Abstract

A further development of a focusing monochromator concept for X-ray energy resolution of 0.1 meV and below is presented. Theoretical analysis of several optical layouts based on this concept was supported by numerical simulations performed in the “Synchrotron Radiation Workshop” software package using the physical-optics approach and careful modeling of partially-coherent synchrotron (undulator) radiation. Along with the energy resolution, the spectral shape of the energy resolution function was investigated. It was shown that under certain conditions the decay of the resolution function tails can be faster than that of the Gaussian function.

© 2015 Optical Society of America

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References

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  1. M. Yabashi, K. Tamasaku, S. Kikuta, and T. Ishikawa, “X-ray monochromator with an energy resolution of 8x10−9 at 14.41 keV,” Rev. Sci. Instrum. 72(11), 4080–4083 (2001).
    [Crossref]
  2. Y. Shvyd’ko, X-Ray Optics: High-Energy-Resolution Applications (Springer-Verlag Berlin Heidelberg, New York, 2004).
  3. T. S. Toellner, A. Alatas, and A. H. Said, “Six-reflection meV-monochromator for synchrotron radiation,” J. Synchrotron Radiat. 18(4), 605–611 (2011).
    [Crossref] [PubMed]
  4. Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
    [Crossref]
  5. Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
    [PubMed]
  6. Y. Shvyd’ko, “Enhanced x-ray angular dispersion and x-ray spectrographs with resolving power beyond 108,” Proc. SPIE 8502, 85020J (2012).
    [Crossref]
  7. J. W. M. DuMond, “Theory of the use of more than two successive X-ray crystal reflections to obtain increased resolving power,” Phys. Rev. 52(8), 872–883 (1937).
    [Crossref]
  8. V. G. Kohn, A. I. Chumakov, and R. Rüffer, “Wave theory of focusing monochromator of synchrotron radiation,” J. Synchrotron Radiat. 16(5), 635–641 (2009).
    [Crossref] [PubMed]
  9. Y. Shvyd’ko, “Theory of angular dispersive imaging hard x-ray spectrographs,” Phys. Rev. A 91(5), 053817 (2015).
    [Crossref]
  10. M. Czerny and A. F. Turner, “Über den Astigmatismus bei Spiegelspektrometern,” Z. Phys. 61(11-12), 792–797 (1930).
    [Crossref]
  11. V. G. Kohn, I. Snigireva, and A. Snigirev, “Theory of imaging a perfect crystal under the conditions of X-ray spherical wave dynamical diffraction,” Phys. Status Solidi222(2), 407–423 (2000) (b).
    [Crossref]
  12. A. Authier, Dynamical Theory of X-ray Diffraction (Oxford University Press, 2001).
  13. A. Suvorov, Y. Q. Cai, J. P. Sutter, and O. Chubar, “Partially-coherent wavefront propagation simulations for inelastic X-ray scattering beamline including crystal optics,” Proc. SPIE 9209, 92090H (2014).
    [Crossref]

2015 (1)

Y. Shvyd’ko, “Theory of angular dispersive imaging hard x-ray spectrographs,” Phys. Rev. A 91(5), 053817 (2015).
[Crossref]

2014 (2)

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

A. Suvorov, Y. Q. Cai, J. P. Sutter, and O. Chubar, “Partially-coherent wavefront propagation simulations for inelastic X-ray scattering beamline including crystal optics,” Proc. SPIE 9209, 92090H (2014).
[Crossref]

2013 (1)

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

2012 (1)

Y. Shvyd’ko, “Enhanced x-ray angular dispersion and x-ray spectrographs with resolving power beyond 108,” Proc. SPIE 8502, 85020J (2012).
[Crossref]

2011 (1)

T. S. Toellner, A. Alatas, and A. H. Said, “Six-reflection meV-monochromator for synchrotron radiation,” J. Synchrotron Radiat. 18(4), 605–611 (2011).
[Crossref] [PubMed]

2009 (1)

V. G. Kohn, A. I. Chumakov, and R. Rüffer, “Wave theory of focusing monochromator of synchrotron radiation,” J. Synchrotron Radiat. 16(5), 635–641 (2009).
[Crossref] [PubMed]

2001 (1)

M. Yabashi, K. Tamasaku, S. Kikuta, and T. Ishikawa, “X-ray monochromator with an energy resolution of 8x10−9 at 14.41 keV,” Rev. Sci. Instrum. 72(11), 4080–4083 (2001).
[Crossref]

1937 (1)

J. W. M. DuMond, “Theory of the use of more than two successive X-ray crystal reflections to obtain increased resolving power,” Phys. Rev. 52(8), 872–883 (1937).
[Crossref]

1930 (1)

M. Czerny and A. F. Turner, “Über den Astigmatismus bei Spiegelspektrometern,” Z. Phys. 61(11-12), 792–797 (1930).
[Crossref]

Alatas, A.

T. S. Toellner, A. Alatas, and A. H. Said, “Six-reflection meV-monochromator for synchrotron radiation,” J. Synchrotron Radiat. 18(4), 605–611 (2011).
[Crossref] [PubMed]

Cai, Y. Q.

A. Suvorov, Y. Q. Cai, J. P. Sutter, and O. Chubar, “Partially-coherent wavefront propagation simulations for inelastic X-ray scattering beamline including crystal optics,” Proc. SPIE 9209, 92090H (2014).
[Crossref]

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Chubar, O.

A. Suvorov, Y. Q. Cai, J. P. Sutter, and O. Chubar, “Partially-coherent wavefront propagation simulations for inelastic X-ray scattering beamline including crystal optics,” Proc. SPIE 9209, 92090H (2014).
[Crossref]

Chumakov, A. I.

V. G. Kohn, A. I. Chumakov, and R. Rüffer, “Wave theory of focusing monochromator of synchrotron radiation,” J. Synchrotron Radiat. 16(5), 635–641 (2009).
[Crossref] [PubMed]

Coburn, D. S.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Collins, S. P.

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

Cunsolo, A.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Czerny, M.

M. Czerny and A. F. Turner, “Über den Astigmatismus bei Spiegelspektrometern,” Z. Phys. 61(11-12), 792–797 (1930).
[Crossref]

DuMond, J. W. M.

J. W. M. DuMond, “Theory of the use of more than two successive X-ray crystal reflections to obtain increased resolving power,” Phys. Rev. 52(8), 872–883 (1937).
[Crossref]

Hiraoka, N.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Honnicke, M. G.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Huang, X. R.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Ishikawa, T.

M. Yabashi, K. Tamasaku, S. Kikuta, and T. Ishikawa, “X-ray monochromator with an energy resolution of 8x10−9 at 14.41 keV,” Rev. Sci. Instrum. 72(11), 4080–4083 (2001).
[Crossref]

Keister, J. W.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Kikuta, S.

M. Yabashi, K. Tamasaku, S. Kikuta, and T. Ishikawa, “X-ray monochromator with an energy resolution of 8x10−9 at 14.41 keV,” Rev. Sci. Instrum. 72(11), 4080–4083 (2001).
[Crossref]

Kodituwakku, C. N.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Kohn, V. G.

V. G. Kohn, A. I. Chumakov, and R. Rüffer, “Wave theory of focusing monochromator of synchrotron radiation,” J. Synchrotron Radiat. 16(5), 635–641 (2009).
[Crossref] [PubMed]

V. G. Kohn, I. Snigireva, and A. Snigirev, “Theory of imaging a perfect crystal under the conditions of X-ray spherical wave dynamical diffraction,” Phys. Status Solidi222(2), 407–423 (2000) (b).
[Crossref]

Mundboth, K.

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

Rüffer, R.

V. G. Kohn, A. I. Chumakov, and R. Rüffer, “Wave theory of focusing monochromator of synchrotron radiation,” J. Synchrotron Radiat. 16(5), 635–641 (2009).
[Crossref] [PubMed]

Said, A. H.

T. S. Toellner, A. Alatas, and A. H. Said, “Six-reflection meV-monochromator for synchrotron radiation,” J. Synchrotron Radiat. 18(4), 605–611 (2011).
[Crossref] [PubMed]

Shu, D.

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

Shvyd’ko, Y.

Y. Shvyd’ko, “Theory of angular dispersive imaging hard x-ray spectrographs,” Phys. Rev. A 91(5), 053817 (2015).
[Crossref]

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

Y. Shvyd’ko, “Enhanced x-ray angular dispersion and x-ray spectrographs with resolving power beyond 108,” Proc. SPIE 8502, 85020J (2012).
[Crossref]

Snigirev, A.

V. G. Kohn, I. Snigireva, and A. Snigirev, “Theory of imaging a perfect crystal under the conditions of X-ray spherical wave dynamical diffraction,” Phys. Status Solidi222(2), 407–423 (2000) (b).
[Crossref]

Snigireva, I.

V. G. Kohn, I. Snigireva, and A. Snigirev, “Theory of imaging a perfect crystal under the conditions of X-ray spherical wave dynamical diffraction,” Phys. Status Solidi222(2), 407–423 (2000) (b).
[Crossref]

Stetsko, Y.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Stoupin, S.

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

Sutter, J.

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

Sutter, J. P.

A. Suvorov, Y. Q. Cai, J. P. Sutter, and O. Chubar, “Partially-coherent wavefront propagation simulations for inelastic X-ray scattering beamline including crystal optics,” Proc. SPIE 9209, 92090H (2014).
[Crossref]

Suvorov, A.

A. Suvorov, Y. Q. Cai, J. P. Sutter, and O. Chubar, “Partially-coherent wavefront propagation simulations for inelastic X-ray scattering beamline including crystal optics,” Proc. SPIE 9209, 92090H (2014).
[Crossref]

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Tamasaku, K.

M. Yabashi, K. Tamasaku, S. Kikuta, and T. Ishikawa, “X-ray monochromator with an energy resolution of 8x10−9 at 14.41 keV,” Rev. Sci. Instrum. 72(11), 4080–4083 (2001).
[Crossref]

Toellner, T. S.

T. S. Toellner, A. Alatas, and A. H. Said, “Six-reflection meV-monochromator for synchrotron radiation,” J. Synchrotron Radiat. 18(4), 605–611 (2011).
[Crossref] [PubMed]

Tolkiehn, M.

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

Tsuei, K. D.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Turner, A. F.

M. Czerny and A. F. Turner, “Über den Astigmatismus bei Spiegelspektrometern,” Z. Phys. 61(11-12), 792–797 (1930).
[Crossref]

Wille, H. C.

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

Yabashi, M.

M. Yabashi, K. Tamasaku, S. Kikuta, and T. Ishikawa, “X-ray monochromator with an energy resolution of 8x10−9 at 14.41 keV,” Rev. Sci. Instrum. 72(11), 4080–4083 (2001).
[Crossref]

J. Phys. Conf. Ser. (1)

Y. Q. Cai, D. S. Coburn, A. Cunsolo, J. W. Keister, M. G. Honnicke, X. R. Huang, C. N. Kodituwakku, Y. Stetsko, A. Suvorov, N. Hiraoka, K. D. Tsuei, and H. C. Wille, “The ultrahigh resolution IXS beamline of NSLS-II: recent advances and scientific opportunities,” J. Phys. Conf. Ser. 425(20), 202001 (2013).
[Crossref]

J. Synchrotron Radiat. (2)

V. G. Kohn, A. I. Chumakov, and R. Rüffer, “Wave theory of focusing monochromator of synchrotron radiation,” J. Synchrotron Radiat. 16(5), 635–641 (2009).
[Crossref] [PubMed]

T. S. Toellner, A. Alatas, and A. H. Said, “Six-reflection meV-monochromator for synchrotron radiation,” J. Synchrotron Radiat. 18(4), 605–611 (2011).
[Crossref] [PubMed]

Nat. Commun. (1)

Y. Shvyd’ko, S. Stoupin, D. Shu, S. P. Collins, K. Mundboth, J. Sutter, and M. Tolkiehn, “High-contrast sub-millivolt inelastic X-ray scattering for nano- and mesoscale science,” Nat. Commun. 5, 4219 (2014).
[PubMed]

Phys. Rev. (1)

J. W. M. DuMond, “Theory of the use of more than two successive X-ray crystal reflections to obtain increased resolving power,” Phys. Rev. 52(8), 872–883 (1937).
[Crossref]

Phys. Rev. A (1)

Y. Shvyd’ko, “Theory of angular dispersive imaging hard x-ray spectrographs,” Phys. Rev. A 91(5), 053817 (2015).
[Crossref]

Proc. SPIE (2)

Y. Shvyd’ko, “Enhanced x-ray angular dispersion and x-ray spectrographs with resolving power beyond 108,” Proc. SPIE 8502, 85020J (2012).
[Crossref]

A. Suvorov, Y. Q. Cai, J. P. Sutter, and O. Chubar, “Partially-coherent wavefront propagation simulations for inelastic X-ray scattering beamline including crystal optics,” Proc. SPIE 9209, 92090H (2014).
[Crossref]

Rev. Sci. Instrum. (1)

M. Yabashi, K. Tamasaku, S. Kikuta, and T. Ishikawa, “X-ray monochromator with an energy resolution of 8x10−9 at 14.41 keV,” Rev. Sci. Instrum. 72(11), 4080–4083 (2001).
[Crossref]

Z. Phys. (1)

M. Czerny and A. F. Turner, “Über den Astigmatismus bei Spiegelspektrometern,” Z. Phys. 61(11-12), 792–797 (1930).
[Crossref]

Other (3)

V. G. Kohn, I. Snigireva, and A. Snigirev, “Theory of imaging a perfect crystal under the conditions of X-ray spherical wave dynamical diffraction,” Phys. Status Solidi222(2), 407–423 (2000) (b).
[Crossref]

A. Authier, Dynamical Theory of X-ray Diffraction (Oxford University Press, 2001).

Y. Shvyd’ko, X-Ray Optics: High-Energy-Resolution Applications (Springer-Verlag Berlin Heidelberg, New York, 2004).

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Figures (8)

Fig. 1
Fig. 1 Schematic of the HRM. Ln, n = 1-3, denotes distances from the corresponding crystals.
Fig. 2
Fig. 2 Diffraction geometry of the first crystal of the HRM. The angles of incidence and diffraction are denoted as θ1i and θ1o, respectively. The angle of asymmetry ϕ1, is counted positively if θ1i > θ1o. For the purpose of clearness the origins of the local coordinate systems are annotated away from the crystal surface at the corresponding beam axes.
Fig. 3
Fig. 3 Schematic of the optical layout I.
Fig. 4
Fig. 4 Schematic of the optical layout II.
Fig. 5
Fig. 5 Schematic of the optical layout III.
Fig. 6
Fig. 6 Energy resolution curve simulated for the optical layout I with fully open slit. Energy FWHM is 0.6 meV.
Fig. 7
Fig. 7 Energy resolution curve simulated for the optical layout I with the vertical slit size of 17 μm. Energy FWHM is 0.1 meV.
Fig. 8
Fig. 8 Energy resolution curve simulated for the optical layout III with the vertical slit size of 25 μm. Energy FWHM is 0.07 meV.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

U 1 ( x 2 )= U ^ 0 ( k 0x ) R 1 ( k 0x )exp( i k 1x x 2 i L 1 k 1x 2 2k ) d k 0x 2π ,
k 1x = b 1 k 0x +K( 1+ b 1 )( φ 1 + δ E tan ϕ 1 ),
U 4 ( x )= U ^ 0 ( k 0x ) R Σ ( k 0x )exp( i k 4x xi n=1 4 L n k nx 2 2k ) d k 0x 2π , R Σ ( k 0x )= n=1 4 R n ( k ( n1 )x ) .
k ( n+1 )x = b n k nx +sign( c n )K( 1+ b n ) Δ n , Δ n = φ n + δ E tan ϕ n ,
U 4 ( x )= 1 b Σ U ^ 0 ( k 4x ) R Σ ( k 4x )exp( i k 4x xi L 4 k 4x 2 2k ) d k 4x 2π ,
α Δ =( 1+ b 4 ) Δ 4 + b 4 ( 1+ b 3 ) Δ 3 b 3 b 4 ( 1+ b 2 ) Δ 2 b 2 b 3 b 4 ( 1+ b 1 ) Δ 1 .
U 0 ( x )= 1 iλq( z ) exp( ik ( x x 0 ) 2 2q( z ) ),
U 0 ( x )= A 0 exp{ i k 0x ( x+ q 1 q 0 x 0 )+i( q 1 z 1 ) k 0x 2 2k } d k 0x 2π , A 0 = q 1 q 0 exp( ik x 0 2 2 q 0 [ 1+ q 1 q 0 ] ), 1 q 1 = 1 f 1 q 0 , q 0 = z 0 i z R ,
U 1 ( x 1 )= A 0 b Σ exp( i q 1 b Σ q 0 α Δ x 0 ) R Σ ( k 1x ) exp{ i( x 1 x ˜ 1 ) k 1x i( z 2 z ) k 1x 2 2K } d k 1x 2π , x ˜ 1 = α Δ z q 1 b Σ q 0 x 0 , z = 1 b Σ 2 ( q 1 z 1 ).
z 2 = 1 b Σ 2 ( z 0 f z 0 f z 1 ).
I 1 ( x 1 )= M 1 b Σ | R Σ ( k 1x ) exp{ i( x ˜ 1 x 1 ) k 1x z R M 1 2 k 1x 2 2K } d k 1x 2π | 2 , M 1 = f b Σ ( z 0 f ) ,
x 1 = α Δ z 2 M 1 x 0 .
U 2 ( x 1 )= A 1 b Σ R Σ ( z 3 z s k 1x )exp{ i( x ˜ 2 x 1 ) k 1x +i( z s z ) z 3 2 z s 2 k 1x 2 2k } d k 1x 2π , A 1 = z 3 z s exp{ ik x 1 2 2 z i ( 1+ z s z 3 )+i 1 b Σ α Δ x 0 }, x ˜ 2 = z 3 b Σ z s ( 1 b Σ α Δ q 1 x 0 ), z = q 1 b Σ 2 + z 2 , 1 z s = 1 f 1 z 3 .
1 z 3 = 1 f 1 z 0 , z 0 = z 1 b Σ 2 + z 2 .
I 2 ( x 1 )= M 2 b Σ | R Σ ( b Σ M 2 k 1x )exp{ i( x ˜ 2 x 1 ) k 1x z R M 2 2 k 1x 2 2k } d k 1x 2π | 2 , M 2 = f b Σ ( z 0 f ) ,
x 1 = M 2 ( 1 b Σ α Δ z 1 x 0 ).
I 3 ( x 1 )=Aexp{ 2 w 0 2 M 3 2 ( x ˜ 3 x 1 ) 2 } | R Σ ( k x 1 f 2 ) | 2 , A=1/ b Σ M 3 π w 0 2 , x ˜ 3 = α Δ f 2 M 3 x 0 , M 3 = b Σ f 2 f 1 ,
x 1 = α Δ f 2 M 3 x 0 .
Δ E HRM E 0 = b 1 b 2 ω s tan θ B ,
D= 2 E 0 [ ( 1+ b 4 )tan ϕ 4 + b 4 ( 1+ b 3 )tan ϕ 3 b 3 b 4 ( 1+ b 2 )tan ϕ 2 b 2 b 3 b 4 ( 1+ b 1 )tan ϕ 1 ],
I Σ ( E )= I n ( x ˜ n x 1 ) u a ( x 1 )d x 1 ,
Δ E I = M 1 S D z 2 , Δ E II = b Σ S D z 1 , Δ E III = M 3 S D f 2 ,

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