Abstract

Four independent procedures were developed and tested to measure the apparatus response function of a VUV spectrometer–detector system for unpolarized 46-nm radiation dispersed in second order. These measurements were made to allow the use of continuum synchrotron radiation for the calibration of the response of the spectrometer–detector system for dispersion of 92-nm radiation in first order with full correction for the effects of synchrotron radiation dispersed in secondorder. In the first method, synchrotron radiation was used in combination with a thin Al foil to block out synchrotron radiation at 92 nm while allowing 46-nm radiation to enter the spectrometer. In the second as well as the third method Ne ii 46-nm line radiation was used to measure the response function in first and second order. The line radiation was produced by (1) an electron beam exciting a Ne gas target for which the resulting VUV light illuminated the entire grating and (2) a duoplasmatron VUV light source operating with Ne gas producing a small spot of radiation that was scanned across the surface of the spectrometer grating. In the fourth method the difference in the spectral distributions of synchrotron radiation produced by electrons with different kinetic energies was employed to deduce the second-order detection efficiency. The ratio of the second- to first-order response function for 46-nm radiation could be determined to a precision of 6% using the bandpass filter and electron-beam methods, 10% using the duoplasmatron method, and 250% using the multiple electron energy method.

© 1987 Optical Society of America

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References

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  1. A. McPherson, N. Rouze, W. B. Westerveld, J. S. Risley, “Calibration of a VUV Spectrometer–Detector System using Synchrotron Radiation,” Appl. Opt. 25, 298 (1986).
    [CrossRef] [PubMed]
  2. R. Kendrick, “A Comparison of Methods for Determining the Second Order Correction to the Detector Efficiency of a VUV Spectrometer,” M.S. Thesis, North Carolina State U. (1984) (unpublished).
  3. W. R. Hunter, D. W. Angel, R. Tousey, “Thin Films and Their Uses for the XUV,” Appl. Opt. 4, 891 (1965).
    [CrossRef]
  4. E. B. Saloman, “Unfolding First and Second Order Diffracted Radiation when Using Synchrotron Radiation Sources: A Technique,” Appl. Opt. 14, 1391 (1975).
    [CrossRef] [PubMed]
  5. W. B. Westerveld, A. McPherson, J. S. Risley, “Synchrotron Radiation Intensity for 50 MeV to 50 GeV Electrons,” At. Data Nucl. Data Tables 28, 21 (1983).
    [CrossRef]
  6. J. S. Risley, A. McPherson, W. B. Westerveld, “Use of a Scaling Relationship for Synchrotron Radiation,” Phys. Rev. A 24, 3255 (1981).
    [CrossRef]
  7. R. Kendrick, N. Rouze, J. S. Risley, “Apple Sync: Synchrotron Flux Calculation,” (1983). A 16,000 number look-up table based on the algorithms developed for Ref. 5 using an Apple II microcomputer.
  8. J. A. R. Samson, Techniques of Vacuum Ultraviolet Radiation (Pied Publishers, Lincoln, NE, 1967), 348 pp.
  9. A. McPherson, “Measurement of the Electron Impact Photoemission Cross Sections of the 92.0 nm and 93.2 nm Emission Lines of Ar II for the VUV Radiometric Project,” Ph.D. Thesis. North Carolina State U. (1984) (unpublished).
  10. B. F. Luyken, F. J. de Heer, R. Ch. Baas, “The Role of the Outer s Shell in Single Ionization of Ne, Ar, Kr, and Xe by Electron Impact,” Physica 61, 200 (1972).
    [CrossRef]
  11. L. R. Hughey, A. R. Schaefer, “Reduced Absolute Uncertainty in the Irradiance of SURF-II and Instrumentation for Measuring Linearity of X-Ray, XUV and UV Detectors,” Nucl. Instrum. Methods 195, 367 (1982).
    [CrossRef]
  12. D. L. Ederer et al., “An Overview of Research at NBS using Synchrotron Radiation at SURF II,” IEEE Trans. Nucl. Sci. NS-301020 (1983).
    [CrossRef]
  13. W. R. Ott, L. R. Canfield, S. C. Ebner, L. R. Hughey, R. P. Madden, “XUV Radiometric Standards at NBS,” Proc. Soc. Photo-Opt. Instrum. Eng. 689, 178 (1986).

1986 (2)

A. McPherson, N. Rouze, W. B. Westerveld, J. S. Risley, “Calibration of a VUV Spectrometer–Detector System using Synchrotron Radiation,” Appl. Opt. 25, 298 (1986).
[CrossRef] [PubMed]

W. R. Ott, L. R. Canfield, S. C. Ebner, L. R. Hughey, R. P. Madden, “XUV Radiometric Standards at NBS,” Proc. Soc. Photo-Opt. Instrum. Eng. 689, 178 (1986).

1983 (2)

D. L. Ederer et al., “An Overview of Research at NBS using Synchrotron Radiation at SURF II,” IEEE Trans. Nucl. Sci. NS-301020 (1983).
[CrossRef]

W. B. Westerveld, A. McPherson, J. S. Risley, “Synchrotron Radiation Intensity for 50 MeV to 50 GeV Electrons,” At. Data Nucl. Data Tables 28, 21 (1983).
[CrossRef]

1982 (1)

L. R. Hughey, A. R. Schaefer, “Reduced Absolute Uncertainty in the Irradiance of SURF-II and Instrumentation for Measuring Linearity of X-Ray, XUV and UV Detectors,” Nucl. Instrum. Methods 195, 367 (1982).
[CrossRef]

1981 (1)

J. S. Risley, A. McPherson, W. B. Westerveld, “Use of a Scaling Relationship for Synchrotron Radiation,” Phys. Rev. A 24, 3255 (1981).
[CrossRef]

1975 (1)

1972 (1)

B. F. Luyken, F. J. de Heer, R. Ch. Baas, “The Role of the Outer s Shell in Single Ionization of Ne, Ar, Kr, and Xe by Electron Impact,” Physica 61, 200 (1972).
[CrossRef]

1965 (1)

Angel, D. W.

Baas, R. Ch.

B. F. Luyken, F. J. de Heer, R. Ch. Baas, “The Role of the Outer s Shell in Single Ionization of Ne, Ar, Kr, and Xe by Electron Impact,” Physica 61, 200 (1972).
[CrossRef]

Canfield, L. R.

W. R. Ott, L. R. Canfield, S. C. Ebner, L. R. Hughey, R. P. Madden, “XUV Radiometric Standards at NBS,” Proc. Soc. Photo-Opt. Instrum. Eng. 689, 178 (1986).

de Heer, F. J.

B. F. Luyken, F. J. de Heer, R. Ch. Baas, “The Role of the Outer s Shell in Single Ionization of Ne, Ar, Kr, and Xe by Electron Impact,” Physica 61, 200 (1972).
[CrossRef]

Ebner, S. C.

W. R. Ott, L. R. Canfield, S. C. Ebner, L. R. Hughey, R. P. Madden, “XUV Radiometric Standards at NBS,” Proc. Soc. Photo-Opt. Instrum. Eng. 689, 178 (1986).

Ederer, D. L.

D. L. Ederer et al., “An Overview of Research at NBS using Synchrotron Radiation at SURF II,” IEEE Trans. Nucl. Sci. NS-301020 (1983).
[CrossRef]

Hughey, L. R.

W. R. Ott, L. R. Canfield, S. C. Ebner, L. R. Hughey, R. P. Madden, “XUV Radiometric Standards at NBS,” Proc. Soc. Photo-Opt. Instrum. Eng. 689, 178 (1986).

L. R. Hughey, A. R. Schaefer, “Reduced Absolute Uncertainty in the Irradiance of SURF-II and Instrumentation for Measuring Linearity of X-Ray, XUV and UV Detectors,” Nucl. Instrum. Methods 195, 367 (1982).
[CrossRef]

Hunter, W. R.

Kendrick, R.

R. Kendrick, “A Comparison of Methods for Determining the Second Order Correction to the Detector Efficiency of a VUV Spectrometer,” M.S. Thesis, North Carolina State U. (1984) (unpublished).

R. Kendrick, N. Rouze, J. S. Risley, “Apple Sync: Synchrotron Flux Calculation,” (1983). A 16,000 number look-up table based on the algorithms developed for Ref. 5 using an Apple II microcomputer.

Luyken, B. F.

B. F. Luyken, F. J. de Heer, R. Ch. Baas, “The Role of the Outer s Shell in Single Ionization of Ne, Ar, Kr, and Xe by Electron Impact,” Physica 61, 200 (1972).
[CrossRef]

Madden, R. P.

W. R. Ott, L. R. Canfield, S. C. Ebner, L. R. Hughey, R. P. Madden, “XUV Radiometric Standards at NBS,” Proc. Soc. Photo-Opt. Instrum. Eng. 689, 178 (1986).

McPherson, A.

A. McPherson, N. Rouze, W. B. Westerveld, J. S. Risley, “Calibration of a VUV Spectrometer–Detector System using Synchrotron Radiation,” Appl. Opt. 25, 298 (1986).
[CrossRef] [PubMed]

W. B. Westerveld, A. McPherson, J. S. Risley, “Synchrotron Radiation Intensity for 50 MeV to 50 GeV Electrons,” At. Data Nucl. Data Tables 28, 21 (1983).
[CrossRef]

J. S. Risley, A. McPherson, W. B. Westerveld, “Use of a Scaling Relationship for Synchrotron Radiation,” Phys. Rev. A 24, 3255 (1981).
[CrossRef]

A. McPherson, “Measurement of the Electron Impact Photoemission Cross Sections of the 92.0 nm and 93.2 nm Emission Lines of Ar II for the VUV Radiometric Project,” Ph.D. Thesis. North Carolina State U. (1984) (unpublished).

Ott, W. R.

W. R. Ott, L. R. Canfield, S. C. Ebner, L. R. Hughey, R. P. Madden, “XUV Radiometric Standards at NBS,” Proc. Soc. Photo-Opt. Instrum. Eng. 689, 178 (1986).

Risley, J. S.

A. McPherson, N. Rouze, W. B. Westerveld, J. S. Risley, “Calibration of a VUV Spectrometer–Detector System using Synchrotron Radiation,” Appl. Opt. 25, 298 (1986).
[CrossRef] [PubMed]

W. B. Westerveld, A. McPherson, J. S. Risley, “Synchrotron Radiation Intensity for 50 MeV to 50 GeV Electrons,” At. Data Nucl. Data Tables 28, 21 (1983).
[CrossRef]

J. S. Risley, A. McPherson, W. B. Westerveld, “Use of a Scaling Relationship for Synchrotron Radiation,” Phys. Rev. A 24, 3255 (1981).
[CrossRef]

R. Kendrick, N. Rouze, J. S. Risley, “Apple Sync: Synchrotron Flux Calculation,” (1983). A 16,000 number look-up table based on the algorithms developed for Ref. 5 using an Apple II microcomputer.

Rouze, N.

A. McPherson, N. Rouze, W. B. Westerveld, J. S. Risley, “Calibration of a VUV Spectrometer–Detector System using Synchrotron Radiation,” Appl. Opt. 25, 298 (1986).
[CrossRef] [PubMed]

R. Kendrick, N. Rouze, J. S. Risley, “Apple Sync: Synchrotron Flux Calculation,” (1983). A 16,000 number look-up table based on the algorithms developed for Ref. 5 using an Apple II microcomputer.

Saloman, E. B.

Samson, J. A. R.

J. A. R. Samson, Techniques of Vacuum Ultraviolet Radiation (Pied Publishers, Lincoln, NE, 1967), 348 pp.

Schaefer, A. R.

L. R. Hughey, A. R. Schaefer, “Reduced Absolute Uncertainty in the Irradiance of SURF-II and Instrumentation for Measuring Linearity of X-Ray, XUV and UV Detectors,” Nucl. Instrum. Methods 195, 367 (1982).
[CrossRef]

Tousey, R.

Westerveld, W. B.

A. McPherson, N. Rouze, W. B. Westerveld, J. S. Risley, “Calibration of a VUV Spectrometer–Detector System using Synchrotron Radiation,” Appl. Opt. 25, 298 (1986).
[CrossRef] [PubMed]

W. B. Westerveld, A. McPherson, J. S. Risley, “Synchrotron Radiation Intensity for 50 MeV to 50 GeV Electrons,” At. Data Nucl. Data Tables 28, 21 (1983).
[CrossRef]

J. S. Risley, A. McPherson, W. B. Westerveld, “Use of a Scaling Relationship for Synchrotron Radiation,” Phys. Rev. A 24, 3255 (1981).
[CrossRef]

Appl. Opt. (3)

At. Data Nucl. Data Tables (1)

W. B. Westerveld, A. McPherson, J. S. Risley, “Synchrotron Radiation Intensity for 50 MeV to 50 GeV Electrons,” At. Data Nucl. Data Tables 28, 21 (1983).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

D. L. Ederer et al., “An Overview of Research at NBS using Synchrotron Radiation at SURF II,” IEEE Trans. Nucl. Sci. NS-301020 (1983).
[CrossRef]

Nucl. Instrum. Methods (1)

L. R. Hughey, A. R. Schaefer, “Reduced Absolute Uncertainty in the Irradiance of SURF-II and Instrumentation for Measuring Linearity of X-Ray, XUV and UV Detectors,” Nucl. Instrum. Methods 195, 367 (1982).
[CrossRef]

Phys. Rev. A (1)

J. S. Risley, A. McPherson, W. B. Westerveld, “Use of a Scaling Relationship for Synchrotron Radiation,” Phys. Rev. A 24, 3255 (1981).
[CrossRef]

Physica (1)

B. F. Luyken, F. J. de Heer, R. Ch. Baas, “The Role of the Outer s Shell in Single Ionization of Ne, Ar, Kr, and Xe by Electron Impact,” Physica 61, 200 (1972).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

W. R. Ott, L. R. Canfield, S. C. Ebner, L. R. Hughey, R. P. Madden, “XUV Radiometric Standards at NBS,” Proc. Soc. Photo-Opt. Instrum. Eng. 689, 178 (1986).

Other (4)

R. Kendrick, “A Comparison of Methods for Determining the Second Order Correction to the Detector Efficiency of a VUV Spectrometer,” M.S. Thesis, North Carolina State U. (1984) (unpublished).

R. Kendrick, N. Rouze, J. S. Risley, “Apple Sync: Synchrotron Flux Calculation,” (1983). A 16,000 number look-up table based on the algorithms developed for Ref. 5 using an Apple II microcomputer.

J. A. R. Samson, Techniques of Vacuum Ultraviolet Radiation (Pied Publishers, Lincoln, NE, 1967), 348 pp.

A. McPherson, “Measurement of the Electron Impact Photoemission Cross Sections of the 92.0 nm and 93.2 nm Emission Lines of Ar II for the VUV Radiometric Project,” Ph.D. Thesis. North Carolina State U. (1984) (unpublished).

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Figures (13)

Fig. 1
Fig. 1

Spectrometer-detector system.

Fig. 2
Fig. 2

Flux NE(λ,Ψ0) into the spectrometer vs wavelength for synchrotron radiation produced by 1 mA of 282-MeV electrons at the SURF II storage ring. The entrance slit of the spectrometer is located 12 m from the tangent point. The opening angle To above (and below) the orbital plane is 2.27 × 10−5 rad, and the acceptance angle in the orbital plane Δϕ is 8.3 × 10−6 rad. The bandwidth Δλ is 0.1 nm.

Fig. 3
Fig. 3

Count rate C syn ( α , β , Λ ) vs spectrometer setting Λ measured at the center of the grating (α = β = 0°) for radiation polarized parallel to the grating grooves. The uncorrected data (a) correspond to the observed count rate for synchrotron radiation from 282-MeV electrons at SURF II. The corrected data (b) correspond to a count rate for which the second-order contribution has been subtracted.

Fig. 4
Fig. 4

Geometrical arrangement of the electron–atom source, the spectrometer entrance slit, and the grating. The pair of angles (α,β) characterizes the direction of VUV radiation emitted into the spectrometer from a section dl of the electron beam.

Fig. 5
Fig. 5

Calibration beam line at SURF II with the spectrometer–detector system and foil holder.

Fig. 6
Fig. 6

Count rate C f ( α , β , Λ ) vs spectrometer setting Λ. A spot near the center of the grating was illuminated with synchrotron radiation from 282-MeV electrons polarized parallel to the grooves of the grating. An aluminum filter with a surface density of ~22 μg/cm2 was used to block out radiation above 80 nm.

Fig. 7
Fig. 7

Map of the count rate C j f ( α , β , Λ L 0 / 2 ) and C j f ( α , β , Λ 0 ) in first and second order vs scanning angles α and β. The polarization indices are defined by the grooves in the grating which are parallel to the α scanning direction. This measurement was made using synchrotron radiation produced by 282-MeV electrons. The 22-μg/cm2 aluminum foil was in place. (a) and (b), 46-nm radiation in first order polarized parallel and perpendicular to grooves, respectively; (c) and (d), 46-nm radiation in second order polarized parallel and perpendicular to the grooves, respectively.

Fig. 8
Fig. 8

Count rate C ea ( n ) ( λ o , Λ ) vs spectrometer setting Λ for Ne ii 46-nm line radiation from 300-eV electrons: (a) first order, n = 1; (b) second order, n = 2.

Fig. 9
Fig. 9

Count rate C duo ( n ) ( α , β , λ 0 , Λ ) vs spectrometer setting Λ observed using a duoplasmatron VUV light source illuminating a spot near the center of the grating: (a) first order, n = 1; (b) second order, n = 2.

Fig. 10
Fig. 10

Count rate C duo ( n ) ( α , β , λ 0 ) vs scanning angles β and β observed using a duoplasmatron VUV light source. A 30-point grid was defined for the grating map: (a) first order, n = 1; (b) second order, n = 2.

Fig. 11
Fig. 11

Flux NE(λ,Ψ0) into the spectrometer vs wavelength of synchrotron radiation produced by 1 mA of 140- and 282-MeV electrons at the SURF II storage ring. The angular acceptance angles are the same as in the caption for Fig. 2.

Fig. 12
Fig. 12

Uncorrected apparatus response function S syn (Λ,E)/F(Λ,E) vs spectrometer setting for synchrotron radiation from electrons with energies of 282 and 140 MeV.

Fig. 13
Fig. 13

Count rate C j syn ( α , β , Λ 0 , E ) vs scanning angles α and β at Λ0 = 92 nm observed using synchrotron radiation polarized (a) parallel || and (b) perpendicular ⊥ to the grating grooves, respectively, for an electron energy of 140 MeV and synchrotron radiation polarized (c) parallel || and (d) perpendicular ⊥ to the grating grooves, respectively, for an electron energy of 282 MeV.

Tables (6)

Tables Icon

Table I Sources of Uncertainty in the Signals S f0) and S f0/2) Measured Using the Bandpass Filter Method

Tables Icon

Table II Sources of Uncertainty in the Signal S ea ( n ) ( λ 0 ) Measured in the Electron-Beam Method

Tables Icon

Table III Sources of Uncertainty in the Signals S duo ( n ) ( λ 0 ) Measured Using the Duoplasmatron Method

Tables Icon

Table IV Sources of Uncertainty in the Synchrotron Flux F(Λ,E) at SURF II for the Wavelength Range from 46 to 92 nma

Tables Icon

Table V Ratio k0/2 = 46 nm) Determined Using the Bandpass Filter Method, the Electron—Beam Method, and the Duoplasmatron Methoda

Tables Icon

Table VI Second-Order Contribution Factor [1 − k0/2) S syn0/2)/ S syn0)] to the Measured Absolute Apparatus Response Function Caused by Second-Order Radiation [see Eq. (14)]

Equations (37)

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C j syn ( α , β , Λ 0 ) = 0 F ( λ ) R j ( α , β , λ , Λ 0 ) d λ ,
N E ( λ , Ψ 0 ) = Δ λ F ( λ ) ,
C j ( α , β , Λ 0 ) = F ( Λ 0 ) 0 R j ( α , β , λ , Λ 0 ) d λ .
S j syn ( Λ 0 ) = Λ 0 α 1 α 2 β 1 β 2 C j syn ( α , β , Λ 0 ) d β d α
= Λ 0 F ( Λ 0 ) α 1 α 2 β 1 β 2 0 R j ( α , β , λ , Λ 0 ) d λ d β d α
C j syn ( α , β , Λ 0 ) = 0 F ( λ ) n = 1 R j ( n ) ( α , β , n λ , Λ 0 ) d λ
= n = 1 F ( Λ 0 n ) 0 R j ( n ) ( α , β , n λ , Λ 0 ) d λ ,
S j syn ( Λ 0 ) = α 1 α 2 β 1 β 2 C j syn ( α , β , Λ 0 ) d β d α = n = 1 F ( Λ 0 / n ) R j ( n ) ( Λ 0 ) ,
R j ( n ) ( Λ 0 ) = α 1 α 2 β 1 β 2 0 R j ( n ) ( α , β , n λ , Λ 0 ) d λ d β d α .
S j syn ( Λ 0 ) = F ( Λ 0 ) R j ( 1 ) ( Λ 0 ) + F ( Λ 0 / 2 ) R j ( 2 ) ( Λ 0 ) .
R ( 1 ) ( Λ 0 ) = S syn ( Λ 0 ) F ( Λ 0 ) - F ( Λ 0 / 2 ) F ( Λ 0 ) R ( 2 ) ( Λ 0 ) .
k ( Λ 0 / 2 ) = R ( 2 ) ( Λ 0 ) R ( 1 ) ( Λ 0 / 2 ) .
S syn ( Λ 0 / 2 ) = F ( Λ 0 / 2 ) R ( 1 ) ( Λ 0 / 2 ) .
R ( 1 ) ( Λ 0 ) = S syn ( Λ 0 ) F ( Λ 0 ) [ 1 - k ( Λ 0 / 2 ) S syn ( Λ 0 / 2 ) S syn ( Λ 0 ) ] .
S f ( Λ 0 ) = n = 1 2 F ( Λ 0 / n ) α 1 α 2 β 1 β 2 0 T ( λ ) × R ( n ) ( α , β , n λ , Λ 0 ) d λ d β d α ,
S f ( Λ 0 ) = n = 1 2 F ( Λ 0 / n ) T ( Λ 0 / n ) R ( n ) ( Λ 0 ) .
S f ( Λ 0 ) = F ( Λ 0 / 2 ) T ( Λ 0 / 2 ) R ( 2 ) ( Λ 0 ) ,
S f ( Λ 0 / 2 ) = F ( Λ 0 / 2 ) T ( Λ 0 / 2 ) R ( 1 ) ( Λ 0 / 2 ) ,
k ( Λ 0 / 2 ) = S f ( Λ 0 ) S f ( Λ 0 / 2 ) ,
S ea ( n ) ( λ 0 ) = N ( λ 0 ) w s α 1 α 2 β 1 β 2 Λ - ( n ) Λ + ( n ) R ( n ) ( α , β , n λ 0 , Λ ) d Λ d β d α ,
S ea ( 1 ) ( λ 0 ) = N ( λ 0 ) α 1 α 2 β 1 β 2 Λ - ( 1 ) Λ + ( 1 ) R ( 1 ) ( α , β , λ 0 , Λ ) d Λ d β d α ,
S ea ( 2 ) ( λ 0 ) = N ( λ 0 ) α 1 α 2 β 1 β 2 Λ - ( 2 ) Λ + ( 2 ) R ( 2 ) ( α , β , 2 λ 0 , Λ ) d Λ d β d α .
S ea ( n ) ( λ 0 ) = n N ( λ 0 ) α 1 α 2 β 1 β 2 Λ - ( n ) Λ + ( n ) R ( n ) ( α , β , n λ , Λ 0 ) d λ d β d α ,
S ea ( 1 ) ( λ 0 ) = N ( λ 0 ) R ( 1 ) ( Λ 0 / 2 ) ,
S ea ( 2 ) ( λ 0 ) = 2 N ( λ 0 ) R ( 2 ) ( Λ 0 ) .
k ( Λ 0 / 2 ) = ( ½ ) S ea ( 2 ) ( λ 0 ) S ea ( 1 ) ( λ 0 ) .
S ea ( n ) ( λ 0 ) = i [ C ea ( n ) ( λ 0 , Λ i + 1 ) - B i + 1 ] + [ C ea ( n ) ( λ 0 , Λ i ) - B i ] 2 p i Q i Δ Λ ,
C duo ( n ) ( α , β , λ 0 ) = N ( λ 0 ) Λ - ( n ) Λ + ( n ) R ( n ) ( α , β , n λ 0 , Λ ) d Λ
k ( Λ 0 / 2 ) = ( 1 / 2 ) S duo ( 2 ) ( λ 0 ) S duo ( 1 ) ( λ 0 ) .
S duo ( n ) ( λ 0 ) = α , β C duo ( n ) ( α , β , λ 0 ) Δ α Δ β .
C syn ( α , β , Λ 0 , E ) = γ E n = 1 0 F ( λ , E ) R ( n ) ( α , β , n λ , Λ 0 ) d λ ,
S syn ( Λ 0 , E H ) = F ( Λ 0 , E H ) R ( 1 ) ( Λ 0 ) + F ( Λ 0 / 2 , E H ) R ( 2 ) ( Λ 0 ) ,
S syn ( Λ 0 , E L ) = γ [ F ( Λ 0 , E L ) R ( 1 ) ( Λ 0 ) + F ( Λ 0 / 2 , E L ) R ( 2 ) ( Λ 0 ) ] .
S syn ( Λ , E H ) = F ( Λ , E H ) R ( 1 ) ( Λ ) ,
S syn ( Λ , E L ) = γ F ( Λ , E L ) R ( 1 ) ( Λ ) ,
γ = F ( Λ , E H ) S syn ( Λ , E L ) F ( Λ , E L ) S syn ( Λ , E H ) .
R ( 1 ) ( Λ 0 ) = [ S syn ( Λ 0 , E H ) F ( Λ 0 / 2 , E H ) - S syn ( Λ 0 , E L ) γ F ( Λ 0 / 2 , E L ) ] × [ F ( Λ 0 , E H ) F ( Λ 0 / 2 , E H ) - F ( Λ 0 , E L ) F ( Λ 0 / 2 , E L ) ] - 1 .

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