Abstract

The development of third-generation synchrotron sources has stimulated efforts toward high-resolution monochromators. A good knowledge of grating efficiency is needed to achieve an optimal compromise between resolution and photon flux. Because simple geometric models fail to describe correctly the gratings properties in the UV–to–soft-X-ray range, we have developed a simulation software based on differential theory. A simplified R-matrix propagation algorithm assures the numerical stability of the code for deep gratings. Our numerical results are compared with previous research on deep gratings. Experimental and numerical studies have been performed on some test cases at a synchrotron source. Very good agreement between numerical prediction and measurement has been found.

© 1998 Optical Society of America

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References

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  1. A. G. Michette, Optical Systems for Soft X Rays (Plenum, New York, 1986).
    [CrossRef]
  2. M. Neviere, J. Flamand, J. M. Lerner, “Optimization of gratings for soft-x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
    [CrossRef]
  3. E. Delcamp, F. Polack, B. Lagarde, “SM-PGM monochromators design for large spectral range,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 48–57 (1997).
    [CrossRef]
  4. E. DelCamp, B. Lagarde, F. Polack, J. Susini, “Optimization software for X-UV monochromators,” in Optics for High Brightness Synchrotron Radiation Beamlines, L. E. Berman, J. Arthur, eds., Proc. SPIE2856, 120–128 (1996).
    [CrossRef]
  5. M. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications á l’optique,” Nouv. Rev. Opt. 8, 65–74 (1974).
  6. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thickness,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  7. M. Neviere, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
    [CrossRef]
  8. F. Montiel, M. Neviere, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3252 (1994).
    [CrossRef]
  9. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  10. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
    [CrossRef]
  11. M. Neviere, “Multilayer coated gratings for x-rays diffraction: differential theory,” J. Opt. Soc. Am. A 8, 1468–1473 (1991).
    [CrossRef]
  12. J. B. West, H. A. Padmore, “Optical engineering,” in Handbook of Synchrotron Radiation, Vol. 2, Ge. Marr, eds. (North-Holland, Amsterdam, 1987).

1994 (3)

1991 (2)

1982 (1)

M. Neviere, J. Flamand, J. M. Lerner, “Optimization of gratings for soft-x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

1974 (1)

M. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications á l’optique,” Nouv. Rev. Opt. 8, 65–74 (1974).

Awada, K. A.

DelCamp, E.

E. DelCamp, B. Lagarde, F. Polack, J. Susini, “Optimization software for X-UV monochromators,” in Optics for High Brightness Synchrotron Radiation Beamlines, L. E. Berman, J. Arthur, eds., Proc. SPIE2856, 120–128 (1996).
[CrossRef]

E. Delcamp, F. Polack, B. Lagarde, “SM-PGM monochromators design for large spectral range,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 48–57 (1997).
[CrossRef]

Flamand, J.

M. Neviere, J. Flamand, J. M. Lerner, “Optimization of gratings for soft-x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

Lagarde, B.

E. DelCamp, B. Lagarde, F. Polack, J. Susini, “Optimization software for X-UV monochromators,” in Optics for High Brightness Synchrotron Radiation Beamlines, L. E. Berman, J. Arthur, eds., Proc. SPIE2856, 120–128 (1996).
[CrossRef]

E. Delcamp, F. Polack, B. Lagarde, “SM-PGM monochromators design for large spectral range,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 48–57 (1997).
[CrossRef]

Lerner, J. M.

M. Neviere, J. Flamand, J. M. Lerner, “Optimization of gratings for soft-x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

Li, L.

Michette, A. G.

A. G. Michette, Optical Systems for Soft X Rays (Plenum, New York, 1986).
[CrossRef]

Montiel, F.

Neviere, M.

F. Montiel, M. Neviere, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3252 (1994).
[CrossRef]

M. Neviere, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

M. Neviere, “Multilayer coated gratings for x-rays diffraction: differential theory,” J. Opt. Soc. Am. A 8, 1468–1473 (1991).
[CrossRef]

M. Neviere, J. Flamand, J. M. Lerner, “Optimization of gratings for soft-x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

M. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications á l’optique,” Nouv. Rev. Opt. 8, 65–74 (1974).

Padmore, H. A.

J. B. West, H. A. Padmore, “Optical engineering,” in Handbook of Synchrotron Radiation, Vol. 2, Ge. Marr, eds. (North-Holland, Amsterdam, 1987).

Pai, D. M.

Petit, R.

M. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications á l’optique,” Nouv. Rev. Opt. 8, 65–74 (1974).

Polack, F.

E. DelCamp, B. Lagarde, F. Polack, J. Susini, “Optimization software for X-UV monochromators,” in Optics for High Brightness Synchrotron Radiation Beamlines, L. E. Berman, J. Arthur, eds., Proc. SPIE2856, 120–128 (1996).
[CrossRef]

E. Delcamp, F. Polack, B. Lagarde, “SM-PGM monochromators design for large spectral range,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 48–57 (1997).
[CrossRef]

Susini, J.

E. DelCamp, B. Lagarde, F. Polack, J. Susini, “Optimization software for X-UV monochromators,” in Optics for High Brightness Synchrotron Radiation Beamlines, L. E. Berman, J. Arthur, eds., Proc. SPIE2856, 120–128 (1996).
[CrossRef]

Vincent, P.

M. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications á l’optique,” Nouv. Rev. Opt. 8, 65–74 (1974).

West, J. B.

J. B. West, H. A. Padmore, “Optical engineering,” in Handbook of Synchrotron Radiation, Vol. 2, Ge. Marr, eds. (North-Holland, Amsterdam, 1987).

J. Opt. Soc. Am. A (4)

Nouv. Rev. Opt. (1)

M. Neviere, P. Vincent, R. Petit, “Sur la theorie du reseau conducteur et ses applications á l’optique,” Nouv. Rev. Opt. 8, 65–74 (1974).

Nucl. Instrum. Methods (1)

M. Neviere, J. Flamand, J. M. Lerner, “Optimization of gratings for soft-x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

Opt. Commun. (1)

M. Neviere, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

Other (5)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

J. B. West, H. A. Padmore, “Optical engineering,” in Handbook of Synchrotron Radiation, Vol. 2, Ge. Marr, eds. (North-Holland, Amsterdam, 1987).

E. Delcamp, F. Polack, B. Lagarde, “SM-PGM monochromators design for large spectral range,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 48–57 (1997).
[CrossRef]

E. DelCamp, B. Lagarde, F. Polack, J. Susini, “Optimization software for X-UV monochromators,” in Optics for High Brightness Synchrotron Radiation Beamlines, L. E. Berman, J. Arthur, eds., Proc. SPIE2856, 120–128 (1996).
[CrossRef]

A. G. Michette, Optical Systems for Soft X Rays (Plenum, New York, 1986).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Differential theory of grating studies of the propagation of the electromagnetic field in a medium (the grating) described by a periodic dielectric function ∊(x, z) = ∊(x + d, z). The grating is bound by a substrate at z = 0 and by vacuum at z = h.

Fig. 2
Fig. 2

In the reflectivity–matrix propagation algorithm the grating is split into a stack of thin layers. Each of them is sufficiently thin for the differential theory to apply to without numerical divergences. The reflectivity matrix is then propagated from the bottom to the top of the stack by the methods described in Subsection 2.B.

Fig. 3
Fig. 3

Evolution of the -1- and 0-order efficiencies as a function of the groove depth for a 2000-groove/mm gold grating in a Littrow mount with a symmetrical triangular profile. The grating is used in TE polarization; the incident wavelength is 0.6 μm. The refractive index of gold at this wavelength is 0.2 + i2.897. This curve should be compared with Fig. 5 of Ref. 8.

Fig. 4
Fig. 4

Efficiency of 0 order as a function of thickness e of the coating for a sinusoidal silver grating with a dielectric coating. The groove spacing is 1 μm, the groove depth is 0.3 μm, the wavelength is 0.8 μm, the incidence angle is 30°, and the polarization is TE. The refractive index is 0.09 + i5.45 for silver and 1.5 for the dielectric coating. This curve should be compared with Fig. 6 of Ref. 8.

Fig. 5
Fig. 5

Efficiency of order -1 as a function of the normalized thickness e for a coated sinusoidal silver grating with a dielectric coating in the Littrow mount and TM polarization. The grating has a 0.3333-μm period, 1200-Å sinusoidal corrugation, and a coating formed by 15 alternating zinc sulfite (ZnS, n = 2.37) and cryolite (Na3AlF6, n = 1.35) layers with the ZnS layer adjacent to the substrate. The wavelength is 0.59 μm.

Fig. 6
Fig. 6

Experimental and numerical efficiencies of the diffracted orders of -1 and +1 for the U2991 sample at 800 eV.

Fig. 7
Fig. 7

Experimental and numerical efficiencies of the diffracted orders of -1 and +1 for the U2991 sample at 1000 eV.

Fig. 8
Fig. 8

Experimental and numerical efficiencies of the diffracted orders of -1 and +1 for the U2286 sample at 1000 eV.

Fig. 9
Fig. 9

Efficiency of -1 and 1 orders at 800 eV versus the grazing angle for the U2991 sample. We compare our code (solid curve) with geometric modeling with (dotted–dashed curve) and without (dashed curve) correction for shadowing effects (Chap. 7 of Ref. 1). The calculations are made without considering the roughness. One can see that the geometric models break down in the most interesting part of the spectrum when grazing geometry is reached.

Tables (2)

Tables Icon

Table 1 Absolute Efficiency of the Blazed Multilayer Grating

Tables Icon

Table 2 Properties of the Two Gratings from Jobin–Yvon

Equations (17)

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x ,   z = x   +   d ,   z ,
g x ,   z = j = - n j = n   g j z exp i k + 2 π j d x .
g ˆ z = g - n z g 0 z g n z .
E ˆ y z = U ˆ z + D ˆ z , z E ˆ y z = iK z U ˆ z - iK z D ˆ z ,
K z m ,   m = k 0 2 - k + 2 m π d 2 1 / 2 ,
K z s m ,   m = s k 0 2 - k + 2 m π d 2 1 / 2 .
E ˆ y z = M E ˆ y + z U ˆ h + M E ˆ y - z D ˆ h , z E ˆ y z = M z E ˆ y + z U ˆ h + M z E ˆ y - z D ˆ h ,
z M E ˆ y ± = M z E ˆ y ± , z M z E ˆ y ± = K z 2 M E ˆ y ± - ¯ z - I k 0 2 M E ˆ y M ± ,
¯ m , n z = 1 d 0 d   x ,   z exp - i n - m 2 π d   x d x .
M E ˆ y ± h = I , M z E ˆ y ± h = ± iK z ,
D ˆ 0 = M E ˆ y + 0 U ˆ h + M E ˆ y - 0 D ˆ h , - iK z s D ˆ 0 = M z E ˆ y + 0 U ˆ h + M z E ˆ y - 0 D ˆ h ,
U ˆ h = R D ˆ h , R = - iK z s M E ˆ y + 0 + M z E ˆ y + 0 - 1 × M z E ˆ y - 0 + iK z s M E ˆ y - 0 .
U ˆ j - 1 = A j + + U ˆ j + A j + - D ˆ j , D ˆ j - 1 = A j - + U ˆ j + A j - - D ˆ j .
A j s 1 , s 2 = M E ˆ y s 2 0 j + s 1 iK z - 1 M z E ˆ y s 2 0 j / 2 ,
R j = A j + + - R j - 1 A j - + - 1 R j - 1 A j - - - A j + - .
ρ = d j n j λ ,
r α + β 2 × 1 d t 1 t 2 exp - i   m 2 π x d d x + exp ih sin α + sin β k 0 b 1 b 2 exp - i   m 2 π x d d x 2 ,

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