Abstract

This paper addresses the task of obtaining the far-field spectrum for a finite structure given the near-field calculated by the aperiodic Fourier modal method in contrast-field formulation (AFMM-CFF). The AFMM-CFF efficiently calculates the solution to Maxwell’s equations for a finite structure by truncating the computational domain with perfectly matched layers (PMLs). However, this limits the far-field solution to a narrow strip between the PMLs. The Green’s function for layered media is used to extend the solution over the whole super- and substrate. The approach is validated by applying it to the problem of scattering from a cylinder for which the analytical solution is available. Moreover, a numerical study is conducted on the accuracy of the approximate far-field computed with the super-cell Fourier modal method by using the AFMM-CFF with near- to far-field transformation as a reference.

© 2013 Optical Society of America

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References

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  1. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
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    [CrossRef]
  3. G. Bao, L. Cowsar, and W. Masters, eds., Mathematical Modeling in Optical Science (Frontiers in Applied Mathematics) (Society for Industrial Mathematics, 2001).
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    [CrossRef]
  5. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
    [CrossRef]
  6. M. Pisarenco, J. M. L. Maubach, I. D. Setija, and R. M. M. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
    [CrossRef]
  7. M. Pisarenco, “Scattering from finite structures: an extended Fourier modal method,” Ph.D. thesis (Eindhoven University of Technology, 2011).
  8. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  9. J. P. Hugonin and P. Lalanne, “Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization,” J. Opt. Soc. Am. A 22, 1844–1849 (2005).
    [CrossRef]
  10. O. M. Bucci, C. Gennarelli, and C. Savarese, “Fast and accurate near-field-far-field transformation by sampling interpolation of plane-polar measurements,” IEEE Trans. Antennas Propag. 39, 48–55 (1991).
    [CrossRef]
  11. A. Taaghol and T. K. Sarkar, “Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current,” IEEE Trans. Electromag. Compat. 38, 536–542 (1996).
  12. T. K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM,” IEEE Trans. Antennas Propag. 47, 566–573 (1999).
    [CrossRef]
  13. P.-W. Zhai, Y.-K. Lee, G. W. Kattawar, and P. Yang, “Implementing the near- to far-field transformation in the finite-difference time-domain method,” Appl. Opt. 43, 3738–3746 (2004).
    [CrossRef]
  14. P. Török, P. R. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A 23, 713–722 (2006).
    [CrossRef]
  15. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).
  16. K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory,” IEEE Trans. Antennas Propag. 38, 335–344 (1990).
    [CrossRef]
  17. J. R. Wait, Electromagnetic Waves in Stratified Media (IEEE/OUP Series on Electromagnetic Wave Theory) (Oxford University, 1996).
  18. R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput. 7, 477–485 (2000).
  19. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1990).
  20. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953).
  21. C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
    [CrossRef]
  22. A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
    [CrossRef]
  23. J. Saarinen, E. Noponen, and J. P. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
    [CrossRef]

2012 (1)

M. Pisarenco, J. M. L. Maubach, I. D. Setija, and R. M. M. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[CrossRef]

2011 (1)

2010 (1)

2006 (2)

P. Török, P. R. Munro, and E. E. Kriezis, “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” J. Opt. Soc. Am. A 23, 713–722 (2006).
[CrossRef]

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

2005 (1)

2004 (1)

2000 (1)

R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput. 7, 477–485 (2000).

1999 (1)

T. K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM,” IEEE Trans. Antennas Propag. 47, 566–573 (1999).
[CrossRef]

1996 (1)

A. Taaghol and T. K. Sarkar, “Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current,” IEEE Trans. Electromag. Compat. 38, 536–542 (1996).

1995 (3)

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

1991 (1)

O. M. Bucci, C. Gennarelli, and C. Savarese, “Fast and accurate near-field-far-field transformation by sampling interpolation of plane-polar measurements,” IEEE Trans. Antennas Propag. 39, 48–55 (1991).
[CrossRef]

1990 (1)

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory,” IEEE Trans. Antennas Propag. 38, 335–344 (1990).
[CrossRef]

1960 (1)

C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
[CrossRef]

Benisty, H.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bucci, O. M.

O. M. Bucci, C. Gennarelli, and C. Savarese, “Fast and accurate near-field-far-field transformation by sampling interpolation of plane-polar measurements,” IEEE Trans. Antennas Propag. 39, 48–55 (1991).
[CrossRef]

Clenshaw, C. W.

C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
[CrossRef]

Cools, R.

R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput. 7, 477–485 (2000).

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953).

Curtis, A. R.

C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
[CrossRef]

David, A.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Gaylord, T. K.

Gennarelli, C.

O. M. Bucci, C. Gennarelli, and C. Savarese, “Fast and accurate near-field-far-field transformation by sampling interpolation of plane-polar measurements,” IEEE Trans. Antennas Propag. 39, 48–55 (1991).
[CrossRef]

Grann, E. B.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953).

Hugonin, J. P.

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1990).

Kattawar, G. W.

Kim, K.

R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput. 7, 477–485 (2000).

Kriezis, E. E.

Lalanne, P.

Lee, Y.-K.

Mattheij, R.

Mattheij, R. M. M.

M. Pisarenco, J. M. L. Maubach, I. D. Setija, and R. M. M. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[CrossRef]

Maubach, J.

Maubach, J. M. L.

M. Pisarenco, J. M. L. Maubach, I. D. Setija, and R. M. M. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[CrossRef]

Michalski, K. A.

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory,” IEEE Trans. Antennas Propag. 38, 335–344 (1990).
[CrossRef]

Moharam, M. G.

Munro, P. R.

Noponen, E.

J. Saarinen, E. Noponen, and J. P. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Pisarenco, M.

M. Pisarenco, J. M. L. Maubach, I. D. Setija, and R. M. M. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[CrossRef]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation,” J. Opt. Soc. Am. A 28, 1364–1371 (2011).
[CrossRef]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
[CrossRef]

M. Pisarenco, “Scattering from finite structures: an extended Fourier modal method,” Ph.D. thesis (Eindhoven University of Technology, 2011).

Pommet, D. A.

Saarinen, J.

J. Saarinen, E. Noponen, and J. P. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Sarkar, T. K.

T. K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM,” IEEE Trans. Antennas Propag. 47, 566–573 (1999).
[CrossRef]

A. Taaghol and T. K. Sarkar, “Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current,” IEEE Trans. Electromag. Compat. 38, 536–542 (1996).

Savarese, C.

O. M. Bucci, C. Gennarelli, and C. Savarese, “Fast and accurate near-field-far-field transformation by sampling interpolation of plane-polar measurements,” IEEE Trans. Antennas Propag. 39, 48–55 (1991).
[CrossRef]

Setija, I.

Setija, I. D.

M. Pisarenco, J. M. L. Maubach, I. D. Setija, and R. M. M. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[CrossRef]

Taaghol, A.

T. K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM,” IEEE Trans. Antennas Propag. 47, 566–573 (1999).
[CrossRef]

A. Taaghol and T. K. Sarkar, “Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current,” IEEE Trans. Electromag. Compat. 38, 536–542 (1996).

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Török, P.

Turunen, J. P.

J. Saarinen, E. Noponen, and J. P. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Wait, J. R.

J. R. Wait, Electromagnetic Waves in Stratified Media (IEEE/OUP Series on Electromagnetic Wave Theory) (Oxford University, 1996).

Weisbuch, C.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Yang, P.

Zhai, P.-W.

Zheng, D.

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory,” IEEE Trans. Antennas Propag. 38, 335–344 (1990).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (3)

K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I—Theory,” IEEE Trans. Antennas Propag. 38, 335–344 (1990).
[CrossRef]

O. M. Bucci, C. Gennarelli, and C. Savarese, “Fast and accurate near-field-far-field transformation by sampling interpolation of plane-polar measurements,” IEEE Trans. Antennas Propag. 39, 48–55 (1991).
[CrossRef]

T. K. Sarkar and A. Taaghol, “Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM,” IEEE Trans. Antennas Propag. 47, 566–573 (1999).
[CrossRef]

IEEE Trans. Electromag. Compat. (1)

A. Taaghol and T. K. Sarkar, “Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current,” IEEE Trans. Electromag. Compat. 38, 536–542 (1996).

J. Appl. Math. Comput. (1)

R. Cools and K. Kim, “A survey of known and new cubature formulas for the unit disk,” J. Appl. Math. Comput. 7, 477–485 (2000).

J. Comput. Phys. (2)

M. Pisarenco, J. M. L. Maubach, I. D. Setija, and R. M. M. Mattheij, “Efficient solution of Maxwell’s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method,” J. Comput. Phys. 231, 8209–8228 (2012).
[CrossRef]

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Opt. Soc. Am. A (6)

Numer. Math. (1)

C. W. Clenshaw and A. R. Curtis, “A method for numerical integration on an automatic computer,” Numer. Math. 2, 197–205 (1960).
[CrossRef]

Opt. Eng. (1)

J. Saarinen, E. Noponen, and J. P. Turunen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2566 (1995).
[CrossRef]

Phys. Rev. B (1)

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Other (6)

G. Bao, L. Cowsar, and W. Masters, eds., Mathematical Modeling in Optical Science (Frontiers in Applied Mathematics) (Society for Industrial Mathematics, 2001).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

M. Pisarenco, “Scattering from finite structures: an extended Fourier modal method,” Ph.D. thesis (Eindhoven University of Technology, 2011).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1990).

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953).

J. R. Wait, Electromagnetic Waves in Stratified Media (IEEE/OUP Series on Electromagnetic Wave Theory) (Oxford University, 1996).

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Figures (8)

Fig. 1.
Fig. 1.

Problems P1 and P2 have equal solutions on Ω0 for reflectionless PMLs (indicated by hatched stripes).

Fig. 2.
Fig. 2.

Slicing of the cylinder with color-coded refraction indices.

Fig. 3.
Fig. 3.

Logarithmic plot (log10E1) of the error in the spatial far-field for the cylinder problem as defined in Eq. (16).

Fig. 4.
Fig. 4.

Logarithmic plot (log10E2) of the error in the spectral far-field for the cylinder problem as defined in Eq. (17).

Fig. 5.
Fig. 5.

Reflective modes for different angle of incidence: θ=0 (solid line) to θ=1/3 (dashed–dotted line).

Fig. 6.
Fig. 6.

Multiple resist lines with color-coded refractive indices.

Fig. 7.
Fig. 7.

Super cell modes (open circle) compared to the continuous modes (solid line).

Fig. 8.
Fig. 8.

Logarithmic plot (log10E3) of the error in the spectral far-field for the super-cell problem as defined in Eq. (18).

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

×et(x)=k0ht(x),
×ht(x)=k0ϵ(x,z)et(x),
einc(x)=aeikinc·x,
er(x,y,z)=n=NNrnei(kxnx+kyyk1,znz),
kxn=kxinc+n2πΛ,n=N,,+N,
ky=kyinc,
k1,zn=k02ϵ1ky2kxn2,n=N,,+N.
××et(x)k02ϵ(x,z)et(x)=0.
××ec(x)k02ϵ(x,z)ec(x)=k02(ϵ(x,z)ϵb(z))eb(x),
××ec(x)k02ϵb(z)ec(x)=k02(ϵ(x,z)ϵb(z))et(x),
Lec(x)=f(x)
L=××k02ϵb(z),
f(x)=k02(ϵ(x,z)ϵb(z))et(x).
LG(x,x)=Iδ(xx)
G(x,x)=(Gxx(x,x)Gxy(x,x)Gxz(x,x)Gyx(x,x)Gyy(x,x)Gyz(x,x)Gzx(x,x)Gzy(x,x)Gzz(x,x))
Iδ(xx)=(δ(xx)000δ(xx)000δ(xx)).
VLG(x,x)f(x)dx=Lec(x).
ec(x)=VG(x,x)f(x)dx.
eyc(x)=VGyy(x,x)(ϵ(x)ϵb(x))ey(x)dx.
e^yc(kx;z)=VG^yy(kx;x,z,z)×(ϵ(x,z)ϵb(z))ey(x,z)dxdz,
G˜yy(kx;z,z)=12ik1,z[eik1,z|zz|+R˜12e+ik1,z0(z+z2d)],
R˜l,l+1=Rl,l+1+R˜l+1,l+2e2ikl+1,z0(dl+1dl)1+Rl,l+1R˜l+1,l+2e2ikl+1,z0(dl+1dl)
Rl,l+1=kl,zkl+1,zkl,z+kl+1,z
kl,z=k02ϵl(kyinc)2kx2,
kl,z0=k02ϵl(kyinc)2(kxinc)2.
R˜M,M+1=0.
Gyy(x,z,x,z)=i4H0(2)(k0(xx)2+(zz)2).
E1=eyeyref2eyref2,
E2=e^ye^yref2e^yref2,
kx(κ)=kxinc+κ,
E3=Ie^ysuper-celle^yref2e^yref2,

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