Abstract

In this work, four fast and rigorous methods for the simulation of light propagation in a homogenous medium are introduced. It is shown that in free-space propagation, the analytical handling of smooth but strong phase terms is very efficient in reducing the computational effort. Therefore, the angular spectrum of plane waves (SPW) operator is reformulated to handle linear, spherical, and general smooth phase terms without limiting the application of the fast-Fourier-transformation algorithm. Especially for nonparaxial field propagation, the proposed techniques can significantly reduce the required number of sampling points. Numerical results are presented to demonstrate the efficiency and the accuracy of the new methods.

© 2014 Optical Society of America

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References

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  1. F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
    [CrossRef]
  2. J. A. C. Veerman, J. J. Rusch, and H. P. Urbach, “Calculation of the Rayleigh-Sommerfeld diffraction integral by exact integration of the fast oscillating factor,” J. Opt. Soc. Am. A 22, 636–646 (2005).
    [CrossRef]
  3. J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51.
  4. D. Asoubar, S. Zhang, F. Wyrowski, and M. Kuhn, “Parabasal field decomposition and its applications to non-paraxial propagation,” Opt. Express 20, 23502–23517 (2012).
    [CrossRef]
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
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    [CrossRef]
  7. C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
    [CrossRef]
  8. E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959).
    [CrossRef]
  9. J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
    [CrossRef]
  10. J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, 1988).
  16. J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), Vol. XI, Chap. 1.
  17. R. J. Hanson and C. L. Lawson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).

2012 (1)

2011 (2)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[CrossRef]

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

2010 (1)

2005 (2)

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. A. C. Veerman, J. J. Rusch, and H. P. Urbach, “Calculation of the Rayleigh-Sommerfeld diffraction integral by exact integration of the fast oscillating factor,” J. Opt. Soc. Am. A 22, 636–646 (2005).
[CrossRef]

2004 (1)

1992 (1)

1989 (1)

1965 (1)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

1959 (1)

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959).
[CrossRef]

Asoubar, D.

Braat, J. J. M.

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, 1988).

Cooley, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Creath, K.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), Vol. XI, Chap. 1.

Dirksen, P.

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51.

Engelberg, Y. M.

Goodmann, W.

W. Goodmann, Introduction to Fourier Optics (McGraw-Hill, 1968).

Hanson, R. J.

R. J. Hanson and C. L. Lawson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).

Hrynevych, M.

Janssen, A. J. E. M.

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51.

Kuhn, M.

Lawson, C. L.

R. J. Hanson and C. L. Lawson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mansuripur, M.

Matsushima, K.

Rusch, J. J.

Ruschin, S.

Sheppard, C. J. R.

Tukey, J. W.

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Turunen, J.

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

Urbach, H. P.

van der Avoort, C.

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

van Haver, S.

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51.

Veerman, J. A. C.

Wolf, E.

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), Vol. XI, Chap. 1.

Wyrowski, F.

Zhang, S.

J. Mod. Opt. (3)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt. 58, 449–466 (2011).
[CrossRef]

C. van der Avoort, J. J. M. Braat, P. Dirksen, and A. J. E. M. Janssen, “Aberration retrieval from the intensity point-spread function in the focal region using the extended Nijboer Zernike approach,” J. Mod. Opt. 52, 1695–1728 (2005).
[CrossRef]

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

J. Opt. Soc. Am. A (4)

Math. Comput. (1)

J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Express (2)

Proc. Phys. Soc. London (1)

E. Wolf, “A scalar representation of electromagnetic fields: II,” Proc. Phys. Soc. London 74, 269–280 (1959).
[CrossRef]

Other (6)

W. Goodmann, Introduction to Fourier Optics (McGraw-Hill, 1968).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

J. J. M. Braat, S. van Haver, A. J. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point-spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), Vol. 51.

E. O. Brigham, The Fast Fourier Transform and Its Applications (Prentice-Hall, 1988).

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” in Applied Optics and Optical Engineering, R. R. Shannon and J. C. Wyant, eds. (Academic, 1992), Vol. XI, Chap. 1.

R. J. Hanson and C. L. Lawson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).

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Figures (12)

Fig. 1.
Fig. 1.

Schematic illustration of the scope for common FFT-based propagation operators in field tracing. Gray areas indicate parameter spaces for which the techniques are numerically feasible. The rigorous SPW operator suffers from high numerical effort for high propagation distances or spatial frequencies. The far-field and Fresnel propagation operators have a limited range of validity due to the nature of their physical approximations.

Fig. 2.
Fig. 2.

Four examples of smooth phase terms (2π-modulo-sampled), which are quite common in optical modeling and design: Spherical phase terms (a) can be handled analytically by the generalization of the Fresnel diffraction integral, as shown in Section 3. Linear phase terms (b) are handled analytically by a modified SPW operator in Section 4. General smooth phase terms like cylindrical waves (c) and astigmatic parabolic waves (d) can be treated analytically in linear approximation by a PDT (Section 6).

Fig. 3.
Fig. 3.

Fitting of the spherical phase function kz according to Taylor, Avoort, and Mansuripur. (a) 1D spherical function and its corresponding parabolic fitting curves. (b) Higher order phase functions, obtained by subtracting the parabolic fitting function from the spherical one.

Fig. 4.
Fig. 4.

Schematic illustration of the combination of forward and inverse semi-analytical SPW operators. At first, the convergent spherical wave is propagated into the focus by the inverse semi-analytical SPW operator over distance z1. After that, the forward semi-analytical SPW operator is applied over distance z2 to propagate the field into the target plane.

Fig. 5.
Fig. 5.

High-NA diffractive beam splitter setup for numerical investigation of forward semi-analytical SPW operator.

Fig. 6.
Fig. 6.

Diffractive beam shaping setup for numerical investigation of inverse semi-analytical SPW operator.

Fig. 7.
Fig. 7.

Propagation of a harmonic field with smooth linear phase term results in an off-axis field in the target plane. In the case of the conventional SPW operator (a), the computational window in the initial plane has to be extended to ensure that the region of interest in the target plane is within the window. The semi-analytical SPW operator (b) can handle the lateral shift analytically, resulting in a destination window with reduced sampling effort.

Fig. 8.
Fig. 8.

Example for a harmonic field with smooth linear phase term: Amplitude (a) and phase (b) of Super Gaussian beam with 50  μm waist radius at a wavelength of 532 nm. An additional linear phase term (c) of 10° in x direction and 4° in y direction replaces the paraxial properties of the Super Gaussian beam by a parabasal field behavior.

Fig. 9.
Fig. 9.

Upper flowchart illustrates the efficient propagation of nonparaxial fields using a combination of a PDT and the semi-analytical SPW operator. The lower flowchart gives the corresponding phase treatment for each step. A general harmonic field is decomposed into a set of parabasal subfields with analytically stored linear phase terms using a PDT technique. After that, each parabasal harmonic subfield is propagated by the semi-analytical SPW operator, resulting in a set of mutually coherent subfields with analytically stored linear phase term per subfield.

Fig. 10.
Fig. 10.

Complete phase (a), (b) of an astigmatic parabolic beam, created by a plane wave propagated through a thin astigmatic paraxial lens. After applying a PDT in space domain, the analytically stored linear phase κ0 is subtracted. The residual phase (c) of the subfield is much smoother than the corresponding complete phase (b) in the same area. Therefore, the effort to sample the residual phase is much lower than for the complete phase. Please note that the artefacts in the phase distribution (a) is a Moiré pattern due to the low resolution of the picture in comparison with the very fine sampling, which is required for the phase.

Fig. 11.
Fig. 11.

Amplitude (a) and phase (b) of a single propagated parabasal subfield with analytically stored linear phase term.

Fig. 12.
Fig. 12.

Upper flowchart illustrates the inverse PDT for an efficient display of the propagated field obtained by the combination of PDT and semi-analytical SPW operator. The lower flowchart gives the corresponding handling of linear phase terms for each step.

Tables (5)

Tables Icon

Table 1. Accuracy and Numerical Effort for Different Free-Space Propagation Techniques to Propagate Field behind High-NA Beam Splitter into the Target Plane

Tables Icon

Table 2. Coefficients of Seidel Aberrations for the Lens behind the Diffractive Beamshapera

Tables Icon

Table 3. Accuracy and Numerical Effort for Different Free-Space Propagation Techniques to Propagate the Field behind an Aberrated Lens into Focus (z=12mm)

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Table 4. Accuracy and Numerical Effort for Different Free-Space Propagation Techniques to Propagate the Super Gaussian Beam with Linear Phase Term

Tables Icon

Table 5. Accuracy and Numerical Effort for Different Free-Space Propagation Techniques to Propagate the Parabolic Astigmatic Beam by 10 mm

Equations (63)

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V(r,ω)=V(r)δ(ωω0)
A(κ)=F[V(ρ,0)]=12πV(ρ,0)exp(iκ·ρ)dxdy,
exp(ikzz)=exp[i(k2kx2ky2)1/2z].
V(ρ,z)=F1[A(κ)exp(ikzz)]=12πA(κ)exp(ikzz+iκ·ρ)dkxdky.
V(ρ,z)=|V(ρ,z)|exp[iΦres(ρ,z)]exp[iΦsmooth(ρ,z)].
kz=kkx2+ky22k+h(kx,ky),
V(x,y,z)=F1{Amod(kx,ky,z)exp(ikz)exp(ikx2+ky22kz)}
Amod(kx,ky,z)=A(kx,ky,0)exp[ih(kx,ky)z].
Vmod(x,y,z)=F1{Amod(kx,ky,z)},
V(x,y,z)=14π2{Vmod(x,y,z)exp(iκ·ρ)dxdy}exp(ikzikx2+ky22kz+iκ·ρ)dkxdky,
V(x,y,z)=eikzk4π2Vmod(x,y,z){exp[iκ·(ρρ)]exp(ikx2+ky22kz)dkxdky}dxdy.
V(x,y,z)=eikzk2πizexp(ik|ρ|22z)Vmod(x,y,z)exp(ikρ·ρz)exp(ik|ρ|22z)dxdy.
β=kρz=(kxz,kyz)T
V(x,y,z)=eikzk2πizexp(ik|ρ|22z)Vmod(x,y,z)exp(iρ·β)exp(ik|ρ|22z)dxdy.
V(x,y,z)=α(x,y,z)Fβ{Vmod(x,y,z)exp(ik|ρ|22z)}
α(x,y,z)=eikzkizexp(ik|ρ|22z),
kz(kx,ky)=kmaxk^z(kx,ky)=ckmax+dkx2+ky2kmax+h(kx,ky)
k^z(kx,ky)=[k2kmax2(kx2+ky2kmax2)]1/2,
c=23k^z(0,0)+23k^z(0.52kmax,0)13k^z(kmax,0)
d=k^z(kmax,0)k^z(0,0).
V(x,y,z)=α(x,y,z)Fβ{Vmod(x,y,z)exp(ikmax|ρ|24dz)}
β=kmaxρ2dz=(kmaxx2dz,kmaxy2dz)T,
α(x,y,z)=eickmaxzkmax2idzexp(ikmax|ρ|24dz).
V(x,y,0)={Fβ1[V(x,y,z)α(x,y,z)]exp(ikmax|ρ|24dz)}*Fx,y1{exp(izh(kx,ky))},
A*B(x,y,0)=A(x^,y^,0)B(xx^,yy^,0)dx^dy^.
V(x,y,z)={Fβ[V(x,y,0)α(x,y,z)]exp(ikmax|ρ|24dz)}*Fx,y1{exp(izh(kx,ky))}
β=kmaxρ2dz=(kmaxx2dz,kmaxy2dz)T
α(x,y,z)=eickmaxzkmax2idzexp(ikmax|ρ|24dz).
Vres,sphere(x,y):=V(x,y,0)α(x,y,z)=|V(x,y,0)|exp[iΦres(x,y,0)]exp[iΦsmooth(x,y,0)]e(ickmaxz)kmax2idzexp(ikmax|ρ|24dz).
σ:=x,y|V,operator(x,y,z)V,reference(x,y,z)|2x,y|V,reference(x,y,z)|2,
V(ρ,z)=V(ρ+zγ1,,z)exp[i(γ0,z+γ1,·κ0,z+ρ·κ0,)]
V(ρ,z)=F1[A(κ,0)eiγ(κ)z]
A(κ,0)=A(κκ0,,0).
γ0,=k0z,+1k0z,(k0x,2+k0y,2)k0x,2k0y,2k0z,3,
γ1,=[k0x,k0z,(k0y,2k0z,21),k0y,k0z,(k0x,2k0z,21)]T
k0z,=(k2k0x,2k0y,2)1/2.
γ(κ)=kz(κ+κ0,)γ0,γ1,·(κ+κ0,).
V(x,y,0)=V(x,y,0)exp(ik0x,x+ik0y,y).
V(x,y,z)=12πA(kxk0x,,kyk0y,,0)exp(ikxx+ikyy+ikzz)dkxdky
A(kx,ky,0)=12πV(x,y,0)exp(ikxxikyy)dxdy.
kz=(k2kx2ky2)1/2=γ0,+γ1,·κ+γx,2,kx2+γy,2,ky2+h(κκ0,)
γ0,=k0z,+k0x,2+k0y,2k0z,k0x,2k0y,2k0z,312k0z,2[(k0z,+k0y,2k0z,)k0y,2+(k0z,+k0x,2k0z,)k0x,2]
γ1,=[k0x,k0z,(k0z,+k0x,2k0z,k0z,+k0y,2k0z,21),k0y,k0z,(k0z,+k0y,2k0z,k0z,+k0x,2k0z,21)]T,
γx,2,=12(k0z,+k0x,2k0z,k0z,2),
γy,2,=12(k0z,+k0y,2k0z,k0z,2),
V(x,y,z)=F1{Amod(κκ0,,z)exp(ik0z,z)exp(i(γ1,·κ)z)exp(i(γx,2,kx2+γy,2,ky2)z)}
Amod(κκ0,,z)=A(κκ0,,0)exp(i(h(κκ0,))z).
Vmod(x,y,z)=F1{Amod(kx,ky,z)},
V(x,y,z)=14π2{Vmod(x,y,z)exp(iκ0,·ρ)exp(iκ·ρ)dxdy}exp(ik0z,z+i(γ1,·κ)z+iγx,2,kx2z+iγy,2,ky2z)exp(iκ·ρ)dkxdky,
V(x,y,z)=exp(ik0z,z)4π2Vmod(x,y,z)exp(iκ0,·ρ){exp[iκ·(ρρ+zγ1,)]exp[i(γx,2,kx2+γy,2,ky2)z]dkxdky}dxdy.
V(x,y,z)=exp(ik0z,z)4πiz(γx,2,γy,2,)1/2exp(i2zxγx,1,+x2+(γx,1,z)24γx,2,z)exp(i2zyγy,1,+y2+(γy,1,z)24γy,2,z)Vmod(x,y,z)exp(iκ0,·ρ)exp(i2x+2γx,1,z4γx,2,zx)exp(i2y+2γy,1,z4γy,2,zy)exp(ix24γx,2,z)exp(iy24γy,2,z)dxdy.
β=(2x+2γx,1,z4γx,2,zk0x,,2y+2γy,1,z4γy,2,zk0y,)T
V(x,y,z)=exp(ik0z,z)4πiz(γx,2,γy,2,)1/2exp(i2zxγx,1,+x2+(γx,1,z)24γx,2,z)exp(i2zyγy,1,+y2+(γy,1,z)24γy,2,z)Fβ{Vmod(x,y,z)exp(ix24γx,2,z)exp(iy24γy,2,z)}.
Φsmooth(x,y,0)=k(x2fx+y2fy).
V(x,y,z)=jV,j(ρρ0,,j,z)exp[iκ0,,j·ρ].
V(x,y,z)=j|V,j(ρρ0,,j,z)|exp[iΦres,j(ρρ0,,j,z)]exp[iκ0,,j·ρ]
f=x,y,j{[d[Φres,,j(ρρ0,,j,z)+κ0,,j·ρ]dxdΦsmooth,fitdx]2+[d[Φres,,j(ρρ0,,j,z)+κ0,,j·ρ]dydΦsmooth,fitdy]2}
f=x^,y^,j{[dΦres,,j(ρ^,z)dx^+k0x,,jdΦsmooth,fit(ρ^+ρ0,,j,z)dx^]2+[dΦres,,j(ρ^,z)dy^+k0y,,jdΦsmooth,fit(ρ^+ρ0,,j,z)dy^]2},
f(Φsmooth,fit(x,y,z))Φsmooth,fitMmin.,
Φsmooth,fit,spher(ρ^+ρ0,,j)={k(|ρ^+ρ0,,j|2+rsph2)1/2ifr<0,+k(|ρ^+ρ0,,j|2+rsph2)1/2ifr>0.
dΦsmooth,fit,spher(ρ^+ρ0,,j)dx^={k(x^+x0,,j)(|ρ^+ρ0,,j|2+rsph2)1/2ifr<0,k(x^+x0,,j)(|ρ^+ρ0,,j|2+rsph2)1/2ifr>0.
dΦsmooth,fit,spher(ρ^+ρ0,,j)dy^={k(y^+y0,,j)(|ρ^+ρ0,,j|2+rsph2)1/2ifr<0,k(y^+y0,,j)(|ρ^+ρ0,,j|2+rsph2)1/2ifr>0.
M={Φsmooth,fit,spher(ρ^+ρ0,,j,rsph):rsphR\0}.

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