Abstract

Aberration effects are studied in parabolic and elliptic multilayer mirrors for hard x-rays, basing on a simple analytical approach. The interpretation of the underlying equations provides insight into fundamental limitations of the focusing properties of curved multilayers. Using realistic values for the multilayer parameters the potential impact on the broadening of the focal spot is evaluated. Within the limits of this model, systematic contributions to the spot size can be described. The work is complemented by a comparison with experimental results obtained with a W/B4C curved multilayer mirror.

© 2008 Optical Society of America

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References

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  1. J-P. Guigay, Ch. Morawe, V. Mocella, and C. Ferrero, "An analytical approach to estimating aberrations in curved multilayer optics for hard x-rays: 1. Derivation of caustic shapes," Opt. Express 16, 12050-12059 (2008) and references therein.
    [CrossRef] [PubMed]
  2. H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
    [CrossRef]
  3. H. Mimura, S. Handa, and K. Yamauchi, Department of Precision Science and Technology, Graduate School of Engineering, Osaka University, Japan (personal communication, 2008)
  4. L. G. Parratt, "Surface studies of solids by total reflection of x-rays," Phys. Rev. 45, 359 (1954).
    [CrossRef]
  5. Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
    [CrossRef]
  6. M. Schuster and H. Göbel, "Parallel beam coupling into channel-cut monochromators using curved graded multilayers," J. Phys. D: Appl. Phys. 28, A270 (1995).
    [CrossRef]
  7. Ch. Morawe, P. Pecci, J-Ch. Peffen, and E. Ziegler, "Design and performance of graded multilayers as focusing elements for x-ray optics," Rev. Sci. Instrum. 70, 3227 (1999).
    [CrossRef]

2008 (1)

2007 (1)

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

2006 (1)

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

1999 (1)

Ch. Morawe, P. Pecci, J-Ch. Peffen, and E. Ziegler, "Design and performance of graded multilayers as focusing elements for x-ray optics," Rev. Sci. Instrum. 70, 3227 (1999).
[CrossRef]

1995 (1)

M. Schuster and H. Göbel, "Parallel beam coupling into channel-cut monochromators using curved graded multilayers," J. Phys. D: Appl. Phys. 28, A270 (1995).
[CrossRef]

1954 (1)

L. G. Parratt, "Surface studies of solids by total reflection of x-rays," Phys. Rev. 45, 359 (1954).
[CrossRef]

Bernard, P.

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

Borel, Ch.

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

Cloetens, P.

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

Ferrero, C.

Göbel, H.

M. Schuster and H. Göbel, "Parallel beam coupling into channel-cut monochromators using curved graded multilayers," J. Phys. D: Appl. Phys. 28, A270 (1995).
[CrossRef]

Guigay, J-P.

Handa, S.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Hignette, O.

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

Ishikawa, T.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Kimura, T.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Ludwig, W.

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

Matsuyama, S.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Mimura, H.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Mocella, V.

Morawe, Ch.

J-P. Guigay, Ch. Morawe, V. Mocella, and C. Ferrero, "An analytical approach to estimating aberrations in curved multilayer optics for hard x-rays: 1. Derivation of caustic shapes," Opt. Express 16, 12050-12059 (2008) and references therein.
[CrossRef] [PubMed]

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

Ch. Morawe, P. Pecci, J-Ch. Peffen, and E. Ziegler, "Design and performance of graded multilayers as focusing elements for x-ray optics," Rev. Sci. Instrum. 70, 3227 (1999).
[CrossRef]

Nishino, Y.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Parratt, L.G.

L. G. Parratt, "Surface studies of solids by total reflection of x-rays," Phys. Rev. 45, 359 (1954).
[CrossRef]

Pecci, P.

Ch. Morawe, P. Pecci, J-Ch. Peffen, and E. Ziegler, "Design and performance of graded multilayers as focusing elements for x-ray optics," Rev. Sci. Instrum. 70, 3227 (1999).
[CrossRef]

Peffen, J-Ch.

Ch. Morawe, P. Pecci, J-Ch. Peffen, and E. Ziegler, "Design and performance of graded multilayers as focusing elements for x-ray optics," Rev. Sci. Instrum. 70, 3227 (1999).
[CrossRef]

Rommeveaux, A.

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

Sano, Y.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Schuster, M.

M. Schuster and H. Göbel, "Parallel beam coupling into channel-cut monochromators using curved graded multilayers," J. Phys. D: Appl. Phys. 28, A270 (1995).
[CrossRef]

Tamasaku, K.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Yabashi, M.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Yamauchi, K.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Yumoto, H.

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Ziegler, E.

Ch. Morawe, P. Pecci, J-Ch. Peffen, and E. Ziegler, "Design and performance of graded multilayers as focusing elements for x-ray optics," Rev. Sci. Instrum. 70, 3227 (1999).
[CrossRef]

J. Phys. D: Appl. Phys. (1)

M. Schuster and H. Göbel, "Parallel beam coupling into channel-cut monochromators using curved graded multilayers," J. Phys. D: Appl. Phys. 28, A270 (1995).
[CrossRef]

Opt. Express (1)

Phys. Rev. (1)

L. G. Parratt, "Surface studies of solids by total reflection of x-rays," Phys. Rev. 45, 359 (1954).
[CrossRef]

Proc. SPIE (2)

Ch. Morawe, O. Hignette, P. Cloetens, W. Ludwig, Ch. Borel, P. Bernard, and A. Rommeveaux, "Graded multilayers for focusing hard x rays below 50 nm," Proc. SPIE 6317, 63170F (2006).
[CrossRef]

H. Mimura, S. Matsuyama, H. Yumoto, S. Handa, T. Kimura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, "Reflective optics for sub-10nm hard X-ray focusing," Proc. SPIE 6705, 67050L (2007).
[CrossRef]

Rev. Sci. Instrum. (1)

Ch. Morawe, P. Pecci, J-Ch. Peffen, and E. Ziegler, "Design and performance of graded multilayers as focusing elements for x-ray optics," Rev. Sci. Instrum. 70, 3227 (1999).
[CrossRef]

Other (1)

H. Mimura, S. Handa, and K. Yamauchi, Department of Precision Science and Technology, Graduate School of Engineering, Osaka University, Japan (personal communication, 2008)

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Figures (9)

Fig. 1.
Fig. 1.

Basic focusing geometries of parabolic (a) and elliptic (b) RMLs. Red lines indicate rays reflected from the upper surface (P or E). Blue lines represent rays undergoing a reflection at the deeper interface (P’ or E’). The rays are refracted at each passage through the upper surface.

Fig. 2.
Fig. 2.

Close-up of the region of interest of Figs. 1. For simplicity’s sake, only the parabolic approximation is shown. The red solid line shows a ray reflected from the upper ML surface reaching the ideal focus F. The blue solid line indicates a ray that is first refracted when crossing P in point C, then reflected in point N on P’, and again refracted when crossing P in point D. The green solid line represents the resulting envelope of all rays reflected from P’. The polar angle φ is defined by the virtual ray reflected in point N without refraction in the ML medium (broken line). The dimensions are not to scale.

Fig. 3.
Fig. 3.

Caustic curves x (blue) and y (red) as a function of the grazing incident angle θ.

Fig. 4.
Fig. 4.

Positive branch of the caustic curve y(x) (red line) and corresponding grazing incidence angle θ(x) (blue line) for the same set of parameters as in Fig. 3. The insert zooms into the zone around the ideal focus and the corresponding angles of incidence.

Fig. 5.
Fig. 5.

Family of rays (coloured arrows), emerging from a curved RML with constant penetration parameter s but variable grazing angles. The black curve is the corresponding caustic. Each broken line intersects the caustic at a given angle of incidence (see colour coded list). By definition, each ray is the tangent line to the caustic at the respective intersection point.

Fig. 6.
Fig. 6.

Intersection points of the emerging rays with the optical axis (red) and with the focal plane (blue) as a function of the grazing angle. Solid curves were derived from Eq. (3). Broken lines indicate exact numerical calculations. The data set corresponds to the case shown in Fig. 5.

Fig. 7.
Fig. 7.

Chromatic blurring of different x-ray optical elements and bandwidth of hard x-ray sources and monochromators at 20 keV. Straight lines indicate RMLs (red), FZPs (blue), and CRLs (brown). The chromaticity of FZPs and CRLs is shown for three different focal lengths (1 mm, 10 mm, and 100 mm). For the RMLs two W/B4C structures with periods of 3 nm and 6 nm are shown. The bandwidths of a typical undulator source (purple line), of a Si(111) reflection (orange line), and of a ML monochromator (shaded red area) are added. Total reflection mirrors are indicated by the green broken line.

Fig. 8.
Fig. 8.

Positive branch of the caustic area (green triangle) spanned by both the angular and the depth variation. The intersections with the y(x(θ)) curves are plotted for s=0.05…0.25 nm (thick lines in colour). The limiting y(x(s)) curves are the broken lines. The open symbols indicate the upper limit due to the penetration effect for various ML d-spacings.

Fig. 9.
Fig. 9.

Large scale view on the zone near the ideal focus. The broken lines show the beam waist as expected due to diffraction from the aperture of the RML compared with the caustic area. The graph illustrates the characteristic tilt of the caustic with respect to the principal beam direction.

Tables (2)

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Table 1. ID19 W/B4C multilayer characteristics for various d-spacings.

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Table 2. Comparison of the tolerated energy bandwidth of various optical elements.

Equations (27)

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x ( t ) = δ · s · ( 1 + s p ) · ( 1 + 6 · t 2 3 · t 4 )
y ( t ) = δ · s · ( 1 + s p ) · 8 · t 3
y ( x ) = 8 · t 3 1 + 6 · t 2 3 · t 4 · x
X ( Y = 0 ) δ · s · ( 1 + 2 · t 2 + t 4 ) δ · s · t 4
Y ( X = 0 ) 2 · δ · s · ( 2 · t + t 3 ) 2 · δ · s · t 3
m · λ = 2 · Λ · n 2 cos 2 θ
2 · s · n 2 cos 2 ( π 2 ) 2 · z ( θ ) · n 2 cos 2 θ
s z ( θ ) · n 2 cos 2 θ z ( θ ) · sin 2 θ 2 · δ z ( θ ) · sin θ const .
a = ( P + Q ) 2
b = P · Q · sin θ
c = a 2 b 2
a c = a a 2 b 2 = p 2
y 2 = 2 · p · x .
R ( N C ) = ( 1 1 e ) · R MAX
δ 1 E 2
f = f 0 + Δ f = f 0 + G E 2 ,
d f d E RML = 2 · Δ f E
d f d E FZP = f E
d f d E CRL = 2 · f E
D tot 2 = D diff 2 + D chrom 2 2 · D diff 2
D diff = 0 . 44 · λ N A
D chrom = 2 · N A · df
d f = D diff 2 0 . 88 · λ = D tot 2 1 . 76 · λ
y = x · tan 2 θ
m C m B 4 3 .
d y = ( m C m B ) · x = 1 3 · tan 2 θ · x
d y = δ · s · ( 1 + s p ) · 2 · t 3 = 1 4 · y ( t )

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