Abstract

We study the electromagnetic beam reflection from layered structures that include the so-called double-negative metamaterials, also called left-handed metamaterials. We predict that such structures can demonstrate a giant lateral Goos-Hänchen shift of the scattered beam accompanied by a splitting of the reflected and transmitted beams due to the resonant excitation of surface waves at the interfaces between the conventional and double-negative materials as well as due to the excitation of leaky modes in the layered structures. The beam shift can be either positive or negative, depending on the type of the guided waves excited by the incoming beam. We also perform finite-difference time-domain simulations and confirm the major effects predicted analytically.

© 2005 Optical Society of America

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References

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  1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Physik 1, 333–346 (1947).
    [Crossref]
  2. T. Tamir, “Leaky waves in planar optical waveguides,” Nouvelle Revue D’Optique 6, 273–284 (1975).
    [Crossref]
  3. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Zeitschrift fur Physik 216, 398–410 (1968).
    [Crossref]
  4. A. Otto, “Spectroscopy of surface polaritons by attenuated total reflection,” in Optical properties of solids-new developments, B. O. Seraphin, ed., (North-Holland, Netherlands, 1975), 677–729.
  5. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [Crossref] [PubMed]
  6. J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today,  57, 37–43 (June 2004).
    [Crossref]
  7. R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
    [Crossref]
  8. I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
    [Crossref]
  9. I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. 83, 2713–2715 (2003).
    [Crossref]
  10. R. W. Ziolkowski, “Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express 11, 662–681 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-662.
    [Crossref] [PubMed]
  11. R. W. Ziolkowski, “Pulsed Gaussian beam interactions with double negative met amaterial slabs: errata,” Opt. Express 11, 1596–1597 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1596.
    [Crossref] [PubMed]
  12. L. M. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1980).
  13. P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E 66, 67603–3 (2002).
    [Crossref]
  14. I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
    [Crossref]
  15. X. Chen and C. F. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617–6 (2004).
    [Crossref]
  16. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
    [Crossref]
  17. I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602–4 (2003).
    [Crossref]
  18. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).

2004 (3)

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
[Crossref]

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today,  57, 37–43 (June 2004).
[Crossref]

X. Chen and C. F. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617–6 (2004).
[Crossref]

2003 (5)

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602–4 (2003).
[Crossref]

I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. 83, 2713–2715 (2003).
[Crossref]

R. W. Ziolkowski, “Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express 11, 662–681 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-662.
[Crossref] [PubMed]

R. W. Ziolkowski, “Pulsed Gaussian beam interactions with double negative met amaterial slabs: errata,” Opt. Express 11, 1596–1597 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1596.
[Crossref] [PubMed]

2002 (1)

P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E 66, 67603–3 (2002).
[Crossref]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

2000 (2)

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[Crossref]

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[Crossref]

1975 (1)

T. Tamir, “Leaky waves in planar optical waveguides,” Nouvelle Revue D’Optique 6, 273–284 (1975).
[Crossref]

1968 (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Zeitschrift fur Physik 216, 398–410 (1968).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Physik 1, 333–346 (1947).
[Crossref]

Berman, P. R.

P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E 66, 67603–3 (2002).
[Crossref]

Boardman, A. D.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
[Crossref]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1980).

Chen, X.

X. Chen and C. F. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617–6 (2004).
[Crossref]

Egan, P.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
[Crossref]

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Physik 1, 333–346 (1947).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Physik 1, 333–346 (1947).
[Crossref]

Kivshar, Yu. S.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
[Crossref]

I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. 83, 2713–2715 (2003).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602–4 (2003).
[Crossref]

Kwok, C. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[Crossref]

Lai, H. M.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[Crossref]

Li, C. F.

X. Chen and C. F. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617–6 (2004).
[Crossref]

Loo, Y. W.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[Crossref]

Otto, A.

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Zeitschrift fur Physik 216, 398–410 (1968).
[Crossref]

A. Otto, “Spectroscopy of surface polaritons by attenuated total reflection,” in Optical properties of solids-new developments, B. O. Seraphin, ed., (North-Holland, Netherlands, 1975), 677–729.

Pendry, J. B.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today,  57, 37–43 (June 2004).
[Crossref]

Ruppin, R.

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[Crossref]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
[Crossref]

I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. 83, 2713–2715 (2003).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602–4 (2003).
[Crossref]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Smith, D. R.

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today,  57, 37–43 (June 2004).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Sukhorukov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602–4 (2003).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[Crossref]

Tamir, T.

T. Tamir, “Leaky waves in planar optical waveguides,” Nouvelle Revue D’Optique 6, 273–284 (1975).
[Crossref]

Xu, B. Y.

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[Crossref]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).

Zharov, A. A.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
[Crossref]

I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. 83, 2713–2715 (2003).
[Crossref]

Ziolkowski, R. W.

Ann. Physik (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler versuch zur totalreflexion,” Ann. Physik 1, 333–346 (1947).
[Crossref]

Appl. Phys. Lett. (2)

I. V. Shadrivov, A. A. Zharov, and Yu. S. Kivshar, “Giant Goos-Hänchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. 83, 2713–2715 (2003).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[Crossref]

Nouvelle Revue D’Optique (1)

T. Tamir, “Leaky waves in planar optical waveguides,” Nouvelle Revue D’Optique 6, 273–284 (1975).
[Crossref]

Opt. Express (2)

Phys. Lett. A (1)

R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277, 61–64 (2000).
[Crossref]

Phys. Rev. E (5)

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 16617–9 (2004).
[Crossref]

X. Chen and C. F. Li, “Lateral shift of the transmitted light beam through a left-handed slab,” Phys. Rev. E 69, 066617–6 (2004).
[Crossref]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7339 (2000).
[Crossref]

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Guided modes in negative-refractive-index waveguides,” Phys. Rev. E 67, 057602–4 (2003).
[Crossref]

P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E 66, 67603–3 (2002).
[Crossref]

Phys. Today (1)

J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today,  57, 37–43 (June 2004).
[Crossref]

Science (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Zeitschrift fur Physik (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Zeitschrift fur Physik 216, 398–410 (1968).
[Crossref]

Other (3)

A. Otto, “Spectroscopy of surface polaritons by attenuated total reflection,” in Optical properties of solids-new developments, B. O. Seraphin, ed., (North-Holland, Netherlands, 1975), 677–729.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).

L. M. Brekhovskikh, Waves in Layered Media (Academic Press, New York, 1980).

Supplementary Material (5)

» Media 1: GIF (665 KB)     
» Media 2: GIF (816 KB)     
» Media 3: GIF (428 KB)     
» Media 4: GIF (1527 KB)     
» Media 5: GIF (365 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Schematic geometry of the excitation of surface waves in a three-layer structure that includes a DNG medium.

Fig. 2.
Fig. 2.

(a,b) Relative beam shift, Δ r , and beam width, Wr , vs. incidence angle (in degrees). (c,d) Relative shift and width of the reflected beam vs. normalized gap width 2πd/λ at a/λ=100/2π. In (c,d) the angle of incidence corresponds to the point of maximum shift in (a).

Fig. 3.
Fig. 3.

(a) Relative beam shift and (b) reflection coefficient vs. the imaginary part of the dielectric permittivity, for a/λ=100/2π and d/λ=3/2π. Insets show the profiles of the reflected beam.

Fig. 4.
Fig. 4.

Distribution of the electric field after the excitation of (a) backward surface wave (665K), and (b) forward surface wave (815K).

Fig. 5.
Fig. 5.

Temporal variation of the amplitudes of the incident (solid) and surface (dashed) waves.

Fig. 6.
Fig. 6.

(a) Geometry of the layered structure. (b) Dependence of the normalized wave number h of the guided modes in the center slab whose thickness is L, for odd (dashed) and even (solid) modes. The vertical dashed line in the lower figure corresponds to the thickness L5λ/2π used in our calculations.

Fig. 7.
Fig. 7.

Dependence of the relative shifts of the (a) reflected and (b) transmitted beams versus the angle of incidence, for L=5λ 0/2π and d=λ 0, and several values of the waist of the incident beam a: 0 (dotted), a=5λ 0 (dashed), and a10λ 0 (solid). The vertical lines indicate the position of the slab eigenmodes. The insert shows an enlargement of the domain marked by the dashed box in the main figure.

Fig. 8.
Fig. 8.

Dependence of the relative shift of the (a) reflected and (b) transmitted beams versus the thickness d of the air gaps between the DNG slab and the high-index slabs when L=5λ 0/2π, a=λ 0, and k x0=1.1862π/λ 0. Dependence of the relative shift of the (c) reflected and (d) transmitted beams versus the waist a of the incident beam when L=5λ 0/2π, 0, and k x0=1.1862π/λ 0.

Fig. 9.
Fig. 9.

Intensity distribution of the electric field for the excitation of (a) backward guided waves (430K) and (b) forward leaky waves guided by the air gaps (1.5M).

Fig. 10.
Fig. 10.

(a) (365K) Contour plot of the x-component of the Poynting vector (blue corresponds to positive values, while yellow corresponds to negative values), (b) Profile of the x-component of the instantaneous Poynting vector as a function of z (normal to the interfaces) at the middle point of the simulation domain.

Equations (4)

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

Δ r = d Φ r d k x ,
E r , t ( x ) 1 2 π { R ( k x ) , T ( k x ) } E ¯ i ( k x ) d k x ,
Δ r , t ( n ) = x n E r , t ( x ) 2 dx a n E r , t ( x ) 2 dx ,
R = ( α 1 + 1 ) ( α 2 + 1 ) ( α 1 1 ) ( α 2 1 ) e 2 i k z 2 d ( α 1 1 ) ( α 2 + 1 ) ( α 1 + 1 ) ( α 2 1 ) e 2 i k z 2 d ,

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