Abstract

Incoherent x rays in the wavelength interval from approximately 0.5–2 Å have been focused with refractive lenses. A single lens would have a long focal length because the refractive index of any material is close to unity; but with a stack of N lens elements the focal length is reduced by the factor N, and such a lens is termed a compound refractive lens (CRL). Misalignment of the parabolic lens elements does not alter the focusing properties and results in only a small reduction in transmission. Based on the principle of spontaneous emission amplification in a FEL wiggler, coherent x-ray sources are being developed with wavelengths of 1–1.5 Å and source diameters of 50–80 µm; and the CRL can be used to provide a small, intense image. Chromatic aberration increases the image size by an amount comparable with the diffraction-limited size, and so chromatic correction is important. Pulse broadening through the lens that is due to material dispersion is negligible. The performance of a CRL used in conjunction with a coherent source is analyzed by means of the Kirchhoff integral. For typical parameters, intensity gain is 105–106, where gain is defined as the intensity ratio in an image plane with and without the lens in place. (There may be some confusion concerning the usage of the word intensity. As employed in this manuscript, intensity, also called irradiance, refers to power per unit area. This is a commonly accepted usage for intensity, although there are places in the literature where the term radiant incidence is reserved for this definition and intensity refers to power per unit solid angle.) The image intensity is maximized when the CRL is placed 100–200 m from the source, and the diameter of the diffraction-limited spot is approximately 0.12 µm.

© 2001 Optical Society of America

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References

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  1. T. Tomie, “X-ray lens,” Japanese patent6-045288 (18February1994); U.S. patents5,594,773 (14January1997) and 5,684,859 (4November1997).
  2. A. Snigirev, V. Kohn, I. Snigireva, B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature (London) 384, 49–51 (1996).
    [CrossRef]
  3. J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 188.
  5. Ref. 4, p. 417.
  6. R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
    [CrossRef]
  7. J. R. Arthur, R. O. Tatchyn, “Radiation properties of the Linac Coherent Light Source: challenges for x-ray optics,” in X-Ray Optics and Instrumentation, H. Schulte-Schrepping, J. R. Arthur, eds., Proc. SPIE4143, 1–8 (2000).
  8. H. Schulte-Schrepping, “Photon beamlines at TESLA x-ray-FEL undulators,” in X-Ray Optics and Instrumentation, H. Schulte-Schrepping, J. R. Arthur, eds., Proc. SPIE4143, 9–13 (2000).

2001 (1)

R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
[CrossRef]

1999 (1)

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
[CrossRef]

1996 (1)

A. Snigirev, V. Kohn, I. Snigireva, B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature (London) 384, 49–51 (1996).
[CrossRef]

Arthur, J. R.

J. R. Arthur, R. O. Tatchyn, “Radiation properties of the Linac Coherent Light Source: challenges for x-ray optics,” in X-Ray Optics and Instrumentation, H. Schulte-Schrepping, J. R. Arthur, eds., Proc. SPIE4143, 1–8 (2000).

Beguiristain, H. R.

R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 188.

Cremer, J. T.

R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
[CrossRef]

Feinstein, J.

R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
[CrossRef]

Gary, C. K.

R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
[CrossRef]

Kohn, V.

A. Snigirev, V. Kohn, I. Snigireva, B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature (London) 384, 49–51 (1996).
[CrossRef]

Lengeler, B.

A. Snigirev, V. Kohn, I. Snigireva, B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature (London) 384, 49–51 (1996).
[CrossRef]

Pantell, R. H.

R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
[CrossRef]

Piestrup, M. A.

R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
[CrossRef]

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
[CrossRef]

Schulte-Schrepping, H.

H. Schulte-Schrepping, “Photon beamlines at TESLA x-ray-FEL undulators,” in X-Ray Optics and Instrumentation, H. Schulte-Schrepping, J. R. Arthur, eds., Proc. SPIE4143, 9–13 (2000).

Snigirev, A.

A. Snigirev, V. Kohn, I. Snigireva, B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature (London) 384, 49–51 (1996).
[CrossRef]

Snigireva, I.

A. Snigirev, V. Kohn, I. Snigireva, B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature (London) 384, 49–51 (1996).
[CrossRef]

Tatchyn, R. O.

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
[CrossRef]

J. R. Arthur, R. O. Tatchyn, “Radiation properties of the Linac Coherent Light Source: challenges for x-ray optics,” in X-Ray Optics and Instrumentation, H. Schulte-Schrepping, J. R. Arthur, eds., Proc. SPIE4143, 1–8 (2000).

Tomie, T.

T. Tomie, “X-ray lens,” Japanese patent6-045288 (18February1994); U.S. patents5,594,773 (14January1997) and 5,684,859 (4November1997).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 188.

Nature (London) (1)

A. Snigirev, V. Kohn, I. Snigireva, B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature (London) 384, 49–51 (1996).
[CrossRef]

Rev. Sci. Instrum. (2)

J. T. Cremer, M. A. Piestrup, H. R. Beguiristain, C. K. Gary, R. H. Pantell, R. O. Tatchyn, “Cylindrical compound refractive X-ray lenses using plastic substrates,” Rev. Sci. Instrum. 70, 3545–3548 (1999).
[CrossRef]

R. H. Pantell, J. Feinstein, H. R. Beguiristain, M. A. Piestrup, C. K. Gary, J. T. Cremer, “The effect of unit lens alignment and surface roughness on x-ray compound lens performance,” Rev. Sci. Instrum. 72, 48–52 (2001).
[CrossRef]

Other (5)

J. R. Arthur, R. O. Tatchyn, “Radiation properties of the Linac Coherent Light Source: challenges for x-ray optics,” in X-Ray Optics and Instrumentation, H. Schulte-Schrepping, J. R. Arthur, eds., Proc. SPIE4143, 1–8 (2000).

H. Schulte-Schrepping, “Photon beamlines at TESLA x-ray-FEL undulators,” in X-Ray Optics and Instrumentation, H. Schulte-Schrepping, J. R. Arthur, eds., Proc. SPIE4143, 9–13 (2000).

T. Tomie, “X-ray lens,” Japanese patent6-045288 (18February1994); U.S. patents5,594,773 (14January1997) and 5,684,859 (4November1997).

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), p. 188.

Ref. 4, p. 417.

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Figures (10)

Fig. 1
Fig. 1

Ratio δ/μ for various materials as functions of photon energy. This ratio is a figure of merit for the CRL, and this graph shows that the most useful range for the x-ray lens is from a few kiloelectron volts to ≈50 keV. In this range, Be is the optimum medium; and above 50 keV, δ/μ is the same for a number of materials.

Fig. 2
Fig. 2

Two-lens arrangement to achieve chromatic correction. The focal length of the combination and the variation of focal length with wavelength depend on the separation between the lenses.

Fig. 3
Fig. 3

Variation of focal length and image size that are due to chromatic aberration as functions of the separation between the lenses illustrated in Fig. 2. To completely eliminate the aberration, the number of lenses must be doubled to maintain a given focal length.

Fig. 4
Fig. 4

Misalignment of lens elements in the CRL. The focusing properties of the CRL for parabolic lenses are unaltered with misalignment and result in only a small effective increase in the base thickness b.

Fig. 5
Fig. 5

Coordinates for the one-dimensional Kirchhoff integral including a lens between the source and the image.

Fig. 6
Fig. 6

Gaussian beam parameters for the LCLS.

Fig. 7
Fig. 7

Gain versus lens position. Gain is defined as the ratio of the on-axis intensity in the image plane with the lens in place to the on-axis intensity in the same plane without the lens. The lens position is measured from the source plane shown in Fig. 6. The base thickness of the lens is b, R is the radius of curvature, and R 0 is the lens radius.

Fig. 8
Fig. 8

On-axis intensity in the image plane for the 2-D case as a function of the distance between the source and the lens. The base thickness of each lens element is 15 µm, the curvature of radius is 1 mm, and the lens radius is 0.4 mm. For the specified parameters, the number of elements is N = 100/f, where f is measured in meters. At the point of maximum intensity the gain is ≈ 6.5 × 105.

Fig. 9
Fig. 9

Gain and focal length as functions of the number of lens elements for the parameters of the LCLS. Focal length varies inversely with N and gain maximizes with 400–500 elements. The ordinate in the figure gives the focal length in meters and the two-dimensional gain multiplied by a factor of 10-6.

Fig. 10
Fig. 10

Image size for several lens positions, assuming no chromatic aberration. For curve a, the lens is 50 m from the source plane; for curve b, the lens is 100 m from the source; for curve c, the lens is 1000 m from the source. Image intensity is maximum for curve b even though the image size is smaller for curve c. For curve b, the FWHM of the image is ≈0.12 µm.

Equations (18)

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n=1-δ-iβ,
Δt=Lc3d2ωdk2 Δω,
Δt=Lc 2δ Δωω.
A=4fδμ.
Δff=-2  Δλλ.
1f=1f1+1f2-df1f2,
d=12f1λ0+f2λ0,
hh0=21-f2f0,
ff0=11-df1+f2,
Δb=σ2R,
2Nkδσ<π2,
Ey=expjπ4-jkzλzsourcedsEs×exp- jk2zs-y2.
Ey  sourceds exp-s2w02lensdl×exp-jk2ri1-y2-jk2r0s-12+jkl22f-kl2μ4kfδ.
Iy  sourcedslensdl2,
Iy  sourceds lensdl2.
w=w01+z2zr21/2,
Iy  0R0dl exp-kl22zrr02+zr2+βfδ×cosklyri2.
1ri=1f-1r01+zr2r02.

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