Abstract

The simulation of the propagation of divergent beams using Fourier-based angular spectrum techniques can pose challenges for ensuring correct sampling in the spatial and reciprocal domains. This challenge can be compounded by the presence of diffracting objects, as is often the case. Here, I give details of a method for robustly simulating the propagation of beams with divergent wavefronts in a coordinate system where the wavefronts become planar. I also show how diffracting objects can be simulated, while guaranteeing that correct sampling is maintained. These two advances allow for numerically efficient and accurate simulations of divergent beams propagating through diffracting structures using the multi-slice approximation. The sampling requirements and numerical implementation are discussed in detail, and I have made the computer code freely available.

© 2019 Optical Society of America

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References

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  1. J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
    [Crossref]
  2. A. Hare and G. Morrison, “Near-field soft x-ray diffraction modelled by the multislice method,” J. Mod. Opt. 41, 31–48 (1994).
    [Crossref]
  3. Y. Wang, “A numerical study of resolution and contrast in soft x-ray contact microscopy,” J. Microsc. 191, 159–169 (1998).
    [Crossref]
  4. A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
    [Crossref]
  5. K. Li, M. Wojcik, and C. Jacobsen, “Multislice does it all—calculating the performance of nanofocusing x-ray optics,” Opt. Express 25, 1831–1846 (2017).
    [Crossref]
  6. D. Paganin, Coherent X-Ray Optics (Oxford University, 2006).
  7. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2005).
  8. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [Crossref]
  9. E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
    [Crossref]
  10. K. S. Morgan, K. K. W. Siu, and D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge,” Opt. Express 18, 9865–9878 (2010).
    [Crossref]
  11. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
    [Crossref]
  12. P. R. T. Munro, “Multi-slice x-ray beam propagation code,” http://prtmunro.net .
  13. P. R. T. Munro, K. Ignatyev, R. Speller, and A. Olivo, “The relationship between wave and geometrical optics models of coded aperture type x-ray phase contrast imaging systems,” Opt. Express 18, 4103–4117 (2010).
    [Crossref]
  14. J. W. Goodman, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

2017 (1)

2012 (1)

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
[Crossref]

2010 (2)

2009 (1)

1998 (1)

Y. Wang, “A numerical study of resolution and contrast in soft x-ray contact microscopy,” J. Microsc. 191, 159–169 (1998).
[Crossref]

1994 (1)

A. Hare and G. Morrison, “Near-field soft x-ray diffraction modelled by the multislice method,” J. Mod. Opt. 41, 31–48 (1994).
[Crossref]

1975 (1)

1974 (1)

E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[Crossref]

1957 (1)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
[Crossref]

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
[Crossref]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2005).

Goodman, J. W.

J. W. Goodman, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

Hare, A.

A. Hare and G. Morrison, “Near-field soft x-ray diffraction modelled by the multislice method,” J. Mod. Opt. 41, 31–48 (1994).
[Crossref]

Ignatyev, K.

Jacobsen, C.

Li, K.

Malecki, A.

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
[Crossref]

Matsushima, K.

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
[Crossref]

Morgan, K. S.

Morrison, G.

A. Hare and G. Morrison, “Near-field soft x-ray diffraction modelled by the multislice method,” J. Mod. Opt. 41, 31–48 (1994).
[Crossref]

Munro, P. R. T.

Olivo, A.

Paganin, D.

D. Paganin, Coherent X-Ray Optics (Oxford University, 2006).

Paganin, D. M.

Pfeiffer, F.

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
[Crossref]

Potdevin, G.

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
[Crossref]

Shimobaba, T.

Siegman, A. E.

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[Crossref]

E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[Crossref]

Siu, K. K. W.

Speller, R.

Sziklas, E. A.

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[Crossref]

E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[Crossref]

Wang, Y.

Y. Wang, “A numerical study of resolution and contrast in soft x-ray contact microscopy,” J. Microsc. 191, 159–169 (1998).
[Crossref]

Wojcik, M.

Acta Crystallogr. (1)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).
[Crossref]

Appl. Opt. (1)

Europhys. Lett. (1)

A. Malecki, G. Potdevin, and F. Pfeiffer, “Quantitative wave-optical numerical analysis of the dark-field signal in grating-based x-ray interferometry,” Europhys. Lett. 99, 48001 (2012).
[Crossref]

J. Microsc. (1)

Y. Wang, “A numerical study of resolution and contrast in soft x-ray contact microscopy,” J. Microsc. 191, 159–169 (1998).
[Crossref]

J. Mod. Opt. (1)

A. Hare and G. Morrison, “Near-field soft x-ray diffraction modelled by the multislice method,” J. Mod. Opt. 41, 31–48 (1994).
[Crossref]

Opt. Express (4)

Proc. IEEE (1)

E. A. Sziklas and A. E. Siegman, “Diffraction calculations using fast Fourier transform methods,” Proc. IEEE 62, 410–412 (1974).
[Crossref]

Other (4)

D. Paganin, Coherent X-Ray Optics (Oxford University, 2006).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2005).

J. W. Goodman, Laser Speckle and Related Phenomena (Springer-Verlag, 1975).

P. R. T. Munro, “Multi-slice x-ray beam propagation code,” http://prtmunro.net .

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Figures (12)

Fig. 1.
Fig. 1. Schematic diagram of the system studied in this paper. A point source is located at the origin of the Cartesian coordinate system. Refractive index inhomogeneities may exist for z>Δzso, one example of which is illustrated in a layer bounded by the planes z=zi and z=zi+1=zi+Δzi.
Fig. 2.
Fig. 2. Diagram illustrating the principal coordinate system notation used in this paper.
Fig. 3.
Fig. 3. Illustration of the region in reciprocal space (the intersection of |fx|<fco and |fy|<fco) where the angular spectrum of the field is permitted to reside, along with a guard band where angular spectrum content is filtered after propagation through each layer.
Fig. 4.
Fig. 4. Schematic diagram of the diffracting aperture and point source upon which all examples in Section 3 are based. Values of W=20μm, za=1.6m, and zo=1.9m were chosen. Note that the diagram is not drawn to scale.
Fig. 5.
Fig. 5. Magnitude of the field diffracted by the aperture found by direct evaluation of the Fresnel–Kirchhoff diffraction integral.
Fig. 6.
Fig. 6. Plots of the magnitude of the error in each calculated complex amplitude relative to that calculated using Fresnel–Kirchhoff diffraction theory for a diffracting aperture.
Fig. 7.
Fig. 7. Plot of the error ϵ as a function of N, where the reference field is that calculated using Fresnel–Kirchhoff diffraction theory.
Fig. 8.
Fig. 8. Magnitude of the field diffracted by the aperture and a single sphere found by direct evaluation of the Fresnel–Kirchhoff diffraction integral.
Fig. 9.
Fig. 9. Plots of the magnitude of the error in each calculated complex amplitude relative to that calculated using Fresnel–Kirchhoff diffraction theory for the case of a single sphere and diffracting aperture.
Fig. 10.
Fig. 10. Magnitude of the field obtained when two spheres are located between the aperture and observation plane (left) and the magnitude of the field obtained by considering the two spheres in isolation (right).
Fig. 11.
Fig. 11. Magnitude of the field that results when an ensemble of spheres is located between a square diffracting aperture and the observation plane. Each group of three images corresponds to a different value of δ for the spheres, including the top left group, which represents free space. The middle image in each group is a magnified view of the region denoted by the outline box in the top image. The lowest image shows a histogram of the magnitudes of the pixels within the magnified region along with a Rayleigh distribution with mean equal to that of the magnified region.
Fig. 12.
Fig. 12. Plots of computation time per sample point in the observation plane for the Fresnel–Kirchhoff and primed coordinate system angular spectrum methods.

Equations (73)

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2ux2+2uy2+2ikuz=0.
U(a,b,z)=u(x,y,z)exp(i2π(x(a/λ)+y(b/λ)))dxdy,
u(x,y,z)=U(a,b,z)exp(i2π(x(a/λ)+y(b/λ)))·d(aλ)d(bλ).
P(a,b,Δz)=exp(ik(a2+b2)Δz/2)
uinc(x,y,z)=exp(ik(x2+y2)/(2z))/z,
uinc(x,y,z)=exp(iϕ(x,y,z))/z,
finc,x=12πxϕ(x,y,z)=xλz,
finc,y=12πyϕ(x,y,z)=yλz.
Uinc(a,b,z)=iλexp(i(a2+b2)kz/2),
finc,a=azλ,
finc,b=bzλ.
x˜=y˜={(jN/2)Δx|0j<N},
a˜=b˜={λ(jN/2)/(NΔx)|0j<N}.
Δx12max(|finc,x|),
Δa12max(|finc,a|),
max(|finc,x|)=ΔxN2λΔzso,
max(|finc,a|)=Δzso+Δzod2Δx.
ΔxλΔzsoN,
Δxλ(Δzso+Δzod)N,
n(x,y,z)=1δ(x,y,z)+iβ(x,y,z).
ϕ(x,y)=kzizi+1(δ(x,y,z)+iβ(x,y,z))dz,
u(x,y,zi)=exp(ik(x2+y2)/(2zi))v(xi,yi,zi)/zi,
xi(x,z)=xMi,
yi(x,z)=yMi,
zi(z)=zziMi,
V(ai,bi,zi(zi+1))=P(ai,bi,zi(zi+1))V(ai,bi,0),
u(x,y,zi+1)=exp(ik(x2+y2)/(2zi+1))v(xi,yi,zi(zi+1))zi+1,
x=(zi+1/zi)xi(x,zi+1)=Mi(zi+1)xi(x,zi+1),
y=(zi+1/zi)yi(y,zi+1)=Mi(zi+1)yi(x,zi+1),
z=zi+1,
tBL(x˜,y˜)=F^1{W(f˜x,f˜y)T(f˜x,f˜y)},
V(a0,b0,Δz0)=P(a0,b0,Δz0)V(a0,b0,0),
x=M0(z1)x0(x,z1),
y=M0(z1)y0(y,z1).
x1(x,z1)=M0(z1)x0(x,z1),
y1(x,z1)=M0(z1)y0(y,z1).
x2(x,z2)=x=M1(z2)M0(z1)x0(x,z1),
y2(x,z2)=y=M1(z2)M0(z1)y0(y,z1).
xi(x,zi)=x=x0(x,z1)j=0i1Mj(zj+1),
yi(y,zi)=y=y0(y,z1)j=0i1Mj(zj+1),
Mi(zi+1)xi(x,zi+1)=x=x0(x,z1)j=0iMj(zj+1),
Mi(zi+1)yi(y,zi+1)=y=y0(y,z1)j=0iMj(zj+1)
v^(x^i,y^i,zi(zi+1))=exp(iϕ(Mi(zi+1)x^i,Mi(zi+1)y^i))·F^1{P(a^i,b^i,zi(zi+1))F^{v^(x^i,y^i,zi(zi))}}.
v^(x^i+1Mi(zi+1),y^i+1Mi(zi+1),zi+1(zi+1))=v^(x^i,y^i,zi(zi+1)).
x^i=x^0j=0i1Mj(zj+1),
y^i=y^0j=0i1Mj(zj+1),
a^i=a^0/j=0i1Mj(zj+1),
b^i=b^0/j=0i1Mj(zj+1),
x˜0=y˜0={(jN/2)Δx|0j<N}
a˜0=b˜0={λ(jN/2)/(NΔx)|0j<N}.
t(Mx,My)=F1{T(fx/M,fy/M)}/M2,
Δx12fco.
ΔfxzNs2λ(zNsz0)z0fco,
N2λ(zNsz0)z0fcoΔxzNs.
n(x,y,z)=1(δ(x,y,z)δi)+i(β(x,y,z)βi),
P(a,b,Δz)=exp(ik1δi+iβi(a2+b2)Δz/2).
Ti=exp(ik(zi+1zi)(δi+iβi)),
Δxλ(zoza)zaNzo,
ϵ=i,j|u(iΔx,jΔy)uref(iΔx,jΔy)|2i,j|uref(iΔx,jΔy)|2,
tsphere(x,y)={exp(ik2(δ+iβ)R2x2y2)x2+y2R21Otherwise,
Tsphere(fx,fy)=tsphere(x,y)exp(i2π(xfx+yfy))dxdy=δ(fx,fy)+R2x2R2x2RR(tsphere(x,y)1)exp(i2π(xfx+yfy))dxdy,
x=ρcosϕ,
y=ρsinϕ,
fx=ξcosϕ,
fy=ξsinϕ,
Tsphere(ξ)=δ(fx,fy)+02π0R(tsphere(ρ)1)exp(i2πρξcos(ϕϕ))ρdρdϕ=δ(fx,fy)+2π0R(tsphere(ρ)1)ρJ0(ρξ)dρ,
tsphere,BL(x˜,y˜)=F^1{W(f˜x,f˜y)Tsphere(f˜x2+f˜y2)},
u(x,y,zo)=K(x,y,x,y,zo(z1+Δz1))uinc(x,y,z1+Δz1)exp(iϕ1(x,y))dxdy,
K(x,y,x,y,z)=iλzexp(ikz)exp(ik(xx)2+(yy)22z).
u(x,y,zo)=uinc(x,y,zo)+Ω1K(x,y,x,y,zo(z1+Δz1))·uinc(x,y,z1+Δz1)(exp(iϕ1(x,y))1)dxdy,
u(x,y,zo)=uinc(x,y,zo)+usc(x,y,zo).
u(x,y,zo)=uinc(x,y,zo)+usc(x,y,zo)+Ω2K(x,y,x,y,zo(z2+Δz2))(uinc(x,y,z2+Δz2)+usc(x,y,z2+Δz2))(exp(iϕ2(x,y))1)dxdy.
u(x,y,zo)=uinc(x,y,zo)+usc,1(x,y,zo)+usc,2(x,y,zo)+usc,sc(x,y,zo),

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