Abstract

We theoretically investigate the homogenization of the dielectric response to transverse electric waves of a transverse grating characterized by the Kapitza condition; i.e., the permittivity is rapidly modulated with a modulation depth scaling as the large wavelength-to-modulation-period ratio. We show that the resulting effective dielectric permittivity, in addition to the standard average of the underlying dielectric profile, has a further contribution arising from the fast and deep dielectric modulation. Such a contribution turns out to be comparable with the other one and hence can provide an additional method for designing dielectric metamaterials. As an example, we discuss an effective metal-to-dielectric transition produced by the Kapitza contribution obtained by changing the grating depth, a remarkable result for applications involving epsilon-near-zero metamaterial design.

© 2013 Optical Society of America

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  6. Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, Adv. Optoelectron. 2012, 1 (2012).
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  8. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. B 84, 045424 (2011).
    [CrossRef]
  9. A. Ciattoni, C. Rizza, and E. Palange, Opt. Express 18, 11911 (2010).
    [CrossRef]
  10. A. Ciattoni, C. Rizza, and E. Palange, Opt. Lett. 35, 2130 (2010).
    [CrossRef]
  11. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Opt. Lett. 36, 2530 (2011).
    [CrossRef]
  12. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Phys. Rev. A 85, 053842 (2012).
    [CrossRef]
  13. C. Guclu, S. Campione, and F. Capolino, Phys. Rev. B 86, 205130 (2012).
    [CrossRef]
  14. C. Rizza and A. Ciattoni, Phys. Rev. Lett. 110, 143901 (2013).
    [CrossRef]
  15. L. D. Landau and E. Lifshitz, Course in Theoretical Physics. Mechanics, 3rd ed. (Pergamon, 1976).
  16. P. L. Kapitza, Sov. Phys. JETP 21, 588 (1951).
  17. A. Hasegawa and Y. Kodama, Phys. Rev. Lett. 66, 161 (1991).
    [CrossRef]
  18. A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687 (2008).
  19. A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, Opt. Lett. 33, 2110 (2008).
    [CrossRef]
  20. H. Ammari, H. Kang, and H. Lee, Layer Potential Techniques in Spectral Analysis, vol. 153 of Mathematical Surveys and Monographs (American Mathematical Society, 2009).
  21. COMSOL, www.comsol.com .
  22. M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006).
    [CrossRef]

2013

C. Rizza and A. Ciattoni, Phys. Rev. Lett. 110, 143901 (2013).
[CrossRef]

2012

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Phys. Rev. A 85, 053842 (2012).
[CrossRef]

C. Guclu, S. Campione, and F. Capolino, Phys. Rev. B 86, 205130 (2012).
[CrossRef]

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, Adv. Optoelectron. 2012, 1 (2012).
[CrossRef]

2011

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. B 84, 045424 (2011).
[CrossRef]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Opt. Lett. 36, 2530 (2011).
[CrossRef]

2010

2008

2007

2006

P. A. Belov and Y. Hao, Phys. Rev. B 73, 113110 (2006).
[CrossRef]

M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef]

2003

J. B. Pendry and S. A. Ramakrishna, Physica B 338, 329 (2003).
[CrossRef]

1997

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[CrossRef]

1991

A. Hasegawa and Y. Kodama, Phys. Rev. Lett. 66, 161 (1991).
[CrossRef]

1951

P. L. Kapitza, Sov. Phys. JETP 21, 588 (1951).

Akozbek, N.

Ammari, H.

H. Ammari, H. Kang, and H. Lee, Layer Potential Techniques in Spectral Analysis, vol. 153 of Mathematical Surveys and Monographs (American Mathematical Society, 2009).

Avrutsky, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, Appl. Phys. Lett. 90, 191109 (2007).
[CrossRef]

Belov, P. A.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. B 84, 045424 (2011).
[CrossRef]

P. A. Belov and Y. Hao, Phys. Rev. B 73, 113110 (2006).
[CrossRef]

Bloemer, M. J.

Cai, W.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

Campione, S.

C. Guclu, S. Campione, and F. Capolino, Phys. Rev. B 86, 205130 (2012).
[CrossRef]

Capolino, F.

C. Guclu, S. Campione, and F. Capolino, Phys. Rev. B 86, 205130 (2012).
[CrossRef]

Cappeddu, M. G.

Centini, M.

Ciattoni, A.

C. Rizza and A. Ciattoni, Phys. Rev. Lett. 110, 143901 (2013).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, Opt. Lett. 35, 2130 (2010).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, Opt. Express 18, 11911 (2010).
[CrossRef]

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, Opt. Lett. 33, 2110 (2008).
[CrossRef]

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687 (2008).

Cortes, C. L.

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, Adv. Optoelectron. 2012, 1 (2012).
[CrossRef]

D’Aguanno, G.

de Ceglia, D.

DelRe, E.

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, Opt. Lett. 33, 2110 (2008).
[CrossRef]

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687 (2008).

Elser, J.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, Appl. Phys. Lett. 90, 191109 (2007).
[CrossRef]

Engheta, N.

M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[CrossRef]

Fowler, M.

Guclu, C.

C. Guclu, S. Campione, and F. Capolino, Phys. Rev. B 86, 205130 (2012).
[CrossRef]

Guo, Y.

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, Adv. Optoelectron. 2012, 1 (2012).
[CrossRef]

Hao, Y.

P. A. Belov and Y. Hao, Phys. Rev. B 73, 113110 (2006).
[CrossRef]

Hasegawa, A.

A. Hasegawa and Y. Kodama, Phys. Rev. Lett. 66, 161 (1991).
[CrossRef]

Haus, J. W.

Jacob, Z.

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, Adv. Optoelectron. 2012, 1 (2012).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[CrossRef]

Kang, H.

H. Ammari, H. Kang, and H. Lee, Layer Potential Techniques in Spectral Analysis, vol. 153 of Mathematical Surveys and Monographs (American Mathematical Society, 2009).

Kapitza, P. L.

P. L. Kapitza, Sov. Phys. JETP 21, 588 (1951).

Kidwai, O.

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Phys. Rev. A 85, 053842 (2012).
[CrossRef]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Opt. Lett. 36, 2530 (2011).
[CrossRef]

Kivshar, Y. S.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Kodama, Y.

A. Hasegawa and Y. Kodama, Phys. Rev. Lett. 66, 161 (1991).
[CrossRef]

Landau, L. D.

L. D. Landau and E. Lifshitz, Course in Theoretical Physics. Mechanics, 3rd ed. (Pergamon, 1976).

Lee, H.

H. Ammari, H. Kang, and H. Lee, Layer Potential Techniques in Spectral Analysis, vol. 153 of Mathematical Surveys and Monographs (American Mathematical Society, 2009).

Lifshitz, E.

L. D. Landau and E. Lifshitz, Course in Theoretical Physics. Mechanics, 3rd ed. (Pergamon, 1976).

Mandatori, A.

Marini, A.

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, Opt. Lett. 33, 2110 (2008).
[CrossRef]

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687 (2008).

Mattiucci, N.

Newman, W.

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, Adv. Optoelectron. 2012, 1 (2012).
[CrossRef]

Orlov, A. A.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Palange, E.

Pendry, J. B.

J. B. Pendry and S. A. Ramakrishna, Physica B 338, 329 (2003).
[CrossRef]

Podolskiy, V. A.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, Appl. Phys. Lett. 90, 191109 (2007).
[CrossRef]

Ramakrishna, S. A.

J. B. Pendry and S. A. Ramakrishna, Physica B 338, 329 (2003).
[CrossRef]

Rizza, C.

C. Rizza and A. Ciattoni, Phys. Rev. Lett. 110, 143901 (2013).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, Opt. Lett. 35, 2130 (2010).
[CrossRef]

A. Ciattoni, C. Rizza, and E. Palange, Opt. Express 18, 11911 (2010).
[CrossRef]

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, Opt. Lett. 33, 2110 (2008).
[CrossRef]

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687 (2008).

Salakhutdinov, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, Appl. Phys. Lett. 90, 191109 (2007).
[CrossRef]

Scalora, M.

Shalaev, V.

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

Sibilia, C.

Silveirinha, M.

M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef]

Sipe, J. E.

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Phys. Rev. A 85, 053842 (2012).
[CrossRef]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Opt. Lett. 36, 2530 (2011).
[CrossRef]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[CrossRef]

Voroshilov, P. M.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Zhukovsky, S. V.

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Phys. Rev. A 85, 053842 (2012).
[CrossRef]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Opt. Lett. 36, 2530 (2011).
[CrossRef]

Adv. Optoelectron.

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, Adv. Optoelectron. 2012, 1 (2012).
[CrossRef]

Appl. Phys. Lett.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, Appl. Phys. Lett. 90, 191109 (2007).
[CrossRef]

Nature

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, Phys. Rev. A 85, 053842 (2012).
[CrossRef]

Phys. Rev. B

C. Guclu, S. Campione, and F. Capolino, Phys. Rev. B 86, 205130 (2012).
[CrossRef]

P. A. Belov and Y. Hao, Phys. Rev. B 73, 113110 (2006).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Phys. Rev. Lett.

A. Hasegawa and Y. Kodama, Phys. Rev. Lett. 66, 161 (1991).
[CrossRef]

C. Rizza and A. Ciattoni, Phys. Rev. Lett. 110, 143901 (2013).
[CrossRef]

M. Silveirinha and N. Engheta, Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef]

Physica B

J. B. Pendry and S. A. Ramakrishna, Physica B 338, 329 (2003).
[CrossRef]

Sov. Phys. JETP

P. L. Kapitza, Sov. Phys. JETP 21, 588 (1951).

Other

A. Ciattoni, E. DelRe, C. Rizza, and A. Marini, “Tailoring wave nonlinearity through spatial composites,” arXiv:0804.3687 (2008).

H. Ammari, H. Kang, and H. Lee, Layer Potential Techniques in Spectral Analysis, vol. 153 of Mathematical Surveys and Monographs (American Mathematical Society, 2009).

COMSOL, www.comsol.com .

W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

L. D. Landau and E. Lifshitz, Course in Theoretical Physics. Mechanics, 3rd ed. (Pergamon, 1976).

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Figures (3)

Fig. 1.
Fig. 1.

Sketch of a TE plane wave impinging onto a Kapitza slab of thickness L with incidence angle θ.

Fig. 2.
Fig. 2.

Comparison between numerical slab transmissivity T and reflectivity R and corresponding Kapitza quantities Teff and Reff. (a) Normally impinging waves for different grating depths. (b) Inclined waves for Re(δϵ)=10. (c) Real part of the effective dielectric permittivity as a function of the grating depth.

Fig. 3.
Fig. 3.

Comparison between (a) numerical transmissivity and (b) reflectivity (solid lines) and those predicted by the Kapitza approach (stars) as functions of θ and 1/η. The semitransparent surface interpolating the results of the Kapitza approach is reported for clarity.

Equations (6)

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

2E+k02ϵ(x)E=0,
ϵ(x)=ϵm+n0(an+bnη)exp(k0ηx),
E(x,X,z)=E¯(x,z)+ηn0E˜n(x,z)exp(ink0X),
2E¯+k02ϵmE¯+k02n0(bn+ηan)En˜=0,(bnE¯n2E˜n)+O(η)=0,
ϵeff=ϵm+n0bnbnn2.
ϵ=ϵm+δϵcos[(2π/Λ)x].

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