Abstract

It is insufficient to consider that hypersingularity is unphysical solely based on energy considerations. With a proper combination of the two degenerate hypersingular modes, the energy-flux edge condition is satisfied. A hyperbolic wedge model is presented that is much simpler than the previous model for the purpose of studying singular characteristics of the edge fields. This model not only reproduces the sharp edge model as the wedge becomes infinitely sharp but also naturally shows how the two degenerate hypersingular modes of the sharp edge model should be combined. In an incidental study of the effect of rounding edges on numerical computation, I show that the converged results for rounded edges do not converge to a fixed value when the radius of curvature tends to zero, if the corresponding sharp edge supports hypersingularity. I also prove that introducing a small amount of absorption loss for the purpose of improving numerical convergence is effective only when the ratio of the real parts of the permittivities of the two media forming the wedge is close to 1. Finally I remark on the possible illposedness of the hypersingularity problem without imposition of the edge condition.

© 2014 Optical Society of America

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References

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  1. J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
    [CrossRef]
  2. J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, 1991), pp. 116–162.
  3. J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
    [CrossRef]
  4. J. Meixner, “Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an vollkommen leitenden ebenen Schirmen,” Ann. Phys. 6, 2–9 (1949).
  5. 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]
  6. M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microw. Guided Wave Lett. 4, 3–5 (1994).
    [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. M. Paggi, “Singular, hypersingular and singular free electromagnetic fields at wedge tips in metamaterials,” Int. J. Solids Struct. 47, 2062–2069 (2010).
    [CrossRef]
  9. L. 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]
  10. 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]
  11. G. Oliveri and M. Raffetto, “A warning about metamaterials for users of frequency-domain numerical simulators,” IEEE Trans. Antennas Propag. 56, 792–798 (2008).
    [CrossRef]
  12. 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]
  13. 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]
  14. W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
    [CrossRef]
  15. 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]
  16. L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523–5525 (1976).
  17. L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).
  18. R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves (MacMillan, 1971), p. 4.
  19. C. J. Bouwkamp, “A note on singularities occurring at sharp edges in electromagnetic diffraction theory,” Physica (Amsterdam) 12, 467–474 (1946).
  20. D. S. Jones, The Theory of Electromagnetism (Pergamon, 1964), Sections. 9.1 and 9.2, pp. 562–569.
  21. S. W. Lee and R. Mittra, “Edge condition and ‘intrinsic loss’ in uniaxial plasma,” Can. J. Phys. 46, 111–120 (1968).
    [CrossRef]
  22. R. A. Hurd, “On the possibility of intrinsic loss occurring at the edges of ferrites,” Can. J. Phys. 40, 1067–1076 (1962).
    [CrossRef]
  23. C. A. Valagiannopoulos and A. Sihvola, “Improving the electrostatic field concentration in a negative-permittivity wedge with a grounded “bowtie” configuration,” Rad. Sci. 48, 316–325 (2013).
  24. A.-S. Bonnet-Ben Dhia, L. Chesnel, and X. Claeys, “Radiation condition for a non-smooth interface between a dielectric and a metamaterial,” Math. Mod. Methods Appl. Sci. 23, 1629–1662 (2013).
    [CrossRef]
  25. A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: analytical solution,” Phys. Rev. B 32, 6045–6047 (1985).
    [CrossRef]
  26. V. Cataudella and G. Iadonisi, “Electrostatic edge modes for a hyperbolic dielectric wedge: analytical solutions,” Solid State Commun. 59, 267–270 (1986).
    [CrossRef]
  27. 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]
  28. L. Chesnel, X. Claeys, and S. Nazarov, “A curious instability phenomenon for a rounded corner in presence of a negative material,” arXiv:1304.4788 [math.AP].
  29. L. Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Phil. Mag. 43(263), 259–272 (1897).
    [CrossRef]
  30. R. Mittra, “Relative convergence of the solution of a doubly infinite set of equations,” J. Res. Nat. Bur. Stand. D. 67, 245–254 (1963).
    [CrossRef]
  31. R. Mittra, T. Itoh, and T.-S. Li, “Analytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation by the moment method,” IEEE Trans. Microw. Theory Tech. 20, 96–104 (1972).
    [CrossRef]

2013 (2)

C. A. Valagiannopoulos and A. Sihvola, “Improving the electrostatic field concentration in a negative-permittivity wedge with a grounded “bowtie” configuration,” Rad. Sci. 48, 316–325 (2013).

A.-S. Bonnet-Ben Dhia, L. Chesnel, and X. Claeys, “Radiation condition for a non-smooth interface between a dielectric and a metamaterial,” Math. Mod. Methods Appl. Sci. 23, 1629–1662 (2013).
[CrossRef]

2012 (1)

2011 (1)

2010 (4)

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

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]

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]

W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

2008 (2)

G. Oliveri and M. Raffetto, “A warning about metamaterials for users of frequency-domain numerical simulators,” IEEE Trans. Antennas Propag. 56, 792–798 (2008).
[CrossRef]

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 (2)

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]

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]

2006 (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]

1994 (1)

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

1986 (1)

V. Cataudella and G. Iadonisi, “Electrostatic edge modes for a hyperbolic dielectric wedge: analytical solutions,” Solid State Commun. 59, 267–270 (1986).
[CrossRef]

1985 (1)

A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: analytical solution,” Phys. Rev. B 32, 6045–6047 (1985).
[CrossRef]

1978 (1)

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[CrossRef]

1976 (1)

L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523–5525 (1976).

1972 (3)

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).

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

R. Mittra, T. Itoh, and T.-S. Li, “Analytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation by the moment method,” IEEE Trans. Microw. Theory Tech. 20, 96–104 (1972).
[CrossRef]

1968 (1)

S. W. Lee and R. Mittra, “Edge condition and ‘intrinsic loss’ in uniaxial plasma,” Can. J. Phys. 46, 111–120 (1968).
[CrossRef]

1963 (1)

R. Mittra, “Relative convergence of the solution of a doubly infinite set of equations,” J. Res. Nat. Bur. Stand. D. 67, 245–254 (1963).
[CrossRef]

1962 (1)

R. A. Hurd, “On the possibility of intrinsic loss occurring at the edges of ferrites,” Can. J. Phys. 40, 1067–1076 (1962).
[CrossRef]

1949 (1)

J. Meixner, “Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an vollkommen leitenden ebenen Schirmen,” Ann. Phys. 6, 2–9 (1949).

1946 (1)

C. J. Bouwkamp, “A note on singularities occurring at sharp edges in electromagnetic diffraction theory,” Physica (Amsterdam) 12, 467–474 (1946).

1897 (1)

L. Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Phil. Mag. 43(263), 259–272 (1897).
[CrossRef]

Andersen, J. B.

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[CrossRef]

Boardman, A. D.

A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: analytical solution,” Phys. Rev. B 32, 6045–6047 (1985).
[CrossRef]

Bonnet-Ben Dhia, A.-S.

A.-S. Bonnet-Ben Dhia, L. Chesnel, and X. Claeys, “Radiation condition for a non-smooth interface between a dielectric and a metamaterial,” Math. Mod. Methods Appl. Sci. 23, 1629–1662 (2013).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, “A note on singularities occurring at sharp edges in electromagnetic diffraction theory,” Physica (Amsterdam) 12, 467–474 (1946).

Bressan, M.

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

Cataudella, V.

V. Cataudella and G. Iadonisi, “Electrostatic edge modes for a hyperbolic dielectric wedge: analytical solutions,” Solid State Commun. 59, 267–270 (1986).
[CrossRef]

Chesnel, L.

A.-S. Bonnet-Ben Dhia, L. Chesnel, and X. Claeys, “Radiation condition for a non-smooth interface between a dielectric and a metamaterial,” Math. Mod. Methods Appl. Sci. 23, 1629–1662 (2013).
[CrossRef]

L. Chesnel, X. Claeys, and S. Nazarov, “A curious instability phenomenon for a rounded corner in presence of a negative material,” arXiv:1304.4788 [math.AP].

Claeys, X.

A.-S. Bonnet-Ben Dhia, L. Chesnel, and X. Claeys, “Radiation condition for a non-smooth interface between a dielectric and a metamaterial,” Math. Mod. Methods Appl. Sci. 23, 1629–1662 (2013).
[CrossRef]

L. Chesnel, X. Claeys, and S. Nazarov, “A curious instability phenomenon for a rounded corner in presence of a negative material,” arXiv:1304.4788 [math.AP].

Davis, L. C.

L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523–5525 (1976).

Dobrzynski, L.

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).

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]

Gamba, P.

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

Garcia-Molina, R.

A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: analytical solution,” Phys. Rev. B 32, 6045–6047 (1985).
[CrossRef]

Granet, G.

Gras-Marti, A.

A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: analytical solution,” Phys. Rev. B 32, 6045–6047 (1985).
[CrossRef]

Gundu, K. M.

Hurd, R. A.

R. A. Hurd, “On the possibility of intrinsic loss occurring at the edges of ferrites,” Can. J. Phys. 40, 1067–1076 (1962).
[CrossRef]

Iadonisi, G.

V. Cataudella and G. Iadonisi, “Electrostatic edge modes for a hyperbolic dielectric wedge: analytical solutions,” Solid State Commun. 59, 267–270 (1986).
[CrossRef]

Itoh, T.

R. Mittra, T. Itoh, and T.-S. Li, “Analytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation by the moment method,” IEEE Trans. Microw. Theory Tech. 20, 96–104 (1972).
[CrossRef]

Jones, D. S.

D. S. Jones, The Theory of Electromagnetism (Pergamon, 1964), Sections. 9.1 and 9.2, pp. 562–569.

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]

Lee, S. W.

S. W. Lee and R. Mittra, “Edge condition and ‘intrinsic loss’ in uniaxial plasma,” Can. J. Phys. 46, 111–120 (1968).
[CrossRef]

R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves (MacMillan, 1971), p. 4.

Li, L.

Li, T.-S.

R. Mittra, T. Itoh, and T.-S. Li, “Analytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation by the moment method,” IEEE Trans. Microw. Theory Tech. 20, 96–104 (1972).
[CrossRef]

Louis, E.

A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: analytical solution,” Phys. Rev. B 32, 6045–6047 (1985).
[CrossRef]

Mafi, A.

Maradudin, A. A.

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).

McPhedran, R. C.

W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

Meixner, J.

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

J. Meixner, “Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an vollkommen leitenden ebenen Schirmen,” Ann. Phys. 6, 2–9 (1949).

Mittra, R.

R. Mittra, T. Itoh, and T.-S. Li, “Analytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation by the moment method,” IEEE Trans. Microw. Theory Tech. 20, 96–104 (1972).
[CrossRef]

S. W. Lee and R. Mittra, “Edge condition and ‘intrinsic loss’ in uniaxial plasma,” Can. J. Phys. 46, 111–120 (1968).
[CrossRef]

R. Mittra, “Relative convergence of the solution of a doubly infinite set of equations,” J. Res. Nat. Bur. Stand. D. 67, 245–254 (1963).
[CrossRef]

R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves (MacMillan, 1971), p. 4.

Nazarov, S.

L. Chesnel, X. Claeys, and S. Nazarov, “A curious instability phenomenon for a rounded corner in presence of a negative material,” arXiv:1304.4788 [math.AP].

Oliveri, G.

G. Oliveri and M. Raffetto, “A warning about metamaterials for users of frequency-domain numerical simulators,” IEEE Trans. Antennas Propag. 56, 792–798 (2008).
[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]

Perrins, W. T.

W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

Raffetto, M.

G. Oliveri and M. Raffetto, “A warning about metamaterials for users of frequency-domain numerical simulators,” IEEE Trans. Antennas Propag. 56, 792–798 (2008).
[CrossRef]

Rayleigh, L.

L. Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Phil. Mag. 43(263), 259–272 (1897).
[CrossRef]

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.

C. A. Valagiannopoulos and A. Sihvola, “Improving the electrostatic field concentration in a negative-permittivity wedge with a grounded “bowtie” configuration,” Rad. Sci. 48, 316–325 (2013).

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]

Solodukhov, V. V.

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[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]

Valagiannopoulos, C. A.

C. A. Valagiannopoulos and A. Sihvola, “Improving the electrostatic field concentration in a negative-permittivity wedge with a grounded “bowtie” configuration,” Rad. Sci. 48, 316–325 (2013).

van Bladel, J.

J. 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]

Ann. Phys. (1)

J. Meixner, “Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an vollkommen leitenden ebenen Schirmen,” Ann. Phys. 6, 2–9 (1949).

Can. J. Phys. (2)

S. W. Lee and R. Mittra, “Edge condition and ‘intrinsic loss’ in uniaxial plasma,” Can. J. Phys. 46, 111–120 (1968).
[CrossRef]

R. A. Hurd, “On the possibility of intrinsic loss occurring at the edges of ferrites,” Can. J. Phys. 40, 1067–1076 (1962).
[CrossRef]

IEEE Microw. Guided Wave Lett. (1)

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

IEEE Trans. Antennas Propag. (3)

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

G. Oliveri and M. Raffetto, “A warning about metamaterials for users of frequency-domain numerical simulators,” IEEE Trans. Antennas Propag. 56, 792–798 (2008).
[CrossRef]

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

R. Mittra, T. Itoh, and T.-S. Li, “Analytical and numerical studies of the relative convergence phenomenon arising in the solution of an integral equation by the moment method,” IEEE Trans. Microw. Theory Tech. 20, 96–104 (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 (5)

J. Res. Nat. Bur. Stand. D. (1)

R. Mittra, “Relative convergence of the solution of a doubly infinite set of equations,” J. Res. Nat. Bur. Stand. D. 67, 245–254 (1963).
[CrossRef]

Math. Mod. Methods Appl. Sci. (1)

A.-S. Bonnet-Ben Dhia, L. Chesnel, and X. Claeys, “Radiation condition for a non-smooth interface between a dielectric and a metamaterial,” Math. Mod. Methods Appl. Sci. 23, 1629–1662 (2013).
[CrossRef]

Metamaterials (2)

W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

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]

Phil. Mag. (1)

L. Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Phil. Mag. 43(263), 259–272 (1897).
[CrossRef]

Phys. Rev. B (3)

A. D. Boardman, R. Garcia-Molina, A. Gras-Marti, and E. Louis, “Electrostatic edge modes of a hyperbolic dielectric wedge: analytical solution,” Phys. Rev. B 32, 6045–6047 (1985).
[CrossRef]

L. C. Davis, “Electrostatic edge modes of a dielectric wedge,” Phys. Rev. B 14, 5523–5525 (1976).

L. Dobrzynski and A. A. Maradudin, “Electrostatic edge modes in a dielectric wedge,” Phys. Rev. B 6, 3810–3815 (1972).

Physica (Amsterdam) (1)

C. J. Bouwkamp, “A note on singularities occurring at sharp edges in electromagnetic diffraction theory,” Physica (Amsterdam) 12, 467–474 (1946).

Rad. Sci. (1)

C. A. Valagiannopoulos and A. Sihvola, “Improving the electrostatic field concentration in a negative-permittivity wedge with a grounded “bowtie” configuration,” Rad. Sci. 48, 316–325 (2013).

Solid State Commun. (1)

V. Cataudella and G. Iadonisi, “Electrostatic edge modes for a hyperbolic dielectric wedge: analytical solutions,” Solid State Commun. 59, 267–270 (1986).
[CrossRef]

Other (4)

D. S. Jones, The Theory of Electromagnetism (Pergamon, 1964), Sections. 9.1 and 9.2, pp. 562–569.

L. Chesnel, X. Claeys, and S. Nazarov, “A curious instability phenomenon for a rounded corner in presence of a negative material,” arXiv:1304.4788 [math.AP].

R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves (MacMillan, 1971), p. 4.

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

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Figures (7)

Fig. 1.
Fig. 1.

Coordinate lines of the elliptic coordinate system given by Eq. (11). The figure is drawn for a=1.

Fig. 2.
Fig. 2.

Geometry and notation for a hyperbolic wedge. The thick solid (red) line marks the boundary of the wedge.

Fig. 3.
Fig. 3.

Convergence of diffraction efficiencies for two symmetrical triangular gratings previously considered in [9], when the sharp edges are rounded with inscribed circles of different radii. Figures (a) and (b) here are related to Figs. 8(a) and 8(b) of [9], respectively.

Fig. 4.
Fig. 4.

Diffraction efficiencies for the two sets of gratings considered in Figs. 3(a) and 3(b) as the radius of the inscribed circle tends to zero.

Fig. 5.
Fig. 5.

Effect of introducing absorption loss on convergence of diffraction efficiency for the grating previously considered in Fig. 8(b) of Ref. [9]. The numbers given in the legend are values of ε2/|ε2|.

Fig. 6.
Fig. 6.

Real part of the solution of Eq. (2) with the least nonnegative value near ε2/ε11 and ε2=0. (b) shows 11 cuts of the surface plot in (a) at constant ε2. The curves from top to bottom have ε2/ε1=1.00050.05m, where m=0,1,2,,10. For the top (red) curve m=0. In the figures ε1=1 is assumed.

Fig. 7.
Fig. 7.

Same as Fig. 5 except that ε1=1.5.

Tables (1)

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Table 1. Singularity Exponents for the Nine Curves in Figs. 5 and 7

Equations (21)

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Hz(j)(ρ,φ)=ρτBz(j)(φ),Eρ(j)(ρ,φ)=ρτ1Aρ(j)(φ),Eφ(j)(ρ,φ)=ρτ1Aφ(j)(φ),
tan[τπ(1±φ)2]/tan[τπ(1φ)2]=ε2ε1,
1|φ|1+|φ|<ε2ε1<1+|φ|1|φ|.
Eρ=ρ˜τ1cosτφcosτφ0,Eφ=ρ˜τ1sinτφcosτφ0,Hz=Zτρ˜τsinτφcosτφ0,(φΩφ),
ρ˜τ1=ρ˜α1exp(iβlnρ˜).
V(ε|E|2+μ|H|2)dv<,
ReS0E×H*·ds=0,
Pρ=Re[EφHz*]=2πρ˜|Eφ|2Re[iε*/τ*].
Eρ=12ρ˜(ρ˜iβ+cρ˜iβ)chβφchβφ0,Eφ=i2ρ˜(ρ˜iβcρ˜iβ)shβφchβφ0,Hz=Z2β(ρ˜iβ+cρ˜iβ)shβφchβφ0.(φΩφ).
Eρ=1ρ˜cos(βlnρ˜ζ)chβφchβφ0,Eφ=1ρ˜sin(βlnρ˜ζ)shβφchβφ0,Hz=Zβcos(βlnρ˜ζ)shβφchβφ0,(φΩφ),
x=acoshξcosη,y=asinhξsinη,z=z.(0ξ<,0η2π).
(ξ2+η2)Hz(ξ,η)=0
Eξ=1Zh˜ηHz,Eη=1Zh˜ξHz,
h˜=a˜(sinh2ξ+sin2η)1/2
f(ξ)=Asinhτξ+Bcoshτξ,g(η)=Csinτη+Dcosτη,
Eξ=1h˜(a˜/2)τsinhτξcosτηcosτη0,Eη=1h˜(a˜/2)τcoshτξsinτηcosτη0,Hz=Zτ(a˜/2)τsinhτξsinτηcosτη0.(ηΩη).
ξ,ηφ,ξ^ρ^,η^φ^,a˜coshξa˜eξ/2ρ˜,a˜sinhξa˜eξ/2ρ˜,
Pξ=2πh˜|(a˜/2)τsinτηcosτη0|2Re[iε2*τ*coshτξsinhτ*ξ],(ηΩη),
U=λ2(a˜/2)2αsinh2αξ02α[ε1(πη0)cos2α(η0π)+ε2η0cos2αη0]
U=λ2sin2βξ02β[ε1(πη0)cosh2β(η0π)+ε2η0cosh2βη0]
dτdε2=tan(ωτ)ωε2sec2(ωτ)+ω+ε1sec2(ω+τ),

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