Abstract

An exact solution of the phase retrieval problem is described as applied to in-situ X-ray reflectometry of a growing layered film. The following statement is proved: if the reflectivity R and the derivative dR/dt are known at the time t, then the real and the imaginary parts of the amplitude reflectivity r(t) are found uniquely at this point t.

©2008 Optical Society of America

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References

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  1. M. Tolan, X-Ray Scattering from Soft-Matter Thin Films (Springer Tracts Mod. Phys., 148, Springer, Berlin, 1999).
  2. X.-L. Zhou and S.-H. Chen, “Theoretical foundation of X-ray and neutron reflectometry,” Phys. Rep. 257, 223–348 (1995).
    [Crossref]
  3. I. V. Kozhevnikov, “Physical analysis of the inverse problem of X-ray reflectometry,” Nucl. Instrum. Methods Phys. Res. A 508, 519–541 (2003).
    [Crossref]
  4. C. F. Majkrzak and N. F. Berk, “Exact determination of the phase in neutron reflectometry by variation of the surrounding media,” Phys. Rev. B 58, 15416–15418 (1998).
    [Crossref]
  5. V. I. Klyatskin, Imbedding Method in Theory of Wave Propagation (Nauka, Moscow,in Russian1986).
  6. E. Lüken, E. Ziegler, and M. Lingham, “In-situ growth control of X-ray multilayers using visible light kinetic ellipsometry and grazing incidence X-ray reflectometry,” Proc. SPIE 2873, 113–118 (1996).
  7. L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72, 045445 (2005).
    [Crossref]
  8. L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76, 045411 (2007).
    [Crossref]

2007 (1)

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76, 045411 (2007).
[Crossref]

2005 (1)

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72, 045445 (2005).
[Crossref]

2003 (1)

I. V. Kozhevnikov, “Physical analysis of the inverse problem of X-ray reflectometry,” Nucl. Instrum. Methods Phys. Res. A 508, 519–541 (2003).
[Crossref]

1998 (1)

C. F. Majkrzak and N. F. Berk, “Exact determination of the phase in neutron reflectometry by variation of the surrounding media,” Phys. Rev. B 58, 15416–15418 (1998).
[Crossref]

1996 (1)

E. Lüken, E. Ziegler, and M. Lingham, “In-situ growth control of X-ray multilayers using visible light kinetic ellipsometry and grazing incidence X-ray reflectometry,” Proc. SPIE 2873, 113–118 (1996).

1995 (1)

X.-L. Zhou and S.-H. Chen, “Theoretical foundation of X-ray and neutron reflectometry,” Phys. Rep. 257, 223–348 (1995).
[Crossref]

Berk, N. F.

C. F. Majkrzak and N. F. Berk, “Exact determination of the phase in neutron reflectometry by variation of the surrounding media,” Phys. Rev. B 58, 15416–15418 (1998).
[Crossref]

Bigault, T.

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76, 045411 (2007).
[Crossref]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72, 045445 (2005).
[Crossref]

Chen, S.-H.

X.-L. Zhou and S.-H. Chen, “Theoretical foundation of X-ray and neutron reflectometry,” Phys. Rep. 257, 223–348 (1995).
[Crossref]

Klyatskin, V. I.

V. I. Klyatskin, Imbedding Method in Theory of Wave Propagation (Nauka, Moscow,in Russian1986).

Kozhevnikov, I.

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76, 045411 (2007).
[Crossref]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72, 045445 (2005).
[Crossref]

Kozhevnikov, I. V.

I. V. Kozhevnikov, “Physical analysis of the inverse problem of X-ray reflectometry,” Nucl. Instrum. Methods Phys. Res. A 508, 519–541 (2003).
[Crossref]

Lingham, M.

E. Lüken, E. Ziegler, and M. Lingham, “In-situ growth control of X-ray multilayers using visible light kinetic ellipsometry and grazing incidence X-ray reflectometry,” Proc. SPIE 2873, 113–118 (1996).

Lüken, E.

E. Lüken, E. Ziegler, and M. Lingham, “In-situ growth control of X-ray multilayers using visible light kinetic ellipsometry and grazing incidence X-ray reflectometry,” Proc. SPIE 2873, 113–118 (1996).

Majkrzak, C. F.

C. F. Majkrzak and N. F. Berk, “Exact determination of the phase in neutron reflectometry by variation of the surrounding media,” Phys. Rev. B 58, 15416–15418 (1998).
[Crossref]

Peverini, L.

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76, 045411 (2007).
[Crossref]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72, 045445 (2005).
[Crossref]

Tolan, M.

M. Tolan, X-Ray Scattering from Soft-Matter Thin Films (Springer Tracts Mod. Phys., 148, Springer, Berlin, 1999).

Zhou, X.-L.

X.-L. Zhou and S.-H. Chen, “Theoretical foundation of X-ray and neutron reflectometry,” Phys. Rep. 257, 223–348 (1995).
[Crossref]

Ziegler, E.

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76, 045411 (2007).
[Crossref]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72, 045445 (2005).
[Crossref]

E. Lüken, E. Ziegler, and M. Lingham, “In-situ growth control of X-ray multilayers using visible light kinetic ellipsometry and grazing incidence X-ray reflectometry,” Proc. SPIE 2873, 113–118 (1996).

Nucl. Instrum. Methods Phys. Res. A (1)

I. V. Kozhevnikov, “Physical analysis of the inverse problem of X-ray reflectometry,” Nucl. Instrum. Methods Phys. Res. A 508, 519–541 (2003).
[Crossref]

Phys. Rep. (1)

X.-L. Zhou and S.-H. Chen, “Theoretical foundation of X-ray and neutron reflectometry,” Phys. Rep. 257, 223–348 (1995).
[Crossref]

Phys. Rev. B (3)

C. F. Majkrzak and N. F. Berk, “Exact determination of the phase in neutron reflectometry by variation of the surrounding media,” Phys. Rev. B 58, 15416–15418 (1998).
[Crossref]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Roughness conformity during tungsten film growth: An in situ synchrotron x-ray scattering study,” Phys. Rev. B 72, 045445 (2005).
[Crossref]

L. Peverini, E. Ziegler, T. Bigault, and I. Kozhevnikov, “Dynamic scaling of roughness at the early stage of tungsten film growth,” Phys. Rev. B 76, 045411 (2007).
[Crossref]

Proc. SPIE (1)

E. Lüken, E. Ziegler, and M. Lingham, “In-situ growth control of X-ray multilayers using visible light kinetic ellipsometry and grazing incidence X-ray reflectometry,” Proc. SPIE 2873, 113–118 (1996).

Other (2)

V. I. Klyatskin, Imbedding Method in Theory of Wave Propagation (Nauka, Moscow,in Russian1986).

M. Tolan, X-Ray Scattering from Soft-Matter Thin Films (Springer Tracts Mod. Phys., 148, Springer, Berlin, 1999).

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Figures (2)

Fig. 1.
Fig. 1. Reflectivity versus deposition time from a growing tungsten film measured at an X-ray energy of 17.5 keV and with a grazing angle θ=0.5° (dots). The solid curve was calculated assuming a constant density of the film.
Fig. 2.
Fig. 2. Two solutions of the phase retrieval problem (curves 1 and 2) found directly from the experimental curve shown in Fig. 1. Curve 3 shows the phase evolution calculated assuming a uniform tungsten film with constant density. For illustrative purpose, we assumed the phase to vary within the [-π/2, 3π/2] interval.

Equations (10)

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dr dh ( h ) = 2 ikr ( h ) sin θ + ik 2 sin θ [ ε ( h ) 1 ] [ 1 + r ( h ) ] 2
dR dh ( h ) = k sin θ { [ ε ( h ) 1 ] r * ( h ) [ 1 + r ( h ) ] 2 }
dR dt ( t ) = Aq ( t ) k sin θ { f 1 ( t ) [ 1 R ( t ) ] [ r ( t ) ] + f 2 ( t ) [ 2 R ( t ) + ( 1 + R ( t ) ) [ r ( t ) ] ] }
R ( t ) = [ ( r ( t ) ) ] 2 + [ ( r ( t ) ) ] 2
E ( z , h ) = { e i κ 0 z + r ( h ) e i κ 0 z 2 i κ 0 h , if z h t ( h ) e i κ s z + i ( κ s κ 0 ) h , if z + ; κ 0 = k sin θ ; κ s = k ε s cos 2 θ
dE dh ( h , h ) = { i κ 0 [ 1 + r ( h ) ] + dr dh ( h ) } e i κ 0 h
dE dh ( h , h ) = E z ( z , h ) | z = h + E h ( z , h ) | z = h
E z ( z , h ) | z = h = i κ 0 [ 1 r ( h ) ] e i κ 0 h
E ( z , h ) = E ( z , h 1 ) k 2 h h 1 g ( z , z ; h 1 ) [ ε ( z ) 1 ] E ( z , h ) d z
E h ( z , h ) | z = h = i k 2 2 κ 0 [ ε ( h ) 1 ] [ 1 + r ( h ) ] 2 e i κ 0 h

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