Abstract

We apply a (rigorous) Green’s function theory to study the Doppler effects of a light source placed on top of a metamaterial slab. When the receiver is in motion with the source and the slab, we find that, in addition to a conventional Doppler mode, there are several other frequency components that do not obey the standard frequency-shift rule. We show that such new effects are caused by the coupling between the radiated electromagnetic waves and the surface modes of the metamaterial slab, whose dispersion relation varies as a function of velocity in the moving reference frame.

© 2008 Optical Society of America

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References

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  1. J. A. Kong, Electromagnetic Wave Theory (Higher Education, 2002).
  2. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
    [CrossRef]
  3. M. Einat and E. Jerby, Phys. Rev. E 56, 5996 (1997).
    [CrossRef]
  4. E. J. Reed, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
    [CrossRef] [PubMed]
  5. N. Seddon and T. Bearpark, Science 302, 1537 (2003).
    [CrossRef] [PubMed]
  6. A. B. Kozyrev and D. W. von der Weide, Phys. Rev. Lett. 94, 203902 (2005).
    [CrossRef] [PubMed]
  7. X. H. Hu, Z. H. Hang, J. Li, J. Zi, and C. T. Chan, Phys. Rev. E 73, 015602 (2006).
    [CrossRef]
  8. C. Y. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, Phys. Rev. Lett. 96, 043903 (2006).
    [CrossRef] [PubMed]
  9. Qualitative conclusions reported here are not affected by the specific forms of ϵr(ω), μr(ω), and the parameters are chosen only for easy illustration. Given the velocity adopted in this paper, we find that calculations based on a Galilean transformation do not lead to a significantly different result.
  10. L. Zhou, X. Q. Huang, and C. T. Chan, Photonics Nanostruct. Fundam. Appl. 3, 100 (2005).
    [CrossRef]
  11. Y. Zhang, T. M. Grzegorzyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
    [CrossRef]
  12. R̃TE(ω̃,k̃x) is obtained by applying a Lorentz transformation to RTE(ω,kx), the reflection coefficient calculated in the static-slab frame following .
  13. In the static-slab frame, D⃗(r⃗,t) is determined by E⃗(r⃗,t′), as the response is spatially local. However, after the Lorentz transformation in a moving frame, the point {r⃗,t} is spatially different from the point {r⃗,t′} as long as t≠t′, so that the response appears spatially nonlocal.
  14. We get ω̃r=ω0(1+βs)/(1−βs) when vr=0 and ω̃r=ω0(1−βr)/(1+βr) when vs=0. The frequency shift is zero (ω̃r=ω0) when βs=βr.
  15. R. Ruppin, J. Phys. Condens. Matter 13, 1811 (2001).
    [CrossRef]
  16. X. Q. Huang, L. Zhou, and C. T. Chan, Phys. Rev. B 74, 045123 (2006).
    [CrossRef]
  17. We note that such states had been observed experimentally, see R. B. Pettit, J. Silcox, and R. Vincent, Phys. Rev. B 11, 3116 (1975).
    [CrossRef]
  18. The spectrum is reduced to ω̃=ω0 when vr=0.
  19. Another cross point is automatically excluded in the calculations by the causality requirement.

2006 (3)

X. H. Hu, Z. H. Hang, J. Li, J. Zi, and C. T. Chan, Phys. Rev. E 73, 015602 (2006).
[CrossRef]

C. Y. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, Phys. Rev. Lett. 96, 043903 (2006).
[CrossRef] [PubMed]

X. Q. Huang, L. Zhou, and C. T. Chan, Phys. Rev. B 74, 045123 (2006).
[CrossRef]

2005 (2)

A. B. Kozyrev and D. W. von der Weide, Phys. Rev. Lett. 94, 203902 (2005).
[CrossRef] [PubMed]

L. Zhou, X. Q. Huang, and C. T. Chan, Photonics Nanostruct. Fundam. Appl. 3, 100 (2005).
[CrossRef]

2003 (2)

E. J. Reed, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
[CrossRef] [PubMed]

N. Seddon and T. Bearpark, Science 302, 1537 (2003).
[CrossRef] [PubMed]

2002 (1)

Y. Zhang, T. M. Grzegorzyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
[CrossRef]

2001 (1)

R. Ruppin, J. Phys. Condens. Matter 13, 1811 (2001).
[CrossRef]

1997 (1)

M. Einat and E. Jerby, Phys. Rev. E 56, 5996 (1997).
[CrossRef]

1975 (1)

We note that such states had been observed experimentally, see R. B. Pettit, J. Silcox, and R. Vincent, Phys. Rev. B 11, 3116 (1975).
[CrossRef]

1968 (1)

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Electromagn. Waves (1)

Y. Zhang, T. M. Grzegorzyk, and J. A. Kong, Electromagn. Waves 35, 271 (2002).
[CrossRef]

J. Phys. Condens. Matter (1)

R. Ruppin, J. Phys. Condens. Matter 13, 1811 (2001).
[CrossRef]

Photonics Nanostruct. Fundam. Appl. (1)

L. Zhou, X. Q. Huang, and C. T. Chan, Photonics Nanostruct. Fundam. Appl. 3, 100 (2005).
[CrossRef]

Phys. Rev. B (2)

X. Q. Huang, L. Zhou, and C. T. Chan, Phys. Rev. B 74, 045123 (2006).
[CrossRef]

We note that such states had been observed experimentally, see R. B. Pettit, J. Silcox, and R. Vincent, Phys. Rev. B 11, 3116 (1975).
[CrossRef]

Phys. Rev. E (2)

X. H. Hu, Z. H. Hang, J. Li, J. Zi, and C. T. Chan, Phys. Rev. E 73, 015602 (2006).
[CrossRef]

M. Einat and E. Jerby, Phys. Rev. E 56, 5996 (1997).
[CrossRef]

Phys. Rev. Lett. (3)

E. J. Reed, M. Soljacic, and J. D. Joannopoulos, Phys. Rev. Lett. 91, 133901 (2003).
[CrossRef] [PubMed]

C. Y. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, Phys. Rev. Lett. 96, 043903 (2006).
[CrossRef] [PubMed]

A. B. Kozyrev and D. W. von der Weide, Phys. Rev. Lett. 94, 203902 (2005).
[CrossRef] [PubMed]

Science (1)

N. Seddon and T. Bearpark, Science 302, 1537 (2003).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
[CrossRef]

Other (7)

J. A. Kong, Electromagnetic Wave Theory (Higher Education, 2002).

Qualitative conclusions reported here are not affected by the specific forms of ϵr(ω), μr(ω), and the parameters are chosen only for easy illustration. Given the velocity adopted in this paper, we find that calculations based on a Galilean transformation do not lead to a significantly different result.

The spectrum is reduced to ω̃=ω0 when vr=0.

Another cross point is automatically excluded in the calculations by the causality requirement.

R̃TE(ω̃,k̃x) is obtained by applying a Lorentz transformation to RTE(ω,kx), the reflection coefficient calculated in the static-slab frame following .

In the static-slab frame, D⃗(r⃗,t) is determined by E⃗(r⃗,t′), as the response is spatially local. However, after the Lorentz transformation in a moving frame, the point {r⃗,t} is spatially different from the point {r⃗,t′} as long as t≠t′, so that the response appears spatially nonlocal.

We get ω̃r=ω0(1+βs)/(1−βs) when vr=0 and ω̃r=ω0(1−βr)/(1+βr) when vs=0. The frequency shift is zero (ω̃r=ω0) when βs=βr.

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Figures (5)

Fig. 1
Fig. 1

Geometry of the system studied in this paper. At t = 0 , the source and receiver are located at (0,0,0) and ( 0.1 d , 0 , 0 ) .

Fig. 2
Fig. 2

(a) Frequency spectra measured by a receiver in vacuum at rest (solid) or with v r = 0.01 c in vacuum (dotted), and by a moving receiver on top of a normal right-handed material (RHM) layer (dashed). (b) Frequency spectrum measured by a receiver moving with v r = 0.01 c on top of a metamaterial slab.

Fig. 3
Fig. 3

Frequencies of all five modes inside the received signal as functions of the source’s working frequency f 0 , with the solid line representing the standard Doppler shift rule. Here, v r = 0.01 c .

Fig. 4
Fig. 4

Frequency shifts Δ f = [ ω ̃ r ( v r ) ω ̃ r ( 0 ) ] 2 π of different modes as functions of v r , with the solid line representing the standard Doppler shift rule. Here, f 0 = 10 GHz .

Fig. 5
Fig. 5

(a) SW dispersion relation (open circles) of the metamaterial slab measured in the receiver’s frame, with the thick solid line representing the source’s spectrum ω ̃ = ω 0 γ r v r k ̃ x . (b) Calculated frequency spectrum of the received signal. Here, v r = 0.01 c .

Equations (3)

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E ̃ y ( r ̃ , ω ̃ ) = i μ 0 I 0 4 π γ r ω ̃ γ s ω ̃ γ r ω 0 + γ s γ r 2 ( c k ̃ x + β r ω ̃ ) ( β r β s ) + i γ r η e i k ̃ x x ̃ k ̃ 0 z ( e i k ̃ 0 z z ̃ + R ̃ TE e i k ̃ 0 z z ̃ ) d k ̃ x ,
ϵ ̃ r = ϵ r ( γ r ω ̃ + γ r v r k ̃ x ) , μ ̃ r = μ r ( γ r ω ̃ + γ r v r k ̃ x ) .
ω ̃ r = ω 0 γ r ( 1 β r ) [ γ s ( 1 β s ) ] .

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