Abstract

We propose a differential phase contrast imaging method in x-ray microscopy by utilizing a biased derivative filter, which is structurally similar to that used in visible optics, except that phase changes by the filter cannot be ignored in the x-ray range. However, it is demonstrated that the filter’s phase retardation does not disturb its function of phase contrast imaging, and even enhances the signals to some extent. Theoretical formulations and corresponding numerical simulations show that the approach is capable of performing characteristic differential microscopic phase imaging with nanometer-scale resolution. Manageable parameters are also examined in detail for pursuing a high image quality.

© 2013 Optical Society of America

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References

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  1. G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, and B. Niemann, Optik 97, 181 (1994).
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    [CrossRef]
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    [CrossRef]
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2006

2001

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

1998

1994

G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, and B. Niemann, Optik 97, 181 (1994).

1978

1972

Anderson, E.

Attwood, D.

Cabrini, S.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Chang, C.

Climent, V.

Di Fabrizio, E.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Fernandez-Alonso, M.

Fischer, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Academic, 1996).

Guttmann, P.

G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, and B. Niemann, Optik 97, 181 (1994).

Horwitz, B. A.

Kaulich, B.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Lancis, J.

Niemann, B.

G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, and B. Niemann, Optik 97, 181 (1994).

Romanato, F.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Rudolph, D.

G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, and B. Niemann, Optik 97, 181 (1994).

Sakdinawat, A.

Schmahl, G.

G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, and B. Niemann, Optik 97, 181 (1994).

Schneider, G.

G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, and B. Niemann, Optik 97, 181 (1994).

Sprague, R. A.

Susini, J.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Szoplik, T.

Tajahuerce, E.

Thompson, B. J.

Wilhein, T.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

T. Wilhein, B. Kaulich, E. Di Fabrizio, F. Romanato, S. Cabrini, and J. Susini, Appl. Phys. Lett. 78, 2082 (2001).
[CrossRef]

Opt. Lett.

Optik

G. Schmahl, D. Rudolph, G. Schneider, P. Guttmann, and B. Niemann, Optik 97, 181 (1994).

Other

J. W. Goodman, Introduction to Fourier Optics (Academic, 1996).

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Figures (3)

Fig. 1.
Fig. 1.

Schematic arrangement of an x-ray phase contrast microscopic system using a derivative filter.

Fig. 2.
Fig. 2.

Numerical simulations of an embedded spine image. (a) Pure phase distribution as the input. (b),(c) Output image and profiles with different design parameters a and b (unit: μm1). (d) Profiles of different co (denotes amplitude variation) of the object.

Fig. 3.
Fig. 3.

Simulations of a Siemens-star-like structure. (a) Input pure phase distribution (phase shift of the spokes: 5π/6). (b),(c) Output phase contrast image and the enlarged view. Sizes of structures: first, 15.6 nm; second, 31.2 nm; third, 62.4 nm.

Equations (9)

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T(x3)=2ln(a+bx3)μ,
tA(x3,y3)=(a+bx3)1+jc,
c(λ)=δ(λ)/β(λ).
g(x4,y4)=exp[jk(d1+d2+d3)]jλf·jλd3exp[jπλd3(x42+y42)]×F{F{f(x1,y1)}|fy=y3/λffx=x3/λf×tA(x3,y3)}|fy=y4/λd3fx=x4/λd3,
tA(x3,y3)=(a+bx3)1+jca1+jc+(1+jc)ajcbx3.
f(x1,y1)=exp[jφ(x1,y1)],
g(x4,y4)=exp[jk(d1+d2+d3)]Mexp[jπλd3(x42+y42)]×exp[jφ(x4M,y4M)]ajc[a+(1+jc)bλf2πφ(x4M,y4M)],
φ(x4M,y4M)=φ(x4M,y4M)(x4M).
I(x4,y4)|a+(1+jc)bλf2πφ(x4M,y4M)|2a2[1+bλfπaφ+(1+c24)(bλfπa)2φ2].

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