Abstract

The interactions of pulsed and continuous wave (CW) Gaussian beams with double negative (DNG) metatmaterials are considered. Sub-wavelength focusing of a diverging, normally incident pulsed Gaussian beam with a planar DNG slab is demonstrated. The negative angle of refraction behavior associated with the negative index of refraction exhibited by DNG metamaterials is demonstrated. The transmitted beam resulting from both 3-cycle and CW Gaussian beams that are obliquely incident on a DNG slab are shown to have this property. Gaussian beams that undergo total internal reflection from a DNG metamaterial slab are also shown to experience a negative Goos-Hänchen (lateral) shift. Several potential applications for these effects in the microwave and optical regimes are discussed.

© 2003 Optical Society of America

Full Article  |  PDF Article

Errata

Richard Ziolkowski, "Pulsed Gaussian beam interactions with double negative metamaterial slabs: errata," Opt. Express 11, 1596-1597 (2003)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-11-13-1596

References

  • View by:
  • |

  1. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering, (Prentice Hall, Englewood Cliffs, NJ, 1991), pp. 36-38.
  2. V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of ε and μ,�?? Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, �??Composite Medium with zimultaneously negative permeability and permittivity,�?? Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  4. J. B. Pendry, �??Negative Refraction Makes a Perfect Lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  5. D. R. Smith and N. Kroll, �??Negative refractive index in left-handed materials,�?? Phys. Rev. Lett. 85, 2933-2936 (2000).
    [CrossRef] [PubMed]
  6. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two dimensional, isotropic, left-handed metamaterial,�?? Appl. Phys. Lett. 78, 489-491 (2001).
    [CrossRef]
  7. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of a negative refractive index of refraction,�?? Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  8. R. W. Ziolkowski and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  9. C. Caloz, C.-C. Chang and T. Itoh, �??Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,�?? J. Appl. Phys. 90, 5483-5486 (2001).
    [CrossRef]
  10. J. A. Kong, B.-I. Wu and Y. Zhang, �??A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability,�?? Microwave Opt. Tech. Lett. 33, 136-139 (2002).
    [CrossRef]
  11. P. M. Valanju, R. M. Walter, and A. P. Valanju, �??Wave refraction in negative-index media: Always positive and very inhomogeneous,�?? Phys. Rev. Lett. 88, 187401 (2002).
    [CrossRef] [PubMed]
  12. G. V. Eleftheriades, A. K. Iyer and P. C. Kremer, �??Planar negative refractive index media using periodically L-C loaded transmission lines,�?? IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
    [CrossRef]
  13. K. G. Balmain, A. A. Luttgen, and P. C. Kremer, �??Resonance cone formation, reflection, refraction and focusing in a planar, anisotropic metamaterial,�?? Proceedings of the URSI National Radio Science Meeting, pp. 45, San Antonio, TX, July 2002.
  14. R. W. Ziolkowski, �??Design, fabrication, and testing of double negative metamaterials,�?? to appear in IEEE Trans. Antennas Propagat., June 2003.
    [CrossRef]
  15. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Inc., Norwood, MA, 1995).
  16. A. Taflove, Ed., Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Inc., Norwood, MA, 1998).
  17. D. C. Wittwer and R. W. Ziolkowski, �??Two time-derivative Lorentz material (2TDLM) formulation of a Maxwellian absorbing layer matched to a lossy media,�?? IEEE Trans. Antennas and Propagat. 48, 192-199 (2000).
    [CrossRef]
  18. D. C. Wittwer and R. W. Ziolkowski, �??Maxwellian material based absorbing boundary conditions for lossy media in 3D,�?? IEEE Trans. Antennas and Propagat. 48, 200-213 (2000).
    [CrossRef]
  19. J. B. Judkins, C. W. Haggans, and R. W. Ziolkowski, �??2D-FDTD simulation for rewritable optical disk surface structure design,�?? Special Issue of Applied Optics on Optical Data Storage Technologies, Appl. Opt. 35, 2477-2487 (1996).
    [CrossRef] [PubMed]
  20. M. W. Feise, P. J. Bevelacqua, and J. B. Schneider, �??Effects of surface waves on behavior of perfect lenses,�?? Phys. Rev. B 66, 035113 (2002).
    [CrossRef]
  21. A. Ishimaru and J. Thomas, �??Transmission and focusing of a slab of negative refractive index,�?? Proceedings of the URSI National Radio Science Meeting, pp. 43, San Antonio, TX, July 2002.
  22. A. Grbic and G. V. Eleftheriades, �??Experimental verification of backward-wave radiation from a negative refractive index metamaterial,�?? J. Appl. Phys. 92, 5930-5935 (2002).
    [CrossRef]
  23. S.Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed., Toronto: John Wiley & Sons, 1994, pp. 257-258.
  24. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering, (Prentice Hall, Englewood Cliffs, NJ, 1991), pp. 165-169.
  25. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, �??Superprism phenomena in photonic crystals,�?? Phys. Rev. B 58, R10096-10099 (1998).
    [CrossRef]
  26. M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696 (2000).
    [CrossRef]
  27. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??All-angle negative refraction without negative effective index,�??�?? Phys. Rev. B 65, 201104(R) (2002).
    [CrossRef]

Appl. Opt.

Appl. Phys. Lett.

R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, �??Microwave transmission through a two dimensional, isotropic, left-handed metamaterial,�?? Appl. Phys. Lett. 78, 489-491 (2001).
[CrossRef]

IEEE Trans. Antennas and Propagat.

D. C. Wittwer and R. W. Ziolkowski, �??Two time-derivative Lorentz material (2TDLM) formulation of a Maxwellian absorbing layer matched to a lossy media,�?? IEEE Trans. Antennas and Propagat. 48, 192-199 (2000).
[CrossRef]

D. C. Wittwer and R. W. Ziolkowski, �??Maxwellian material based absorbing boundary conditions for lossy media in 3D,�?? IEEE Trans. Antennas and Propagat. 48, 200-213 (2000).
[CrossRef]

IEEE Trans. Antennas Propagat.

R. W. Ziolkowski, �??Design, fabrication, and testing of double negative metamaterials,�?? to appear in IEEE Trans. Antennas Propagat., June 2003.
[CrossRef]

IEEE Trans. Microwave Theory Tech.

G. V. Eleftheriades, A. K. Iyer and P. C. Kremer, �??Planar negative refractive index media using periodically L-C loaded transmission lines,�?? IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

J. Appl. Phys.

C. Caloz, C.-C. Chang and T. Itoh, �??Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations,�?? J. Appl. Phys. 90, 5483-5486 (2001).
[CrossRef]

A. Grbic and G. V. Eleftheriades, �??Experimental verification of backward-wave radiation from a negative refractive index metamaterial,�?? J. Appl. Phys. 92, 5930-5935 (2002).
[CrossRef]

Microwave Opt. Tech. Lett.

J. A. Kong, B.-I. Wu and Y. Zhang, �??A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability,�?? Microwave Opt. Tech. Lett. 33, 136-139 (2002).
[CrossRef]

Phys. Rev. B

M. W. Feise, P. J. Bevelacqua, and J. B. Schneider, �??Effects of surface waves on behavior of perfect lenses,�?? Phys. Rev. B 66, 035113 (2002).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, �??Superprism phenomena in photonic crystals,�?? Phys. Rev. B 58, R10096-10099 (1998).
[CrossRef]

M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696 (2000).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, �??All-angle negative refraction without negative effective index,�??�?? Phys. Rev. B 65, 201104(R) (2002).
[CrossRef]

Phys. Rev. E

R. W. Ziolkowski and E. Heyman, �??Wave propagation in media having negative permittivity and permeability,�?? Phys. Rev. E 64, 056625 (2001).
[CrossRef]

Phys. Rev. Lett.

P. M. Valanju, R. M. Walter, and A. P. Valanju, �??Wave refraction in negative-index media: Always positive and very inhomogeneous,�?? Phys. Rev. Lett. 88, 187401 (2002).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, �??Composite Medium with zimultaneously negative permeability and permittivity,�?? Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

J. B. Pendry, �??Negative Refraction Makes a Perfect Lens,�?? Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

D. R. Smith and N. Kroll, �??Negative refractive index in left-handed materials,�?? Phys. Rev. Lett. 85, 2933-2936 (2000).
[CrossRef] [PubMed]

Science

A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of a negative refractive index of refraction,�?? Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, �??The electrodynamics of substances with simultaneously negative values of ε and μ,�?? Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering, (Prentice Hall, Englewood Cliffs, NJ, 1991), pp. 36-38.

A. Ishimaru and J. Thomas, �??Transmission and focusing of a slab of negative refractive index,�?? Proceedings of the URSI National Radio Science Meeting, pp. 43, San Antonio, TX, July 2002.

K. G. Balmain, A. A. Luttgen, and P. C. Kremer, �??Resonance cone formation, reflection, refraction and focusing in a planar, anisotropic metamaterial,�?? Proceedings of the URSI National Radio Science Meeting, pp. 45, San Antonio, TX, July 2002.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech House, Inc., Norwood, MA, 1995).

A. Taflove, Ed., Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Inc., Norwood, MA, 1998).

S.Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics, 3rd ed., Toronto: John Wiley & Sons, 1994, pp. 257-258.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering, (Prentice Hall, Englewood Cliffs, NJ, 1991), pp. 165-169.

Supplementary Material (9)

» Media 1: GIF (404 KB)     
» Media 2: GIF (333 KB)     
» Media 3: GIF (414 KB)     
» Media 4: GIF (416 KB)     
» Media 5: GIF (261 KB)     
» Media 6: GIF (458 KB)     
» Media 7: GIF (206 KB)     
» Media 8: GIF (626 KB)     
» Media 9: GIF (528 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1.
Fig. 1.

Reflection and transmission configuration

Fig. 2.
Fig. 2.

Frequency response of the real part of the index of refraction associated with the lossy Drude models used in the FDTD simulations.

Fig. 3.
Fig. 3.

(413 KB) Electric field intensity distribution for the normally incident Gaussian beam propagation in free space. The location of the slab region in the DNG simulations is shown.

Fig. 4.
Fig. 4.

(341 KB) Electric field intensity distribution for the normally incident Gaussian beam interaction with a DNG slab having n=-1. Focusing at the back surface is observed.

Fig. 5.
Fig. 5.

Gaussian beam interaction with a DNG slab having n=-1. The intensity of the electric field along the beam axis is shown. The DNG slab front and back face locations are indicated by the green lines. Sharp discontinuities in the derivatives of the field across the DPS-DNG interface and focusing at the back surface are observed.

Fig. 6.
Fig. 6.

Gaussian beam interaction with a DNG slab having n=-1. The intensity of the electric field along the front and back faces, orthogonal to the beam axis, are shown. Focusing and narrowing of the beam at the back surface are observed.

Fig. 7.
Fig. 7.

(424 KB) Electric field intensity distribution for the normally incident Gaussian beam interaction with a DNG slab having n=-6. Channeling of the beam in the DNG slab is observed. The energy in the wings of the beam is seen to be directed towards the axis of the beam.

Fig. 8.
Fig. 8.

Gaussian beam interaction with a DNG slab having n=-6. The intensity of the electric field along the beam axis is shown. The DNG slab front and back face locations are indicated by the green lines. Sharp discontinuities in the derivatives of the field across the DPS-DNG interface and maintenance of the center intensity are observed.

Fig. 9.
Fig. 9.

Gaussian beam interaction with a DNG slab having n=-6. The intensity of the electric field along the front and back faces, orthogonal to the beam axis, are shown. There is only a slight narrowing in the width of the beam after its propagation through the entire DNG slab.

Fig. 10.
Fig. 10.

(426 KB) Electric field intensity distribution for the interaction of a CW Gaussian beam that is incident at 20° to a DNG slab having n=-1. A negative angle of refraction equal and opposite to the angle of incidence is observed.

Fig. 11.
Fig. 11.

(267 KB) Electric field intensity distribution for the interaction of a 3-cycle pulsed Gaussian beam that is incident at 20° to a DNG slab having n=-1. A negative angle of refraction of the transmitted pulsed beam is observed. A backward wave is generated at the front interface.

Fig. 12.
Fig. 12.

(469 KB) Electric field intensity distribution for the interaction of a Gaussian beam that is incident at 20° to a DNG slab having n=-6. A shallow negative angle of refraction is observed. The beam is compressed in the DNG slab because of the higher refractive index.

Fig. 13.
Fig. 13.

(211 KB) Electric field intensity distribution for the interaction of a 3-cycle pulsed Gaussian beam that is incident at 20° to a DNG slab having n=-6. A negative angle of refraction and compression of the transmitted pulsed beam is observed.

Fig. 14.
Fig. 14.

A Gaussian beam obliquely incident from a higher refractive index magnitude medium to a lower one with an angle of incidence beyond the critical angle (blue) will generate a reflected beam that experiences a positive Goos-Hänchen lateral shift (green) in a DPS medium or a negative Goos-Hänchen lateral shift (red) in a DNG medium.

Fig. 15.
Fig. 15.

(641 KB) Electric field intensity distribution for the interaction of a CW Gaussian beam that is incident at 40° in a DPS medium with ε r =9.0 and µ r =+1, i.e., n=+3 onto a DPS slab having ε r =+3.0 and µ r =+1, i.e., n=+√3. The positive Goös-Hachen shift of this beam is observed. Some penetration of the beam into the slab occurs. The total field-scattered field boundary where the beam is generated is apparent. The total-internal-reflected beam propagates out of the FDTD total field region into the FDTD scattered field region as it passes this boundary.

Fig. 16.
Fig. 16.

(541 KB) Electric field intensity distribution for the interaction of a CW Gaussian beam that is incident at 40° in a DPS medium with ε r =9.0 and µ r =+1, i.e., n=+3 onto a DNG slab having ε r =-3.0 and µ r =-1, i.e., n=-√3. The total field-scattered field boundary where the beam is generated is apparent. The total-internal-reflected beam propagates out of the FDTD total field region into the FDTD scattered field region as it passes this boundary.

Fig. 17.
Fig. 17.

The electric field intensity distribution measured at t=6000Δt at two cells in front of the TF-SF plane for the total internal reflection DPS slab case. The positions of the incident beam center and the specularly-reflected beam center are indicated by the vertical black lines. The theoretical positive Goös-Hachen shift is indicated by the vertical green line.

Fig. 18.
Fig. 18.

The electric field intensity distribution measured at t=6000Δt at two cells in front of the TF-SF plane for the total internal reflection DNG slab case. The positions of the incident beam center and the specularly-reflected beam center are indicated by the vertical black lines. The theoretical negative Goäs-Hachen shift is indicated by the vertical green line.

Equations (28)

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

θ refl = θ inc
n trans sin θ trans = n inc sin θ inc
n i = ε i ε 0 μ i μ 0 = ε r μ r
R = η trans cos θ inc η inc cos θ trans η trans cos θ inc + η inc cos θ trans
T = 2 η trans cos θ inc η trans cos θ inc + η inc cos θ trans
η i = μ i ε i
ε ( ω ) = ε 0 ( 1 ω pe 2 ω ( ω + i Γ e ) )
μ ( ω ) = μ 0 ( 1 ω pm 2 ω ( ω + i Γ m ) )
t 2 P y + Γ e t P y = ε 0 ω pe 2 E y
t 2 M nx + Γ m t M nx = μ 0 ω pm 2 H x
t 2 M nz + Γ m t M nz = μ 0 ω pm 2 H z
K x = t M nx
K z = t M nz
J y = t P y
t H x = + 1 μ 0 ( z E y K x )
t K x + Γ m K x = μ 0 ω pm 2 H x
t H z = 1 μ 0 ( x E y + K z )
t K z + Γ m K z = μ 0 ω pm 2 H z
t E y = 1 ε 0 [ ( z H x x H z ) J y ]
t J y + Γ e J y = ε 0 ω pe 2 E y
t H x = + 1 μ z E y
t H z = 1 μ x E y
t E y = 1 ε ( z H x x H z )
f ( t ) = { 0 for t < 0 g on ( t ) sin ( ω t ) for 0 < t < m T p sin ( ω t ) for m T p < t < ( m + n ) T p g off ( t ) sin ( ω t ) for ( m + n ) T p < t < ( m + n + m ) T p 0 for ( m + n + m ) T p < t
g on ( t ) = 10 x on 3 ( t ) 15 x on 4 ( t ) + 6 x on 5 ( t )
g off ( t ) = 1 [ 10 x off 3 ( t ) 15 x off 4 ( t ) + 6 x off 5 ( t ) ]
E y x z = R ( K x 0 ) e i k x 0 Φ ' ( k x 0 ) E y , inc ( x Φ ' ( k x 0 ) , z )
R ( k x ) = exp [ i Φ ( k x ) ]

Metrics