,” J. Synchrotron Radiat. 5, 1187–1194 (1998).
[CrossRef]

Opt. Commun. (1)

C. Chang, P. Naulleau, E. Anderson, D. Attwood, “Spatial coherence characterization of undulator radiation,” Opt. Commun. 182, 25–34 (2000).
[CrossRef]

Opt. Eng. (1)

S. Marchesini, R. Coïsson, “Two-dimensional coherence measurements with Fresnel mirrors,” Opt. Eng. 35, 3597–3601 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

A. C. Price, L. B. Sorensen, S. D. Kevan, J. Toner, A. Poniewierski, R. Holyst, “Coherent soft-X-ray dynamic light scattering from smectic-A films,” Phys. Rev. Lett. 82, 755–758 (1999).
[CrossRef]

Physica (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica V, 785–795 (1938).
[CrossRef]

Proc. SPIE (1)

P. Naulleau, K. A. Goldberg, S. H. Lee, C. Chang, C. Bresloff, P. Batson, D. Attwood, J. Bokor, “Characterization of the accuracy of EUV phase-shifting point diffraction interferometry,” Proc. SPIE 3331, 114–123 (1998).
[CrossRef]

Rev. Sci. Instrum. (1)

F. Polack, D. Joyeux, J. Svatos, D. Phalippou, “Applications of wavefront division interferometers in soft X-rays,” Rev. Sci. Instrum. 66, 2180–2183 (1995).
[CrossRef]

Other (4)

Joseph W. Goodman, Statistical Optics, (Wiley, New York, 1985), critical illumination: Sec. 7.2.1, pp. 306; Zernike approximation: Sec. 7.2.2, pp. 311; Van Cittert-Zernike theorem: Sec. 5.6, pp. 207–211; generalized Van Cittert-Zernike theorem: Sec. 5.6.4, pp. 218–222; Huygens-Fresnel principle: Sec. 5.4.1, pp. 196–198 and Sec. 7.1.4, pp. 296–300.

D. T. Attwood, Soft X-rays and Extreme Ultraviolet Radiation: Principles and Applications, (Cambridge University, Cambridge, UK, 1999).
[CrossRef]

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, (Cambridge University, Cambridge, UK, 1997), Zernike approximation: Sec. 10.5.2, pp. 522–524; Van Cittert-Zernike theorem: Sec. 10.4.2, pp. 508–512; displacement theorem: Sec. 9.1, pp. 460–464.

Bernard R. A. Nijboer, The Diffraction Theory of Aberrations, (Batavia, Groningen, the Netherlands, 1942). Chap. 4.

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Figures (4)

Fig. 1
Fig. 1

Coordinate system.

Fig. 2
Fig. 2

Simulation results for a large (1.6 mm × 1.6 mm) uniform source. Intensity (left-hand side column) and coherence (right-hand side column) distributions resulting from the various pupil aberrations (a) defocus, (b) astigmatism, (c) coma. The coherence distributions are all essentially Airy patterns as predicted by the Zernike approximation.

Fig. 3
Fig. 3

Simulation results: Intensity and coherence distribution at the condenser image plane resulting from the unaberrated condenser pupil and the Gaussian-shaped incoherent source of (σ x , σ y ) = (260 μm, 16 μm).

Fig. 4
Fig. 4

Simulation results that show the effect of aberrations for the Gaussian-shaped incoherent source: Intensity (left-hand side column) and coherence (right-hand side column) distribution at resulting from the various aberrations (a) defocus, (b) astigmatism, and (c) coma.

Equations (8)

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dsource  dcoheff12πλθaccept,
σxC, σyC=λz12πσξ, λz12πση0.14 mm, 2.23 mm.
Jiu1, v1; u2, v2=-+ Joξ1, η1; ξ2, η2×Ku1, v1; ξ1, η1×K*u2, v2; ξ2, η2dξ1dη1dξ2dη2,
Ku, v; ξ, η=expj πλz2u2+v2expj πλz1ξ2+η2λ2z2z1×-+Px, yexp-j 2πλz2u+z2z1 ξx+v+z2z1 ηydxdy,
Jo=κIsξ, ηδΔξ, Δη
Jiu1, v1; u2, v2=κ -+ Isξ, ηKu1, v1; ξ, η×K*u2, v2; ξ, ηdξdη.
Gu, v -+Px, yexp-j 2πλz2×ux+vydxdy.
Jiu1, v1; u2, v2=κ expj πλz2u12+v12-u22-v22λ4z22z12×-+ Isξ, ηGu1+Mξ, v1+Mη×G*u2+Mξ, v2+Mηdξdη.

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