Abstract

In this paper, we study two surface wave control scenarios at microwave frequencies. The first is a surface wave traveling along an uneven interface with a triangular obstruction present on a grounded dielectric slab. The other is a surface wave that circumvents a metallic rhombus-shaped obstacle, which is partially buried in a flat grounded dielectric slab. With a consideration of the eigenmode properties of the surface wave, our proposed technique – based on transformation optics – offers an efficient and accurate way to perform the filed manipulation. On the one hand, we see that the surface wave is guided along the uneven interface with no scattering into the air, as the grounded dielectric slab is flat. On the other hand, we observe that the surface wave is capable of traversing the rhombus obstacle with no shadow left behind, as the obstacle is cloaked. This technique for surface wave control is also valid at higher frequency ranges, and can easily be extended to encompass other propagating modes.

© 2012 OSA

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References

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  1. R. E. Collin, Field Theory of Guided Waves (Wiley-IEEE Press, 1990).
  2. D. M. Pozar, Microwave Engineering (Wiley, 2005).
  3. G. Goubau, “Surface waves and their application to transmission lines,” J. Appl. Phys. 21(11), 1119–1128 (1950).
    [CrossRef]
  4. S. S. Attwood, “Surface wave propagation over a coated plane conductor,” J. Appl. Phys. 22(4), 504–509 (1951).
    [CrossRef]
  5. H. Barlow and A. Cullen, “Surface waves,” Proc. IEE. 100, 329–347 (1953).
  6. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  7. U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
    [CrossRef] [PubMed]
  8. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
    [CrossRef] [PubMed]
  9. J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
    [CrossRef] [PubMed]
  10. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
    [CrossRef] [PubMed]
  11. E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79(6), 063825 (2009).
    [CrossRef]
  12. D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
    [CrossRef]
  13. H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 21 (2010).
    [CrossRef] [PubMed]
  14. P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
    [CrossRef] [PubMed]
  15. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
    [CrossRef] [PubMed]
  16. M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18(11), 12027–12032 (2010).
    [CrossRef] [PubMed]
  17. J. Renger, M. Kadic, G. Dupont, S. S. Aćimović, S. Guenneau, R. Quidant, and S. Enoch, “Hidden progress: broadband plasmonic invisibility,” Opt. Express 18(15), 15757–15768 (2010).
    [CrossRef] [PubMed]
  18. P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mappings,” New J. Phys. 13(3), 033011 (2011).
    [CrossRef]
  19. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
    [CrossRef] [PubMed]
  20. J. Zhang, S. Xiao, M. Wubs, and N. A. Mortensen, “Surface plasmon wave adapter designed with transformation optics,” ACS Nano 5(6), 4359–4364 (2011).
    [CrossRef] [PubMed]
  21. K. Muamer, D. Guillaume, T. M. Chang, S. Guenneau, and S. Enoch, “Curved trajectories on transformed metal surfaces: Luneburg lens, beam-splitter, invisibility carpet and black hole for surface plasmon polaritons,” http://arxiv.org/abs/1102.0900 .

2011

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mappings,” New J. Phys. 13(3), 033011 (2011).
[CrossRef]

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[CrossRef] [PubMed]

J. Zhang, S. Xiao, M. Wubs, and N. A. Mortensen, “Surface plasmon wave adapter designed with transformation optics,” ACS Nano 5(6), 4359–4364 (2011).
[CrossRef] [PubMed]

2010

M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18(11), 12027–12032 (2010).
[CrossRef] [PubMed]

J. Renger, M. Kadic, G. Dupont, S. S. Aćimović, S. Guenneau, R. Quidant, and S. Enoch, “Hidden progress: broadband plasmonic invisibility,” Opt. Express 18(15), 15757–15768 (2010).
[CrossRef] [PubMed]

D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
[CrossRef]

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 21 (2010).
[CrossRef] [PubMed]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

2009

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79(6), 063825 (2009).
[CrossRef]

2008

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

2006

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

1951

S. S. Attwood, “Surface wave propagation over a coated plane conductor,” J. Appl. Phys. 22(4), 504–509 (1951).
[CrossRef]

1950

G. Goubau, “Surface waves and their application to transmission lines,” J. Appl. Phys. 21(11), 1119–1128 (1950).
[CrossRef]

Acimovic, S. S.

Argyropoulos, C.

D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
[CrossRef]

E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79(6), 063825 (2009).
[CrossRef]

Attwood, S. S.

S. S. Attwood, “Surface wave propagation over a coated plane conductor,” J. Appl. Phys. 22(4), 504–509 (1951).
[CrossRef]

Bao, D.

D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
[CrossRef]

Bartal, G.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Chin, J. Y.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Cui, T. J.

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 21 (2010).
[CrossRef] [PubMed]

D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
[CrossRef]

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Dupont, G.

Enoch, S.

García-Vidal, F. J.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mappings,” New J. Phys. 13(3), 033011 (2011).
[CrossRef]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[CrossRef] [PubMed]

Goubau, G.

G. Goubau, “Surface waves and their application to transmission lines,” J. Appl. Phys. 21(11), 1119–1128 (1950).
[CrossRef]

Guenneau, S.

Hao, Y.

D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
[CrossRef]

E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79(6), 063825 (2009).
[CrossRef]

Huidobro, P. A.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mappings,” New J. Phys. 13(3), 033011 (2011).
[CrossRef]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[CrossRef] [PubMed]

Ji, C.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Kadic, M.

Kallos, E.

D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
[CrossRef]

E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79(6), 063825 (2009).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Li, J.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Liu, R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Ma, H. F.

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 21 (2010).
[CrossRef] [PubMed]

Martín-Moreno, L.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mappings,” New J. Phys. 13(3), 033011 (2011).
[CrossRef]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[CrossRef] [PubMed]

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[CrossRef] [PubMed]

Mock, J. J.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

Mortensen, N. A.

J. Zhang, S. Xiao, M. Wubs, and N. A. Mortensen, “Surface plasmon wave adapter designed with transformation optics,” ACS Nano 5(6), 4359–4364 (2011).
[CrossRef] [PubMed]

Nesterov, M. L.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mappings,” New J. Phys. 13(3), 033011 (2011).
[CrossRef]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[CrossRef] [PubMed]

Pendry, J. B.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

Quidant, R.

Renger, J.

Schurig, D.

Smith, D. R.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

Tang, W. X.

D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
[CrossRef]

Valentine, J.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[CrossRef] [PubMed]

Wubs, M.

J. Zhang, S. Xiao, M. Wubs, and N. A. Mortensen, “Surface plasmon wave adapter designed with transformation optics,” ACS Nano 5(6), 4359–4364 (2011).
[CrossRef] [PubMed]

Xiao, S.

J. Zhang, S. Xiao, M. Wubs, and N. A. Mortensen, “Surface plasmon wave adapter designed with transformation optics,” ACS Nano 5(6), 4359–4364 (2011).
[CrossRef] [PubMed]

Zentgraf, T.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Zhang, J.

J. Zhang, S. Xiao, M. Wubs, and N. A. Mortensen, “Surface plasmon wave adapter designed with transformation optics,” ACS Nano 5(6), 4359–4364 (2011).
[CrossRef] [PubMed]

Zhang, X.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

ACS Nano

J. Zhang, S. Xiao, M. Wubs, and N. A. Mortensen, “Surface plasmon wave adapter designed with transformation optics,” ACS Nano 5(6), 4359–4364 (2011).
[CrossRef] [PubMed]

Front. Phys. China

D. Bao, E. Kallos, W. X. Tang, C. Argyropoulos, Y. Hao, and T. J. Cui, “A broadband simplified free space cloak realized by nonmagnetic dielectric cylinders,” Front. Phys. China 5(3), 319–323 (2010).
[CrossRef]

J. Appl. Phys.

G. Goubau, “Surface waves and their application to transmission lines,” J. Appl. Phys. 21(11), 1119–1128 (1950).
[CrossRef]

S. S. Attwood, “Surface wave propagation over a coated plane conductor,” J. Appl. Phys. 22(4), 504–509 (1951).
[CrossRef]

Nano Lett.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Transformation optics for plasmonics,” Nano Lett. 10(6), 1985–1990 (2010).
[CrossRef] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[CrossRef] [PubMed]

Nat Commun

H. F. Ma and T. J. Cui, “Three-dimensional broadband ground-plane cloak made of metamaterials,” Nat Commun 1(3), 21 (2010).
[CrossRef] [PubMed]

Nat. Nanotechnol.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6(3), 151–155 (2011).
[CrossRef] [PubMed]

New J. Phys.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mappings,” New J. Phys. 13(3), 033011 (2011).
[CrossRef]

Opt. Express

Phys. Rev. A

E. Kallos, C. Argyropoulos, and Y. Hao, “Ground-plane quasicloaking for free space,” Phys. Rev. A 79(6), 063825 (2009).
[CrossRef]

Phys. Rev. Lett.

J. Li and J. B. Pendry, “Hiding under the carpet: a new strategy for cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[CrossRef] [PubMed]

Science

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323(5912), 366–369 (2009).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Other

H. Barlow and A. Cullen, “Surface waves,” Proc. IEE. 100, 329–347 (1953).

R. E. Collin, Field Theory of Guided Waves (Wiley-IEEE Press, 1990).

D. M. Pozar, Microwave Engineering (Wiley, 2005).

K. Muamer, D. Guillaume, T. M. Chang, S. Guenneau, and S. Enoch, “Curved trajectories on transformed metal surfaces: Luneburg lens, beam-splitter, invisibility carpet and black hole for surface plasmon polaritons,” http://arxiv.org/abs/1102.0900 .

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Figures (6)

Fig. 1
Fig. 1

Surface wave propagation on an ideal lossless isotropic homogeneous grounded slab. (a) Configuration of the surface wave guiding structure. The surface wave is propagating along the air-dielectric interface which is depicted by the orange line. In this paper, we consider μ r1 =1 and μ r2 =1 for all cases studied. (b) Demonstration of the TM0 mode propagating along the air-dielectric interface of the surface wave guiding structure, at 5 GHz with d=5mm , ε r1 =3 , and ε r2 =1 . (c) Graphical representation of the relationship between the effective refractive index n eff of TM0 mode and substrate material property ε r1 at 5 GHz with d=5mm and ε r2 =1 . (d) Graphical representation of the relationship between the effective refractive index n eff of the TM0 mode and the relative permittivity of the superstrate ε r2 at 5 GHz with d=5mm and ε r1 =3 . (e) Graphical representation of the relationship between the effective refractive index n eff of the TM0 mode and the substrate thickness d at 5 GHz with ε r1 =3 and ε r2 =1 .

Fig. 2
Fig. 2

The schematic showing how potentially ε r1 , ε r2 and d can be used to steer the surface wave ray trace. The changing of these three parameters actually leads to a change in the corresponding effective refractive index. The surface wave is assumed to propagate along the interface of ABC, which is depicted in orange. (a) Use ε r1 to change ray trace. (b) Use ε r2 to change ray trace. (c) Used d to change ray trace. (d) Graphical representation of refraction relationship between the surface waves at the boundary B from region AB to region BC with the effective refractive index n effi and n effj respectively.

Fig. 3
Fig. 3

Surface wave propagating along an uneven interface ABCDEFGHIJ with a triangular bump on a grounded dielectric slab. The dielectric bump has a dimension of 16mm×144mm in the z ^ and y ^ directions. The substrate is assumed to be infinite in the x ^ direction. In this case, the grounded dielectric slab has a thickness ranging from 5 mm to 21 mm, and material with ε r1 =3 . (a) Configuration of the considered structure. (b) Surface wave traveling on the uneven air-dielectric interface showing significant energy scattering into the air. (c) Configuration of the considered structure with a carpet cloak device. The cloaking device is the same one as used in Ref. [12], with dimensions of 60mm×137mm in the z ^ and y ^ directions, and make up of 8 dielectric blocks. The unit-cell size is thus 30mm×37.25mm . The materials employed in the cloak have ε ri ' =[ 1.17,1.30,1.02,1.46 ] . (d) Surface wave traveling on the uneven interface with carpet cloak device covering the dielectric bump. However, significant energy is still scattered into the air.

Fig. 4
Fig. 4

Surface wave propagating on the uneven interface ABCDEFGHIJ with a self-cloaking scheme by altering the dielectric property of the bump. (a) Configuration of the considered structure with the proposed surface wave cloaking design, where the bump has been divided into four homogeneous blocks. The average thickness of the blocks are d i =[ 9 mm,17 mm ] . (b) Magnified picture of the surface wave self-cloaking dielectric bump with two materials of ε ri_sub sw =[ 1.74,1.38 ] according to the corresponding average thickness of the divided blocks. (c) Normalized E-field distribution of the surface wave TM0 mode propagating at the interface ABCDEFGHIJ with the self-cloaking dielectric bump. This shows that no scattering is present in the air, and the surface waves propagate well along the uneven interface.

Fig. 5
Fig. 5

Surface wave propagating on the air-dielectric interface ABCD with a metallic rhombus obstacle partially buried in a flat grounded dielectric slab. The conductor has a size of l 1 =32mm , l 2 =144mm . The substrate has the relative permittivity ε r1 =3 with a thickness d=5mm . (a) 3D view of the considered structure. (b) Lateral view of the considered structure. (c) Normalized E-field distribution of the surface wave TM0 mode propagation on the air-dielectric interface ABCD when propagating and impinging upon the rhombus conductor. Clearly, when the wave impinges upon the metallic rhombus obstacle, the surface wave split into two parts and forms a shadow.

Fig. 6
Fig. 6

Two schemes of surface wave cloaking device design with n i sw =[ 1.16,1.22,1.08,1.30 ] for the corresponding blocks in the transformed device. The cloaking device has the dimensions of 120mm×137mm in x ^ and y ^ directions and is composed of 16 dielectric blocks. The intact unit size is thus 30mm×37.25mm . (a) Cloaking device embedded in the substrate with the material property of ε ri_sub sw =[ 4.93,5.95,3.27,6.86 ] . (b) Normalized E-field distribution of the surface wave TM0 mode propagating on the interface ABCD with the cloaking device embedded in the substrate, showing surface wave successfully circumventing the rhombus conductor, and with no shadow left. (c) Cloaking device embedded in the superstrate layer with material properties of ε ri_top sw =[ 1.18,1.33,1.02,1.53 ] . (d) Normalized E-field distribution of the surface wave TM0 mode propagation on the interface ABCD with the cloaking device in the air superstrate, showing surface wave successfully circumventing the rhombus conductor.

Equations (9)

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TE: μ r2 k z1 dcot( k z1 d )+i μ r1 k z2 d=0
TM: ε r2 k z1 dtan( k z1 d )i ε r1 k z2 d=0
( k z1 d ) 2 ( k z2 d ) 2 =( ε r1 μ r1 ε r2 μ r2 ) ( k 0 d ) 2
n eff = k ρ / k 0
k ρ 2 + k z1 2 = ε r1 μ r1 k 0 2
k ρ 2 + k z2 2 = ε r2 μ r2 k 0 2
ε ¯ ¯ '= J ε ¯ ¯ J T / det( J )
μ ¯ ¯ '= J μ ¯ ¯ J T / det( J )
Z TM = η 0 k ρ / k 0 = η 0 n eff [1.07 η 0 ,1.61 η 0 ]

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