Abstract

The performances of a spectrometer for the observation of laser plasma absorption with high spectral and spatial resolution are described. Aspherical optics are used to correct the astigmatism in an extended spectral region. In this way only a small portion of the absorbing medium is probed, thus giving a good selection of the ionization stage acting as the absorption. Moreover, in the focal plane the plasma emission from the absorbing medium is spatially separated from the probe beam, with a consequent enhancement of the measurement sensitivity. The predicted optical performances from a ray tracing are compared with experimental observations for both spectral and spatial resolution.

© 2000 Optical Society of America

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References

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  1. J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967).
  2. P. Villoresi, P. Bidoli, P. Nicolosi, “Absorption spectra and oscillator strength ratio measurements for Δn = 1 transitions from excited levels of Be I and Be II,” J. Quant. Spectrosc. Radiat. Transfer 57, 847–857 (1997).
    [CrossRef]
  3. J. T. Costello, J. P. Mosnier, E. T. Kennedy, P. K. Carrol, G. O’Sullivan, “XUV absorption spectroscopy with laser-produced plasmas: a review,” Phys. Scr. T34, 77–92 (1991).
    [CrossRef]
  4. P. Nicolosi, E. Jannitti, G. Tondello, “A review of experimental studies of the photoabsorption spectra of low Z ions,” J. Phys. (IV) C 1, Suppl. 1, 89–98 (1991).
  5. E. Jannitti, M. Gaye, M. Mazzoni, P. Nicolosi, P. Villoresi, “K-shell photoabsorption spectrum of C ii,” Phys. Rev. A 47, 4033–4041 (1993).
    [CrossRef] [PubMed]
  6. E. Jannitti, P. Nicolosi, P. Villoresi, F. Xianping, “Measurement of the K-shell photoionization cross section of C IV through the L-shell photoabsorption spectra,” Phys. Rev. A 49, 314–323 (1995).
    [CrossRef]
  7. P. Nicolosi, P. Villoresi, “Experimental measurement of the Cii L-shell photoabsorption spectrum,” Phys. Rev. A 58, 4985–4988 (1998).
    [CrossRef]
  8. P. Villoresi, P. Nicolosi, “C+ and C++ ion density scaling in laser plasma by ultraviolet photoabsorption spectroscopy,” Hyperfine Interact. 114, 213–216 (1998).
    [CrossRef]
  9. P. Villoresi, P. Nicolosi, “Light ions absorption probed with high spatial, temporal and spectral resolution: the spectrum of CII,” NIST Spec. Publ. 926, 190–193 (1998).
  10. A. Rubenchik, S. Witkowski, Physics of Laser Plasma (North-Holland, Amsterdam, 1991).
  11. W. L. Kruer, The Physics of Laser Plasma Interaction (Addison-Wesley, New York, 1988).
  12. Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: IV. Photoionization cross section for C II,” J. Phys. B 20, 6409–6429 (1987).
    [CrossRef]
  13. Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: III. Oscillator strengths for C II,” J. Phys. B 20, 6399–6408 (1987).
    [CrossRef]
  14. P. Villoresi, G. Naletto, P. Nicolosi, G. Tondello, E. Jannitti, “A new test facility for reflectivity measurements in the extreme ultraviolet spectral region,” in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc SPIE1742, 314–323 (1992).
  15. G. Bonfante, “Light path simulation—ray tracing code,” Master of Science thesis (Universitá degli Studi di Padova, Padova, Italia, 1989).
  16. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1991).
  17. H. Haber, “The torus grating,” J. Opt. Soc. Am. 40, 153–165 (1950).
    [CrossRef]
  18. R. L. Kelly, “Atomic and ionic spectrum below 2000 Å,” J. Phys. Chem. Ref. Data 16, Suppl. 1, Part I, 1–649 (1987).
  19. L. J. Radziemsky, D. A. Cremers, eds., Laser-induced plasmas and applications (Marcel Dekker, New York, 1989).
  20. P. Villoresi, G. Naletto, P. Nicolosi, E. Pace, G. Tondello, “Laser-produced plasma stigmatic observations in the extreme ultraviolet by means of a CCD detector Il,” Nuovo Cimento 19, 759–777 (1997).
    [CrossRef]
  21. P. A. Jansson, Deconvolution of Images and Spectra (Academic, San Diego, Calif., 1997).

1998

P. Nicolosi, P. Villoresi, “Experimental measurement of the Cii L-shell photoabsorption spectrum,” Phys. Rev. A 58, 4985–4988 (1998).
[CrossRef]

P. Villoresi, P. Nicolosi, “C+ and C++ ion density scaling in laser plasma by ultraviolet photoabsorption spectroscopy,” Hyperfine Interact. 114, 213–216 (1998).
[CrossRef]

P. Villoresi, P. Nicolosi, “Light ions absorption probed with high spatial, temporal and spectral resolution: the spectrum of CII,” NIST Spec. Publ. 926, 190–193 (1998).

1997

P. Villoresi, P. Bidoli, P. Nicolosi, “Absorption spectra and oscillator strength ratio measurements for Δn = 1 transitions from excited levels of Be I and Be II,” J. Quant. Spectrosc. Radiat. Transfer 57, 847–857 (1997).
[CrossRef]

P. Villoresi, G. Naletto, P. Nicolosi, E. Pace, G. Tondello, “Laser-produced plasma stigmatic observations in the extreme ultraviolet by means of a CCD detector Il,” Nuovo Cimento 19, 759–777 (1997).
[CrossRef]

1995

E. Jannitti, P. Nicolosi, P. Villoresi, F. Xianping, “Measurement of the K-shell photoionization cross section of C IV through the L-shell photoabsorption spectra,” Phys. Rev. A 49, 314–323 (1995).
[CrossRef]

1993

E. Jannitti, M. Gaye, M. Mazzoni, P. Nicolosi, P. Villoresi, “K-shell photoabsorption spectrum of C ii,” Phys. Rev. A 47, 4033–4041 (1993).
[CrossRef] [PubMed]

1991

J. T. Costello, J. P. Mosnier, E. T. Kennedy, P. K. Carrol, G. O’Sullivan, “XUV absorption spectroscopy with laser-produced plasmas: a review,” Phys. Scr. T34, 77–92 (1991).
[CrossRef]

P. Nicolosi, E. Jannitti, G. Tondello, “A review of experimental studies of the photoabsorption spectra of low Z ions,” J. Phys. (IV) C 1, Suppl. 1, 89–98 (1991).

1987

Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: IV. Photoionization cross section for C II,” J. Phys. B 20, 6409–6429 (1987).
[CrossRef]

Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: III. Oscillator strengths for C II,” J. Phys. B 20, 6399–6408 (1987).
[CrossRef]

R. L. Kelly, “Atomic and ionic spectrum below 2000 Å,” J. Phys. Chem. Ref. Data 16, Suppl. 1, Part I, 1–649 (1987).

1950

Bidoli, P.

P. Villoresi, P. Bidoli, P. Nicolosi, “Absorption spectra and oscillator strength ratio measurements for Δn = 1 transitions from excited levels of Be I and Be II,” J. Quant. Spectrosc. Radiat. Transfer 57, 847–857 (1997).
[CrossRef]

Bonfante, G.

G. Bonfante, “Light path simulation—ray tracing code,” Master of Science thesis (Universitá degli Studi di Padova, Padova, Italia, 1989).

Carrol, P. K.

J. T. Costello, J. P. Mosnier, E. T. Kennedy, P. K. Carrol, G. O’Sullivan, “XUV absorption spectroscopy with laser-produced plasmas: a review,” Phys. Scr. T34, 77–92 (1991).
[CrossRef]

Costello, J. T.

J. T. Costello, J. P. Mosnier, E. T. Kennedy, P. K. Carrol, G. O’Sullivan, “XUV absorption spectroscopy with laser-produced plasmas: a review,” Phys. Scr. T34, 77–92 (1991).
[CrossRef]

Gaye, M.

E. Jannitti, M. Gaye, M. Mazzoni, P. Nicolosi, P. Villoresi, “K-shell photoabsorption spectrum of C ii,” Phys. Rev. A 47, 4033–4041 (1993).
[CrossRef] [PubMed]

Haber, H.

Jannitti, E.

E. Jannitti, P. Nicolosi, P. Villoresi, F. Xianping, “Measurement of the K-shell photoionization cross section of C IV through the L-shell photoabsorption spectra,” Phys. Rev. A 49, 314–323 (1995).
[CrossRef]

E. Jannitti, M. Gaye, M. Mazzoni, P. Nicolosi, P. Villoresi, “K-shell photoabsorption spectrum of C ii,” Phys. Rev. A 47, 4033–4041 (1993).
[CrossRef] [PubMed]

P. Nicolosi, E. Jannitti, G. Tondello, “A review of experimental studies of the photoabsorption spectra of low Z ions,” J. Phys. (IV) C 1, Suppl. 1, 89–98 (1991).

P. Villoresi, G. Naletto, P. Nicolosi, G. Tondello, E. Jannitti, “A new test facility for reflectivity measurements in the extreme ultraviolet spectral region,” in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc SPIE1742, 314–323 (1992).

Jansson, P. A.

P. A. Jansson, Deconvolution of Images and Spectra (Academic, San Diego, Calif., 1997).

Kelly, R. L.

R. L. Kelly, “Atomic and ionic spectrum below 2000 Å,” J. Phys. Chem. Ref. Data 16, Suppl. 1, Part I, 1–649 (1987).

Kennedy, E. T.

J. T. Costello, J. P. Mosnier, E. T. Kennedy, P. K. Carrol, G. O’Sullivan, “XUV absorption spectroscopy with laser-produced plasmas: a review,” Phys. Scr. T34, 77–92 (1991).
[CrossRef]

Kruer, W. L.

W. L. Kruer, The Physics of Laser Plasma Interaction (Addison-Wesley, New York, 1988).

Mazzoni, M.

E. Jannitti, M. Gaye, M. Mazzoni, P. Nicolosi, P. Villoresi, “K-shell photoabsorption spectrum of C ii,” Phys. Rev. A 47, 4033–4041 (1993).
[CrossRef] [PubMed]

Mosnier, J. P.

J. T. Costello, J. P. Mosnier, E. T. Kennedy, P. K. Carrol, G. O’Sullivan, “XUV absorption spectroscopy with laser-produced plasmas: a review,” Phys. Scr. T34, 77–92 (1991).
[CrossRef]

Naletto, G.

P. Villoresi, G. Naletto, P. Nicolosi, E. Pace, G. Tondello, “Laser-produced plasma stigmatic observations in the extreme ultraviolet by means of a CCD detector Il,” Nuovo Cimento 19, 759–777 (1997).
[CrossRef]

P. Villoresi, G. Naletto, P. Nicolosi, G. Tondello, E. Jannitti, “A new test facility for reflectivity measurements in the extreme ultraviolet spectral region,” in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc SPIE1742, 314–323 (1992).

Nicolosi, P.

P. Nicolosi, P. Villoresi, “Experimental measurement of the Cii L-shell photoabsorption spectrum,” Phys. Rev. A 58, 4985–4988 (1998).
[CrossRef]

P. Villoresi, P. Nicolosi, “C+ and C++ ion density scaling in laser plasma by ultraviolet photoabsorption spectroscopy,” Hyperfine Interact. 114, 213–216 (1998).
[CrossRef]

P. Villoresi, P. Nicolosi, “Light ions absorption probed with high spatial, temporal and spectral resolution: the spectrum of CII,” NIST Spec. Publ. 926, 190–193 (1998).

P. Villoresi, P. Bidoli, P. Nicolosi, “Absorption spectra and oscillator strength ratio measurements for Δn = 1 transitions from excited levels of Be I and Be II,” J. Quant. Spectrosc. Radiat. Transfer 57, 847–857 (1997).
[CrossRef]

P. Villoresi, G. Naletto, P. Nicolosi, E. Pace, G. Tondello, “Laser-produced plasma stigmatic observations in the extreme ultraviolet by means of a CCD detector Il,” Nuovo Cimento 19, 759–777 (1997).
[CrossRef]

E. Jannitti, P. Nicolosi, P. Villoresi, F. Xianping, “Measurement of the K-shell photoionization cross section of C IV through the L-shell photoabsorption spectra,” Phys. Rev. A 49, 314–323 (1995).
[CrossRef]

E. Jannitti, M. Gaye, M. Mazzoni, P. Nicolosi, P. Villoresi, “K-shell photoabsorption spectrum of C ii,” Phys. Rev. A 47, 4033–4041 (1993).
[CrossRef] [PubMed]

P. Nicolosi, E. Jannitti, G. Tondello, “A review of experimental studies of the photoabsorption spectra of low Z ions,” J. Phys. (IV) C 1, Suppl. 1, 89–98 (1991).

P. Villoresi, G. Naletto, P. Nicolosi, G. Tondello, E. Jannitti, “A new test facility for reflectivity measurements in the extreme ultraviolet spectral region,” in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc SPIE1742, 314–323 (1992).

O’Sullivan, G.

J. T. Costello, J. P. Mosnier, E. T. Kennedy, P. K. Carrol, G. O’Sullivan, “XUV absorption spectroscopy with laser-produced plasmas: a review,” Phys. Scr. T34, 77–92 (1991).
[CrossRef]

Pace, E.

P. Villoresi, G. Naletto, P. Nicolosi, E. Pace, G. Tondello, “Laser-produced plasma stigmatic observations in the extreme ultraviolet by means of a CCD detector Il,” Nuovo Cimento 19, 759–777 (1997).
[CrossRef]

Rubenchik, A.

A. Rubenchik, S. Witkowski, Physics of Laser Plasma (North-Holland, Amsterdam, 1991).

Samson, J. A. R.

J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967).

Seaton, M. J.

Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: IV. Photoionization cross section for C II,” J. Phys. B 20, 6409–6429 (1987).
[CrossRef]

Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: III. Oscillator strengths for C II,” J. Phys. B 20, 6399–6408 (1987).
[CrossRef]

Tondello, G.

P. Villoresi, G. Naletto, P. Nicolosi, E. Pace, G. Tondello, “Laser-produced plasma stigmatic observations in the extreme ultraviolet by means of a CCD detector Il,” Nuovo Cimento 19, 759–777 (1997).
[CrossRef]

P. Nicolosi, E. Jannitti, G. Tondello, “A review of experimental studies of the photoabsorption spectra of low Z ions,” J. Phys. (IV) C 1, Suppl. 1, 89–98 (1991).

P. Villoresi, G. Naletto, P. Nicolosi, G. Tondello, E. Jannitti, “A new test facility for reflectivity measurements in the extreme ultraviolet spectral region,” in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc SPIE1742, 314–323 (1992).

Villoresi, P.

P. Nicolosi, P. Villoresi, “Experimental measurement of the Cii L-shell photoabsorption spectrum,” Phys. Rev. A 58, 4985–4988 (1998).
[CrossRef]

P. Villoresi, P. Nicolosi, “C+ and C++ ion density scaling in laser plasma by ultraviolet photoabsorption spectroscopy,” Hyperfine Interact. 114, 213–216 (1998).
[CrossRef]

P. Villoresi, P. Nicolosi, “Light ions absorption probed with high spatial, temporal and spectral resolution: the spectrum of CII,” NIST Spec. Publ. 926, 190–193 (1998).

P. Villoresi, P. Bidoli, P. Nicolosi, “Absorption spectra and oscillator strength ratio measurements for Δn = 1 transitions from excited levels of Be I and Be II,” J. Quant. Spectrosc. Radiat. Transfer 57, 847–857 (1997).
[CrossRef]

P. Villoresi, G. Naletto, P. Nicolosi, E. Pace, G. Tondello, “Laser-produced plasma stigmatic observations in the extreme ultraviolet by means of a CCD detector Il,” Nuovo Cimento 19, 759–777 (1997).
[CrossRef]

E. Jannitti, P. Nicolosi, P. Villoresi, F. Xianping, “Measurement of the K-shell photoionization cross section of C IV through the L-shell photoabsorption spectra,” Phys. Rev. A 49, 314–323 (1995).
[CrossRef]

E. Jannitti, M. Gaye, M. Mazzoni, P. Nicolosi, P. Villoresi, “K-shell photoabsorption spectrum of C ii,” Phys. Rev. A 47, 4033–4041 (1993).
[CrossRef] [PubMed]

P. Villoresi, G. Naletto, P. Nicolosi, G. Tondello, E. Jannitti, “A new test facility for reflectivity measurements in the extreme ultraviolet spectral region,” in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc SPIE1742, 314–323 (1992).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1991).

Witkowski, S.

A. Rubenchik, S. Witkowski, Physics of Laser Plasma (North-Holland, Amsterdam, 1991).

Xianping, F.

E. Jannitti, P. Nicolosi, P. Villoresi, F. Xianping, “Measurement of the K-shell photoionization cross section of C IV through the L-shell photoabsorption spectra,” Phys. Rev. A 49, 314–323 (1995).
[CrossRef]

Yan, Y.

Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: IV. Photoionization cross section for C II,” J. Phys. B 20, 6409–6429 (1987).
[CrossRef]

Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: III. Oscillator strengths for C II,” J. Phys. B 20, 6399–6408 (1987).
[CrossRef]

Hyperfine Interact.

P. Villoresi, P. Nicolosi, “C+ and C++ ion density scaling in laser plasma by ultraviolet photoabsorption spectroscopy,” Hyperfine Interact. 114, 213–216 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Phys. (IV)

P. Nicolosi, E. Jannitti, G. Tondello, “A review of experimental studies of the photoabsorption spectra of low Z ions,” J. Phys. (IV) C 1, Suppl. 1, 89–98 (1991).

J. Phys. B

Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: IV. Photoionization cross section for C II,” J. Phys. B 20, 6409–6429 (1987).
[CrossRef]

Y. Yan, M. J. Seaton, “Atomic data for opacity calculations: III. Oscillator strengths for C II,” J. Phys. B 20, 6399–6408 (1987).
[CrossRef]

J. Phys. Chem. Ref. Data

R. L. Kelly, “Atomic and ionic spectrum below 2000 Å,” J. Phys. Chem. Ref. Data 16, Suppl. 1, Part I, 1–649 (1987).

J. Quant. Spectrosc. Radiat. Transfer

P. Villoresi, P. Bidoli, P. Nicolosi, “Absorption spectra and oscillator strength ratio measurements for Δn = 1 transitions from excited levels of Be I and Be II,” J. Quant. Spectrosc. Radiat. Transfer 57, 847–857 (1997).
[CrossRef]

NIST Spec. Publ.

P. Villoresi, P. Nicolosi, “Light ions absorption probed with high spatial, temporal and spectral resolution: the spectrum of CII,” NIST Spec. Publ. 926, 190–193 (1998).

Nuovo Cimento

P. Villoresi, G. Naletto, P. Nicolosi, E. Pace, G. Tondello, “Laser-produced plasma stigmatic observations in the extreme ultraviolet by means of a CCD detector Il,” Nuovo Cimento 19, 759–777 (1997).
[CrossRef]

Phys. Scr.

J. T. Costello, J. P. Mosnier, E. T. Kennedy, P. K. Carrol, G. O’Sullivan, “XUV absorption spectroscopy with laser-produced plasmas: a review,” Phys. Scr. T34, 77–92 (1991).
[CrossRef]

Phys. Rev. A

P. Nicolosi, P. Villoresi, “Experimental measurement of the Cii L-shell photoabsorption spectrum,” Phys. Rev. A 58, 4985–4988 (1998).
[CrossRef]

Phys. Rev. A

E. Jannitti, M. Gaye, M. Mazzoni, P. Nicolosi, P. Villoresi, “K-shell photoabsorption spectrum of C ii,” Phys. Rev. A 47, 4033–4041 (1993).
[CrossRef] [PubMed]

E. Jannitti, P. Nicolosi, P. Villoresi, F. Xianping, “Measurement of the K-shell photoionization cross section of C IV through the L-shell photoabsorption spectra,” Phys. Rev. A 49, 314–323 (1995).
[CrossRef]

Other

A. Rubenchik, S. Witkowski, Physics of Laser Plasma (North-Holland, Amsterdam, 1991).

W. L. Kruer, The Physics of Laser Plasma Interaction (Addison-Wesley, New York, 1988).

J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (Wiley, New York, 1967).

L. J. Radziemsky, D. A. Cremers, eds., Laser-induced plasmas and applications (Marcel Dekker, New York, 1989).

P. Villoresi, G. Naletto, P. Nicolosi, G. Tondello, E. Jannitti, “A new test facility for reflectivity measurements in the extreme ultraviolet spectral region,” in Multilayer and Grazing Incidence X-Ray/EUV Optics for Astronomy and Projection Lithography, R. B. Hoover, A. B. Walker, eds., Proc SPIE1742, 314–323 (1992).

G. Bonfante, “Light path simulation—ray tracing code,” Master of Science thesis (Universitá degli Studi di Padova, Padova, Italia, 1989).

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1991).

P. A. Jansson, Deconvolution of Images and Spectra (Academic, San Diego, Calif., 1997).

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Figures (7)

Fig. 1
Fig. 1

Ray tracing simulation of the image projected by the toroidal mirror for a point source and parameters reported in Table 1.

Fig. 2
Fig. 2

Scheme of the experiment setup. BR, backradiator plasma; TM, toroidal mirror; PC, absorbing plasma; TG, toroidal grating; RCC, Rowland circle center; D, detector.

Fig. 3
Fig. 3

Emission from a tungsten laser plasma. Horizontal axis, spectral direction (0.91 nm); vertical axis, spatial direction (3.3 mm). (a) image taken at 49 nm, best stigmatic position; (b) image taken at 69 nm.

Fig. 4
Fig. 4

Spatial FWHM of the continuous emission from a tungsten plasma versus wavelength.

Fig. 5
Fig. 5

Ray-tracing simulation of three spectral images of a point source.

Fig. 6
Fig. 6

False-colors image of the Ciii transition at 45.95 nm. The image spans 0.8 nm of spectral width (horizontal) and 2.9 mm along the slit (vertical).

Fig. 7
Fig. 7

Spectral profile of the Ciii transition reported in Fig. 6, taken 300 µm above target. Solid curve, experiment; dotted curve, synthesis.

Tables (4)

Tables Icon

Table 1 Parameters of the Toroidal Mirrora

Tables Icon

Table 2 Optical Parameters for the Characteristic Position in the Sagittal Focusing Analysis

Tables Icon

Table 3 Root Mean Square Widths of the Ray Tracing and Experimental Spot, in Y (Spectral) and Z (Spatial) Directions

Tables Icon

Table 4 Identification of the Transitions Used for the Analysis of the Spectral Resolutiona

Equations (13)

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

I λ = I 0 λ exp - k λ ,
k λ = σ λ plasma n l d l ,
k λ = π e 2 λ 2 mc 2   f   ϕ λ plasma n l d l ,
Δ k k 2 = Δ I 0 / I 0 2 + Δ I / I 2 k 2 ,
1 p + 1 q = 2 R   cos   α
1 p + 1 q = 2   cos   α ρ ,
cos 2 α p - cos α R + cos 2 β q - cos β R = 0
1 p - cos α ρ + 1 q - cos β ρ = 0 ,
ρ R = cos   α   cos   β .
A = L / ρ cos α [ cos α + cos β ] ρ - R   cos α cos β ,
m λ d = 2   cos Φ 2 sin α - Φ 2 cos γ 2 - cos α - Φ 2 sin γ 2 ,
λ c nm = 37.22 ± 0.04 + 0.882 ± 0.001 RR .
S λ = 0.2   0.095 2 π λ - 45.97 2 + ¼ 0.095 2 × exp i a i γ i 2 π λ - λ i 2 + ¼ γ i 2

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