Abstract

In a recent work [J. Opt. Soc. Am. A 28, 738 (2011)], Lifeng Li and Gerard Granet investigate nonconvergence cases of the Fourier modal method (FMM). They demonstrate that the nonconvergence is due to the irregular field singularities at lossless metal-dielectric right-angle edges. Here we make further investigations on the problem and find that the FMM surprisingly converges for deep sub-wavelength gratings (grating period being much smaller than the illumination wavelength). To overcome the nonconvergence for gratings that are not deep sub-wavelength, we approximately replace the lossless metal-dielectric right-angle edges by a medium with a gradually varied refraction index, so as to remove the irregular field singularities. With such treatment, convergence is observed as the region of the approximate medium approaches vanishing.

© 2014 Optical Society of America

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References

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  1. T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  2. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  3. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  4. L. F. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  5. K. M. Gundu and A. Mafi, “Reliable computation of scattering from metallic binary gratings using Fourier-based modal methods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
    [CrossRef]
  6. K. M. Gundu and A. Mafi, “Constrained least squares Fourier modal method for computing scattering from metallic binary gratings,” J. Opt. Soc. Am. A 27, 2375–2380 (2010).
    [CrossRef]
  7. L. F. Li and G. Granet, “Field singularities at lossless metal-dielectric right-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A 28, 738–746 (2011).
    [CrossRef]
  8. L. F. Li, “Field singularities at lossless metal-dielectric arbitrary-angle edges and their ramifications to the numerical modeling of gratings,” J. Opt. Soc. Am. A 29, 593–604 (2012).
    [CrossRef]
  9. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  10. J. F. Tang and P. F. Gu, Thin-Film Optics and Technology (Mechanical Industry, 1989).
  11. G. Granet, “Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution,” J. Opt. Soc. Am. A 16, 2510–2516 (1999).
    [CrossRef]

2012 (1)

2011 (1)

2010 (2)

1999 (1)

1996 (3)

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Granet, G.

Gu, P. F.

J. F. Tang and P. F. Gu, Thin-Film Optics and Technology (Mechanical Industry, 1989).

Guizal, B.

Gundu, K. M.

Lalanne, P.

Li, L. F.

Mafi, A.

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Morris, G. M.

Tang, J. F.

J. F. Tang and P. F. Gu, Thin-Film Optics and Technology (Mechanical Industry, 1989).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

J. Opt. Soc. Am. A (8)

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

J. F. Tang and P. F. Gu, Thin-Film Optics and Technology (Mechanical Industry, 1989).

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Figures (9)

Fig. 1.
Fig. 1.

Geometry of the considered periodic binary grating, where εc, εs, εm, and εd are relative permittivities, and d, t, and h are geometric parameters defined in the main text.

Fig. 2.
Fig. 2.

Convergence test of the 0th-order reflection efficiency R0 for the nonconvergence case in Fig. 2(b) in [7] [also reproduced in (a)], which shows convergence for deep sub-wavelength gratings. The black solid curves show the FMM results, the red circles show the theoretical value given by Eq. (2), and N represents the truncated harmonic number. In (b)–(f), we keep the same parameters as those in Fig. 2(b) of [7] except the grating period d, which gradually decreases by taking λ/10, λ/50, λ/500, λ/1000, λ/2000, respectively.

Fig. 3.
Fig. 3.

Same as Fig. 2 but for the nonconvergence case in Fig. 7(c) in [7] [also reproduced in (a)]. In (b)–(f), the grating period d gradually decreases by taking λ/20, λ/50, λ/100, λ/1000, λ/2000, respectively.

Fig. 4.
Fig. 4.

Same as Fig. 2 but for the nonconvergence case in Fig. 7(d) in [7] [also reproduced in (a)]. In (b)–(d), the grating period d gradually decreases by taking λ/5, λ/20, λ/100, respectively.

Fig. 5.
Fig. 5.

Treatment to overcome the nonconvergence of the FMM with the use of an approximate medium with a gradually varied refractive index. (a) Regions of the approximate medium with a width δ that replaces the lossless metal-dielectric right-angle edges. (b1) to (b4) Real and imaginary part of the complex refractive index ε1/2, which changes linearly between εm1/2 and εd1/2 within the left (b1) to (b2) and the right (b3) to (b4) δ-width regions of the approximate medium.

Fig. 6.
Fig. 6.

Real coordinate transformation to enhance the convergence for the calculation of R0 (δ). Within the gradually varied regions (centered at x=w/2 and w/2, w=0.1μm in the figure), a small change of the physical coordinate x is mapped to a big change of the numerical coordinate x.

Fig. 7.
Fig. 7.

Convergence test of the 0th-order reflection efficiency R0 for the nonconvergence case in Fig. 2(b) in [7] [also reproduced in (a)], which shows convergence for our treatment with the use of an approximate medium with a gradually varied refractive index. In (b)–(f), the width δ of the approximate medium gradually decreases by taking 103, 104, 105, 106, 107μm, respectively.

Fig. 8.
Fig. 8.

Same as Fig. 7 but for the nonconvergence case in Fig. 7(c) in [7] [also reproduced in (a)]. In (b)–(f), the width δ of the approximate medium takes 103, 104, 105, 106, 107μm, respectively.

Fig. 9.
Fig. 9.

Same as Fig. 7 but for the nonconvergence case in Fig. 7(d) in [7] [also reproduced in (a)]. In (b)–(f), the width δ of the approximate medium takes 103, 104, 105, 106, 107μm, respectively.

Equations (4)

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εeff=εmεd(1f)εm+fεd,
R0=|r1+exp(2ik0εeffh)t1t1r21exp(2ik0εeffh)r1r2|2,
R0(δ=0)=limδ0+R0(δ).
τ=2πarcsinΔ,

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