Abstract

I extend a previous work [J. Opt. Soc. Am. A [CrossRef]  28, 738 (2011)] on field singularities at lossless metal–dielectric right-angle edges and their ramifications to the numerical modeling of gratings to the case of arbitrary metallic wedge angles. Simple criteria are given that allow one knowing the lossless permittivities and the arbitrary wedge angles to determine if the electric field at the edges is nonsingular, can be regularly singular, or can be irregularly singular without calculating the singularity exponent. Furthermore, the knowledge of the singularity type enables one to predict immediately if a numerical method that uses Fourier expansions of the transverse electric field components at the edges will converge or not without making any numerical tests. All conclusions of the previous work about the general relationships between field singularities, Fourier representation of singular fields, and convergence of numerical methods for modeling lossless metal–dielectric gratings have been reconfirmed.

© 2012 Optical Society of America

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References

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  1. L. 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]
  2. J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
    [CrossRef]
  3. J. G. Van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, 1991), pp. 116–162.
  4. 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]
  5. 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]
  6. V. V. Fisanov, “Singularity index of the electromagnetic field on the edge in the presence of the Veselago medium,” Russ. Phys. J. 49, 853–856 (2006).
    [CrossRef]
  7. V. V. Fisanov, “The singularity of the electromagnetic field on the edge of a wedge-shaped structure containing media with negative permittivities and permeabilities,” J. Commun. Technol. Electron. 52, 991–995 (2007).
    [CrossRef]
  8. A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Numer. Model. 19, 105–117 (2006).
    [CrossRef]
  9. H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
    [CrossRef]
  10. M. Paggi, “Singular, hypersingular and singular free electromagnetic fields at wedge tips in metamaterials,” Int. J. Solids Struct. 47, 2062–2069 (2010).
    [CrossRef]
  11. M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microwave Guided Wave Lett. 4, 3–5 (1994).
    [CrossRef]
  12. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  13. M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232 (1973).
    [CrossRef]
  14. L. Li, “Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am. A 16, 2521–2531 (1999).
    [CrossRef]
  15. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788 (2007).
    [CrossRef]
  16. L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
    [CrossRef]
  17. J. Chandezon, G. Granet, and N. P. Yashina, “C-method: from the beginnings to recent advances,” in Modern Theory of Gratings, Y. K. Sirenko and S. Ström, eds., Springer Series in Optical Sciences, Vol. 153, (Springer, 2010), pp. 173–210.
  18. L. Li, “Using symmetries of grating groove profiles to reduce computation cost of the C method,” J. Opt. Soc. Am. A 24, 1085–1096 (2007).
    [CrossRef]

2011 (1)

2010 (3)

2008 (1)

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[CrossRef]

2007 (3)

2006 (2)

A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Numer. Model. 19, 105–117 (2006).
[CrossRef]

V. V. Fisanov, “Singularity index of the electromagnetic field on the edge in the presence of the Veselago medium,” Russ. Phys. J. 49, 853–856 (2006).
[CrossRef]

1999 (1)

1996 (1)

1994 (1)

M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microwave Guided Wave Lett. 4, 3–5 (1994).
[CrossRef]

1973 (1)

M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232 (1973).
[CrossRef]

1972 (1)

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[CrossRef]

Bobrovnikov, M. S.

M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232 (1973).
[CrossRef]

Bressan, M.

M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microwave Guided Wave Lett. 4, 3–5 (1994).
[CrossRef]

Chandezon, J.

L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

J. Chandezon, G. Granet, and N. P. Yashina, “C-method: from the beginnings to recent advances,” in Modern Theory of Gratings, Y. K. Sirenko and S. Ström, eds., Springer Series in Optical Sciences, Vol. 153, (Springer, 2010), pp. 173–210.

Fisanov, V. V.

V. V. Fisanov, “The singularity of the electromagnetic field on the edge of a wedge-shaped structure containing media with negative permittivities and permeabilities,” J. Commun. Technol. Electron. 52, 991–995 (2007).
[CrossRef]

V. V. Fisanov, “Singularity index of the electromagnetic field on the edge in the presence of the Veselago medium,” Russ. Phys. J. 49, 853–856 (2006).
[CrossRef]

Gamba, P.

M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microwave Guided Wave Lett. 4, 3–5 (1994).
[CrossRef]

Granet, G.

L. 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]

J. Chandezon, G. Granet, and N. P. Yashina, “C-method: from the beginnings to recent advances,” in Modern Theory of Gratings, Y. K. Sirenko and S. Ström, eds., Springer Series in Optical Sciences, Vol. 153, (Springer, 2010), pp. 173–210.

Gundu, K. M.

Kettunen, H.

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[CrossRef]

Kivshar, Y. S.

A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Numer. Model. 19, 105–117 (2006).
[CrossRef]

Li, L.

Lyndin, N. M.

Mafi, A.

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Meixner, J.

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[CrossRef]

Paggi, M.

M. Paggi, “Singular, hypersingular and singular free electromagnetic fields at wedge tips in metamaterials,” Int. J. Solids Struct. 47, 2062–2069 (2010).
[CrossRef]

Parriaux, O.

Shadrivov, I. V.

A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Numer. Model. 19, 105–117 (2006).
[CrossRef]

Sihvola, A.

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[CrossRef]

Sukhorukov, A. A.

A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Numer. Model. 19, 105–117 (2006).
[CrossRef]

Tishchenko, A. V.

Van Bladel, J. G.

J. G. Van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, 1991), pp. 116–162.

Wallén, H.

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[CrossRef]

Yashina, N. P.

J. Chandezon, G. Granet, and N. P. Yashina, “C-method: from the beginnings to recent advances,” in Modern Theory of Gratings, Y. K. Sirenko and S. Ström, eds., Springer Series in Optical Sciences, Vol. 153, (Springer, 2010), pp. 173–210.

Zamaraeva, V. P.

M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232 (1973).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microwave Guided Wave Lett. 4, 3–5 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[CrossRef]

Int. J. Numer. Model. (1)

A. A. Sukhorukov, I. V. Shadrivov, and Y. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Numer. Model. 19, 105–117 (2006).
[CrossRef]

Int. J. Solids Struct. (1)

M. Paggi, “Singular, hypersingular and singular free electromagnetic fields at wedge tips in metamaterials,” Int. J. Solids Struct. 47, 2062–2069 (2010).
[CrossRef]

J. Commun. Technol. Electron. (1)

V. V. Fisanov, “The singularity of the electromagnetic field on the edge of a wedge-shaped structure containing media with negative permittivities and permeabilities,” J. Commun. Technol. Electron. 52, 991–995 (2007).
[CrossRef]

J. Opt. Soc. Am. A (7)

Metamaterials (1)

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[CrossRef]

Russ. Phys. J. (1)

V. V. Fisanov, “Singularity index of the electromagnetic field on the edge in the presence of the Veselago medium,” Russ. Phys. J. 49, 853–856 (2006).
[CrossRef]

Sov. Phys. J. (1)

M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232 (1973).
[CrossRef]

Other (3)

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

J. Chandezon, G. Granet, and N. P. Yashina, “C-method: from the beginnings to recent advances,” in Modern Theory of Gratings, Y. K. Sirenko and S. Ström, eds., Springer Series in Optical Sciences, Vol. 153, (Springer, 2010), pp. 173–210.

J. G. Van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, 1991), pp. 116–162.

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Figures (11)

Fig. 1.
Fig. 1.

Two wedge assemblies considered in this work: (a) 2-wedge assembly, (b) 3-wedge assembly. (a)  0 < φ * < 2 π , (b)  0 < φ * < π , and the lower boundary of the medium with ε c is flat.

Fig. 2.
Fig. 2.

Plots of r ± ( α , φ ) and r ± ( i β , φ ) for φ = ± 2 / 3 . Note that α and β share the same horizontal axis. The interception point on the α axis is 1 / ( 1 + | φ | ) , and the two interception points on the r axis are ( 1 ± φ ) / ( 1 φ ) . The two β curves share a common horizontal asymptote: r = 1 (dot-dashed line).

Fig. 3.
Fig. 3.

Four possible sign combinations of lossless permittivities in a 3-wedge assembly. There are four other combinations that can be obtained by reversing the signs of all permittivities.

Fig. 4.
Fig. 4.

(a) Contours of f ( α , r c , r d ; φ ) = 0 (blue) and f ( i β , r c , r d ; φ ) = 0 (red) for 0 < α < 1 and 0 < β < , respectively, and φ = 3 4 ( φ * = π / 4 ) ; (b) a zoom-in view of the lower left corner of (a). The black dots labeled with uppercase letters mark the characteristic points explained in the text. The gray dots in (b) labeled with lowercase letters give the permittivity combinations of the parallelogram gratings considered in Table 1 and Fig. 6. (Matching between colors and regions: blue, OAB and SKJ; red, ABDEC and KJHEI; white, DEIG and HECF.)

Fig. 5.
Fig. 5.

Same as Fig. 4 except that φ = 1 4 ( φ * = 3 π / 4 ) . Note the existence of an envelope that connects points R, Q, E, M, and N. (Matching between colors and regions: blue, OAPD and SKLH; red, APEC and KLEI; white, EQRI and EMNC; double-blue, BGRQ and JFNM; double-red, EPQ and ELM; blue–red, BDP and JHL.)

Fig. 6.
Fig. 6.

Convergence and nonconvergence of the FMM for seven parallelogram gratings whose parameters are listed in Table 1, shown in Figs. 4(b) and 5(b), and given in the text. The inset in (a) shows the grating parameters shared by all seven gratings. Note that the curves for gratings f and g appear in (f) using the left and right vertical axes, respectively.

Fig. 7.
Fig. 7.

(a) Convergence and (b) nonconvergence of the C method for two asymmetric triangular gratings. The inset in (a) shows the grating parameters shared by both gratings.

Fig. 8.
Fig. 8.

Convergence and nonconvergence of the C method for four symmetric triangular gratings in normal incidence. The inset in (a) shows the grating parameters shared by all four gratings. Note the different vertical axis scales in the four subfigures.

Fig. 9.
Fig. 9.

Convergence and nonconvergence of the C method for two symmetric gratings with one cusp per period in normal incidence: (a) upward cusp, (b) downward cusp.

Fig. 10.
Fig. 10.

α (blue) lines and β (red) lines for 0 < α < 1 and 0 < β < , respectively, (a) for φ = 3 4 ( φ * = π / 4 ) and (b) for φ = 1 4 ( φ * = 3 π / 4 ) . The cyan, yellow, white, and magenta colored regions correspond to the first through fourth quadrants of the ( r c , r d ) space, respectively, and the gray region is nonphysical. Mathematically, the five regions are given (in the same order) by ( x 1 + 1 ) ( x 1 + x 2 ) > 0 and ( x 1 + 1 ) ( x 2 + 1 ) < 0 , ( x 1 + 1 ) ( x 1 + x 2 ) < 0 and ( x 1 + 1 ) ( x 2 + 1 ) < 0 , ( x 1 + 1 ) ( x 1 + x 2 ) < 0 and 0 < ( x 1 + 1 ) ( x 2 + 1 ) < 1 , ( x 1 + 1 ) ( x 1 + x 2 ) > 0 and 0 < ( x 1 + 1 ) ( x 2 + 1 ) < 1 , and ( x 1 + 1 ) ( x 2 + 1 ) > 1 . In (a) [(b)], all α lines have slopes greater (less) than that of the line 1 + x 1 φ 2 + x 2 φ 2 = 0 , and all β lines have slopes less (greater) than that of the line 1 + x 1 φ 2 + x 2 φ 2 = 0 .

Fig. 11.
Fig. 11.

Integration contour for verifying the number of solutions of Eq. (8) or (A1) in domain D for a point in a region in Fig. 10 where the red lines exist but do not cross each other.

Tables (1)

Tables Icon

Table 1. Permittivities a , Permittivity Ratios, and Singularity Exponents of the Lossless Metal–Dielectric Parallelogram Gratings Considered in Fig. 6

Equations (23)

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E ρ ( j ) ( ρ , φ ) = ρ τ 1 A ρ ( j ) ( φ ) , E φ ( j ) ( ρ , φ ) = ρ τ 1 A φ ( j ) ( φ ) ,
sin τ π sin τ ( π φ * ) = ± ε m ε c ε m + ε c .
sin τ π sin τ π φ = ± 1 + r 1 r ,
r ± = tan [ τ π ( 1 φ ) 2 ] / tan [ τ π ( 1 ± φ ) 2 ] ,
r ± ( τ , φ ) = 1 r ( τ , φ ) = r ( τ , φ ) ,
r + ( 0 , φ ) = 1 φ 1 + φ , r + ( 1 1 + φ , φ ) = 0 , r + ( 1 , φ ) = 1 , r + ( i , φ ) = 1 ,
sin 2 τ π + ( ε d ε m ) ( ε m ε c ) ( ε d + ε m ) ( ε m + ε c ) sin 2 τ ( π φ * ) + ( ε d ε m ) ( ε c ε d ) ( ε d + ε m ) ( ε c + ε d ) sin 2 τ φ * = 0 .
sin 2 τ π ( 1 + r c ) ( 1 + r d ) ( 1 r c ) ( 1 r d ) sin 2 τ π φ ( r c r d ) ( 1 + r d ) ( r c + r d ) ( 1 r d ) sin 2 τ π φ = 0 ,
1 φ r c + φ φ r d = 1 ,
1 φ 1 r c + φ φ 1 r d = 1 .
r c | r d 0 = tan τ π φ tan τ π .
f ( α , r c , r d ; φ ) = 0 , α f ( α , r c , r d ; φ ) = 0 .
f ( i β , r c , r d ; φ ) = 0 , β f ( i β , r c , r d ; φ ) = 0 .
τ 2 f ( τ , r c , r d , φ ) | τ = 0 = 0 , τ 4 f ( τ , r c , r d , φ ) | τ = 0 = 0 ,
( r c Q , r d Q ) = ( φ φ 1 + φ , 1 + φ 1 + φ ) , ( r c M , r d M ) = ( 1 r c Q , 1 r d Q ) ( φ < 1 / 2 ) .
r c N = tan α N π φ tan α N π ,
φ sin 2 α N π sin 2 α N π φ = 0 .
f u ( x ) = h | 2 x / d | σ ,
f d ( x ) = h ( 1 | 2 x / d | σ ) ( | x | d / 2 ) ,
sin 2 τ π + x 1 sin 2 τ π φ + x 2 sin 2 τ π φ = 0 ,
x 1 = ( 1 + r c ) ( 1 + r d ) ( 1 r c ) ( 1 r d ) , x 2 = ( r c r d ) ( 1 + r d ) ( r c + r d ) ( 1 r d ) .
r c = [ x 1 1 ( x 1 + 1 ) ( x 2 + 1 ) ] 2 ( x 1 + 1 ) ( x 1 + x 2 ) , r d = [ 1 ± 1 ( x 1 + 1 ) ( x 2 + 1 ) ] 2 ( x 1 + 1 ) ( x 2 + 1 ) ,
sin 2 τ π + x 1 φ sin 2 τ π φ + x 2 φ sin 2 τ π φ = 0 ,

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