Abstract

A method aimed to minimize the impact of spectral aberrations in a monochromator is proposed in which the spectrum of the source of radiation under study is scanned by the rectilinear translation of a plane chirped grating. The chirped grating, which has a spatially variable groove spacing, is used to diffract and to spectrally focus the radiation. Imaging properties of the chirped grating were analyzed in order to develop the expression of the aberration coefficients of the system and the expression of the width of the instrument line shape due to aberrations. The optimal rectilinear trajectory required to operate the monochromator without significant spectral aberrations in measurements has been obtained numerically and tested in the laboratory. Experimental measurements of the emission spectrum of a seven- wavelength helium–neon laser are presented, as well as the sensitivity of the monochromator performance to different geometrical parameters.

© 2008 Optical Society of America

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References

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  1. S. Singh, “Diffraction gratings: aberrations and applications,” Opt. Laser Technol. 31, 195-218 (1999).
    [CrossRef]
  2. C. Palmer, Diffraction Gratings Handbook, 5th ed. (Richardson Grating Laboratory, 2002).
  3. M. Born and E. Wolf, Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 5th ed. (Pergamon Press, 1975).
    [PubMed]
  4. F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1993).
  5. J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (John Wiley & Sons, 1967).
  6. M. C. Hettrick and S. Bowyer, “Variable line-space gratings: new designs for use in grazing incidence spectrometers,” Appl. Opt. 22, 3921-3924 (1983).
    [CrossRef] [PubMed]
  7. M. Itou, T. Harada, and T. Kita, “Soft x-ray monochromator with a varied-space plane grating for synchrotron radiation: design and evaluation,” Appl. Opt. 28, 146-153 (1989).
    [CrossRef] [PubMed]
  8. T. Harada, “Design and application of a varied-space plane grating monochromator for synchrotron radiation,” Nucl. Instrum. Methods Phys. Res. A 291, 179-184 (1990).
    [CrossRef]
  9. E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
    [CrossRef]
  10. M. Koike and T. Namioka, “Grazing-incidence Monk-Gillieson monochromator based on surface normal rotation of a varied-line-spacing grating,” Appl. Opt. 41, 245-257 (2002).
    [CrossRef] [PubMed]
  11. L. Poletto, “Off-axis pivot mounting for aberration-corrected concave gratings at normal incidence,” Appl. Opt. 39, 1084-1093 (2000).
    [CrossRef]
  12. L. Poletto and R. J. Thomas, “Stigmatic spectrometers for extended sources: design with toroidal varied-line-space gratings,” Appl. Opt. 43, 2029-2038 (2004).
    [CrossRef] [PubMed]
  13. R. Grange, “Aberration-reduced holographic spherical gratings for Rowland circle spectrographs,” Appl. Opt. 31, 3744-3749 (1992).
    [CrossRef] [PubMed]
  14. L.-J. Lu, “Coma correction and extension of the focusing geometry of a soft-x-ray monochromator,” Appl. Opt. 34, 5780-5786 (1995).
    [CrossRef] [PubMed]
  15. T. Harada, H. Sakuma, K. Takahashi, T. Watanabe, H. Hara, and T. Kita, “Design of a high-resolution extreme-ultraviolet imaging spectrometer with aberration-corrected concave gratings,” Appl. Opt. 37, 6803-6810 (1998).
    [CrossRef]
  16. G. Fortin and N. McCarthy, “Chirped holographic grating used as the dispersive element in an optical spectrometer,” Appl. Opt. 44, 4874-4883 (2005).
    [CrossRef] [PubMed]
  17. A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. , 271, 327-331 (2007).
    [CrossRef]
  18. H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031-1036 (1974).
    [CrossRef]
  19. C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham, Wash.) 33, 820-829 (1994).
    [CrossRef]
  20. CRC Handbook Of Laser Science And Technology, Vol. II, Gas Lasers (CRC Press, 1982), pp. 58-60.
  21. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566-1573 (1996).
    [CrossRef] [PubMed]

2007

A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. , 271, 327-331 (2007).
[CrossRef]

2005

2004

2002

2000

1999

S. Singh, “Diffraction gratings: aberrations and applications,” Opt. Laser Technol. 31, 195-218 (1999).
[CrossRef]

E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
[CrossRef]

1998

1996

1995

1994

C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham, Wash.) 33, 820-829 (1994).
[CrossRef]

1992

1990

T. Harada, “Design and application of a varied-space plane grating monochromator for synchrotron radiation,” Nucl. Instrum. Methods Phys. Res. A 291, 179-184 (1990).
[CrossRef]

1989

1983

1974

April, A.

A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. , 271, 327-331 (2007).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 5th ed. (Pergamon Press, 1975).
[PubMed]

Bowyer, S.

Ciddor, P. E.

Fortin, G.

Grange, R.

Hara, H.

Harada, T.

Hettrick, M. C.

Ishiguro, E.

E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
[CrossRef]

Ishikawa, T.

E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
[CrossRef]

Itou, M.

Kamizato, M.

E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
[CrossRef]

Kita, T.

Koike, M.

Lu, L.-J.

Lu, Li-jun

E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
[CrossRef]

McCarthy, N.

A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. , 271, 327-331 (2007).
[CrossRef]

G. Fortin and N. McCarthy, “Chirped holographic grating used as the dispersive element in an optical spectrometer,” Appl. Opt. 44, 4874-4883 (2005).
[CrossRef] [PubMed]

McKinney, W. R.

C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham, Wash.) 33, 820-829 (1994).
[CrossRef]

Namioka, T.

Noda, H.

Ohashi, H.

E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
[CrossRef]

Palmer, C.

C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham, Wash.) 33, 820-829 (1994).
[CrossRef]

C. Palmer, Diffraction Gratings Handbook, 5th ed. (Richardson Grating Laboratory, 2002).

Pedrotti, F. L.

F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1993).

Pedrotti, L. S.

F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1993).

Poletto, L.

Sakuma, H.

Samson, J. A. R.

J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (John Wiley & Sons, 1967).

Seya, M.

Singh, S.

S. Singh, “Diffraction gratings: aberrations and applications,” Opt. Laser Technol. 31, 195-218 (1999).
[CrossRef]

Takahashi, K.

Thomas, R. J.

Watanabe, T.

Watari, W.

E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 5th ed. (Pergamon Press, 1975).
[PubMed]

Appl. Opt.

M. Koike and T. Namioka, “Grazing-incidence Monk-Gillieson monochromator based on surface normal rotation of a varied-line-spacing grating,” Appl. Opt. 41, 245-257 (2002).
[CrossRef] [PubMed]

L. Poletto, “Off-axis pivot mounting for aberration-corrected concave gratings at normal incidence,” Appl. Opt. 39, 1084-1093 (2000).
[CrossRef]

L. Poletto and R. J. Thomas, “Stigmatic spectrometers for extended sources: design with toroidal varied-line-space gratings,” Appl. Opt. 43, 2029-2038 (2004).
[CrossRef] [PubMed]

R. Grange, “Aberration-reduced holographic spherical gratings for Rowland circle spectrographs,” Appl. Opt. 31, 3744-3749 (1992).
[CrossRef] [PubMed]

L.-J. Lu, “Coma correction and extension of the focusing geometry of a soft-x-ray monochromator,” Appl. Opt. 34, 5780-5786 (1995).
[CrossRef] [PubMed]

T. Harada, H. Sakuma, K. Takahashi, T. Watanabe, H. Hara, and T. Kita, “Design of a high-resolution extreme-ultraviolet imaging spectrometer with aberration-corrected concave gratings,” Appl. Opt. 37, 6803-6810 (1998).
[CrossRef]

G. Fortin and N. McCarthy, “Chirped holographic grating used as the dispersive element in an optical spectrometer,” Appl. Opt. 44, 4874-4883 (2005).
[CrossRef] [PubMed]

M. C. Hettrick and S. Bowyer, “Variable line-space gratings: new designs for use in grazing incidence spectrometers,” Appl. Opt. 22, 3921-3924 (1983).
[CrossRef] [PubMed]

M. Itou, T. Harada, and T. Kita, “Soft x-ray monochromator with a varied-space plane grating for synchrotron radiation: design and evaluation,” Appl. Opt. 28, 146-153 (1989).
[CrossRef] [PubMed]

P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566-1573 (1996).
[CrossRef] [PubMed]

J. Electron Spectrosc. Relat. Phenom.

E. Ishiguro, H. Ohashi, Li-jun Lu, W. Watari, M. Kamizato, and T. Ishikawa, “Design of a monochromator with varied line space plane gratings for a soft X-ray undulator beamline of SPring-8,” J. Electron Spectrosc. Relat. Phenom. 101-103, 979-984 (1999).
[CrossRef]

J. Opt. Soc. Am.

Nucl. Instrum. Methods Phys. Res. A

T. Harada, “Design and application of a varied-space plane grating monochromator for synchrotron radiation,” Nucl. Instrum. Methods Phys. Res. A 291, 179-184 (1990).
[CrossRef]

Opt. Commun.

A. April and N. McCarthy, “ABCD-matrix elements for a chirped diffraction grating,” Opt. Commun. , 271, 327-331 (2007).
[CrossRef]

Opt. Eng. (Bellingham, Wash.)

C. Palmer and W. R. McKinney, “Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. (Bellingham, Wash.) 33, 820-829 (1994).
[CrossRef]

Opt. Laser Technol.

S. Singh, “Diffraction gratings: aberrations and applications,” Opt. Laser Technol. 31, 195-218 (1999).
[CrossRef]

Other

C. Palmer, Diffraction Gratings Handbook, 5th ed. (Richardson Grating Laboratory, 2002).

M. Born and E. Wolf, Principles of Optics Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 5th ed. (Pergamon Press, 1975).
[PubMed]

F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice-Hall, 1993).

J. A. R. Samson, Techniques of Vacuum Ultraviolet Spectroscopy (John Wiley & Sons, 1967).

CRC Handbook Of Laser Science And Technology, Vol. II, Gas Lasers (CRC Press, 1982), pp. 58-60.

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of the experimental setup used to write the CG.

Fig. 2
Fig. 2

Local groove spacing Λ ( x ) of the holographic CG as a function of position x. The theoretical curve is calculated with Eqs. (1a, 1b, 1c) and data of Table 1.

Fig. 3
Fig. 3

Geometry of the incident rays and diffracted rays on a CG in the dispersion plane ( y = 0 ).

Fig. 4
Fig. 4

Optical monochromator (a) using a constant-period grating with collimating and focusing components, (b) using a CG without any other component. The translation axis makes an angle Θ with the CG axis.

Fig. 5
Fig. 5

Representation of the geometrical parameters in the CG monochromator for both the reference position and an arbitrary position.

Fig. 6
Fig. 6

Ratio of the ILS width due to aberrations to the geometrical ILS width as a function of the wavelength in the optimal configuration of the CG monochromator.

Fig. 7
Fig. 7

Spectrum of the He–Ne tunable laser measured with the CG monochromator.

Fig. 8
Fig. 8

Normalized spectral lines of the He–Ne measured with the CG monochromator.

Fig. 9
Fig. 9

Experimental ILS width as a function of the difference between (a) the angle of incidence and its optimal value and (b) the translation angle and its optimal value.

Tables (4)

Tables Icon

Table 1 Parameters of Fabrication of the Holographic Grating

Tables Icon

Table 2 Configuration of the CG Monochromator

Tables Icon

Table 3 Wavelengths of the He–Ne Tunable Laser

Tables Icon

Table 4 Measured and Theoretical FWHM of the Five Wavelengths under Study

Equations (41)

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Λ ( x , y ) = λ w | G 1 G 2 | ,
G j = sin θ j + x cos 2 θ j R j ( x ) sin θ j 2 R j 2 ( x ) ( x 2 cos 2 θ j + y 2 ) ,
R j ( x ) = z o j + x sin θ j ,
sin θ m = sin θ i + m λ Λ ( x ) .
Λ ( x ) = Λ ( 0 ) ( 1 + Λ 1 x + Λ 2 x 2 + Λ 3 x 3 ) ,
cos 2 θ i r i + cos 2 θ m r m = m λ Λ ( x o ) Λ 1 ( x o ) ,
Δ x m = r m cos θ m d W d x = r m cos θ m Σ n = 2 n F n ( x x o ) n 1 ,
P m 1 r m λ θ m = Λ ( x o ) cos θ m m r m ,
Δ λ a = 2 | P m Δ x m | ,
Δ λ a = | 2 Λ ( x o ) m n = 2 n F n ( L 2 ) n 1 | | Λ ( x o ) m ( 2 F 2 L + 3 2 F 3 L 2 + F 4 L 3 ) | .
I ( β ) = I max sinc 2 ( w e λ sin β ) ,
L r i Δ β cos θ i = r i λ w e cos θ i = r i Λ ( x o ) m w e [ sin θ m cos θ i tan θ i ] ,
sin θ m , ref = sin θ i + m λ ref Λ ( x ref ) ,
cos 2 θ i r i , ref + cos 2 θ m , ref r m , ref = m λ ref Λ ( x ref ) Λ 1 ( x ref ) ,
Λ 1 ( x ref ) [ 1 Λ ( x ) d Λ ( x ) d x ] x = x ref = Λ ( 0 ) ( Λ 1 + 2 Λ 2 x ref + 3 Λ 3 x ref 2 ) Λ ( x ref ) ,
x o ( t ) = x ref ( t t ref ) cos ( Θ θ i ) cos θ i ,
r i = r i , ref ( t t ref ) sin Θ cos θ i ,
r m = [ r m , ref 2 + ( ( t t ref ) sin Θ cos θ i ) 2 2 r m , ref ( t t ref ) sin Θ cos θ i cos ( θ i + θ m , ref ) ] 1 / 2
θ m = arcsin [ r m , ref sin ( θ i + θ m , ref ) r m ] θ i .
λ = Λ ( x o ) m ( sin θ m sin θ i ) .
g ( θ i , Θ ) t min t max Δ λ a ( θ i , Θ , t ) d t = t min t max | Λ ( x o ) m ( 2 F 2 L + 3 2 F 3 L 2 + F 4 L 3 ) | d t .
g ˜ ( θ i , Θ ) = k Δ λ a ( θ i , Θ , t k ) ,
Δ λ g = { [ w s Λ ( x o ) cos θ m ] / | m | r m for w s > | w e | [ w e Λ ( x o ) cos θ i ] / | m | f for w s < | w e | ,
| w e | = r m cos θ i f cos θ m w e .
Δ λ d = λ Λ ( x o ) | m | D ,
W = APB ¯ AOB ¯ + N m λ ,
AOB ¯ = r i + r m ,
APB ¯ = [ ( r i sin θ i + x ) 2 + r i 2 cos 2 θ i ] 1 / 2 + [ ( r m sin θ m x ) 2 + r m 2 cos 2 θ m ] 1 / 2 .
N ( x ) x o x 1 Λ ( x ) d x = ( x x o ) Λ ( x o ) + n = 1 b n Λ ( x o ) ( x x o ) n + 1 ( n + 1 ) ,
b n = ( 1 ) n | Λ 1 ( x o ) 1 0 ... 0 Λ 2 ( x o ) Λ 1 ( x o ) 1 ... 0 . . . . . Λ n 1 ( x o ) Λ n 2 ( x o ) Λ n 3 ( x o ) ... 1 Λ n ( x o ) Λ n 1 ( x o ) Λ n 2 ( x o ) ... Λ 1 ( x o ) | n × n ,
Λ n ( x o ) 1 n ! Λ ( x o ) d n Λ ( x ) d x n | x = x o ,
Λ ( x o ) = Λ ( 0 ) ( 1 + Λ 1 x o + Λ 2 x o 2 + Λ 3 x o 3 ) ,
Λ 1 ( x o ) = [ 1 Λ ( x ) d Λ ( x ) d x ] x = x o = Λ ( 0 ) ( Λ 1 + 2 Λ 2 x o + 3 Λ 3 x o 2 ) Λ ( x o ) ,
Λ 2 ( x o ) = 1 2 [ 1 Λ ( x ) d 2 Λ ( x ) d x 2 ] x = x o = Λ ( 0 ) ( Λ 2 + 3 Λ 3 x o ) Λ ( x o ) ,
Λ 3 ( x o ) = 1 3 ! [ 1 Λ ( x ) d 3 Λ ( x ) d x 3 ] x = x o = Λ ( 0 ) Λ 3 Λ ( x o ) .
W ( x ) = n = 0 F n ( x x o ) n ,
F n = 1 n ! d n W d x n | x = x o .
F 2 = 1 2 ( cos 2 θ i r i + cos 2 θ m r m ) m λ 2 Λ ( x o ) Λ 1 ( x o ) ,
F 3 = 1 2 ( sin θ i cos 2 θ i r i 2 + sin θ m cos 2 θ m r m 2 ) + m λ 3 Λ ( x o ) ( Λ 1 2 ( x o ) Λ 2 ( x o ) ) ,
F 4 = 1 8 ( cos 2 θ i ( 4 5 cos 2 θ i ) r i 3 + cos 2 θ m ( 4 5 cos 2 θ m ) r m 3 ) + m λ 4 Λ ( x o ) ( Λ 1 3 ( x o ) + 2 Λ 1 ( x o ) Λ 2 ( x o ) Λ 3 ( x o ) )
m λ Λ ( x o ) = sin θ m sin θ i .

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