Abstract

We show how the sign of the refractive index in any medium may be derived using a rigorous analysis based on Einstein causality. In particular, we consider left-handed materials, i.e., media that have negative permittivities and permeabilities at the frequency of interest. We find that the consideration of gain in such media can give rise to a positive refractive index.

© 2006 Optical Society of America

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References

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  1. R. W. Ziolkowski and E. Heyman, "Wave propagation in media having negative permittivity and permeability," Phys. Rev. E 64, 056625 (2001).
    [CrossRef]
  2. S. A. Cummer, "Dynamics of causal beam refraction in negative refractive index materials," Appl. Phys. Lett. 82, 2008-2010 (2003).
    [CrossRef]
  3. V. G. Veselago, "Electrodynamics of substances with simultaneously negative values of ϵ and µ," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  4. D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933 (2000).
    [CrossRef] [PubMed]
  5. M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
    [CrossRef]
  6. R. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-316 (2004).
    [CrossRef]
  7. P. W. Milonni, "Controlling the speed of light pulses," J. Phys. B 35, R31-R56 (2002).
    [CrossRef]
  8. R. Y. Chiao and A. M. Steinberg, "Tunneling times and superluminality," in Progress in Optics, E.Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 347-406.
    [CrossRef]
  9. M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, "Abnormal wave propagation in passive media," IEEE J. Sel. Top. Quantum Electron. 9, 30-39 (2003).
    [CrossRef]
  10. In the present context left-handed and right-handed are unrelated to chirality.
  11. L. Brillouin and A. Sommerfeld, Wave Propagation and Group Velocity (Academic, 1960).
  12. G. Arfken, Mathematical Methods for Physicists (Academic, 1937).
  13. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford U. Press, 1937).
  14. Because of a material's finite bandwidth, ϵ̃(omega)-->ϵ0 and µ̃(omega)-->µ0 as ∣omega∣-->∞.
  15. G. Diener, "Superluminal group velocities and information transfer," Phys. Lett. A 223, 327-331 (1996).
    [CrossRef]
  16. The field psi(z,t) due to an arbitrary source s(t) located at z=0 is given by psi(z,t)=(1/2pi)∫−∞∞s(t′)g(z,t−t′)dt′. Given that g(z,t)=0 for t<(z/c), the field psi(z,t) depends only on the source s(t) for a time later than t>(z/c), such that the onset of the disturbance (see Ref. ) propagates at exactly the speed c.
  17. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).
  18. To simplify the presentation in Figs. , we have shown only the zero-pole pair for positive frequencies. More generally, phivϵ and phivµ are determined by both the positive and the negative frequency zero-pole pairs. However, since the zero-pole structure is always symmetric with respect to Re[omega]=0, the contribution to phivϵ and phivµ from the zero-pole pair on the negative frequency side is always smaller in magnitude and is of opposite sign when omega>0. The sign of phivϵ and phivµ is thus fully determined by the zero-pole pair on the positive frequency side. Furthermore, the magnitude of the contribution to phivϵ and phivµ from the zero-pole pair on the negative frequency side is ∣tan−1[Gamma/(omegapole+omega)]−tan−1[Gamma/(omegazero+omega)]∣ and it is negligibly small if omegazero⪢Gamma and omegapole⪢Gamma. In this case, phivϵ and phivµ are entirely determined by the zero-pole pair on the positive frequency side. Our results do not rely on this particular simplification and they apply to the general case where the contribution from the negative frequency zero-pole pairs is included.
  19. K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
    [CrossRef]
  20. V. Yannopapas and A. Moroz, "Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges," J. Phys.: Condens. Matter 17, 3717-3734 (2005).
    [CrossRef]
  21. Y.-F. Chen, P. Fischer, and F. W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95, 067402 (2005).
    [CrossRef] [PubMed]
  22. R. F. Cybulski and M. P. Silverman, "Investigation of light amplification by enhanced internal reflection. I. Theoretical reflectance and transmittance of an exponentially nonuniform gain region," J. Opt. Soc. Am. B 73, 1732-1738 (1983).
    [CrossRef]
  23. M. P. Silverman and R. F. Cybulski, "Investigation of light amplification by enhanced internal reflection. II. Experimental determination of the single-pass reflectance of an optically pumped gain region," J. Opt. Soc. Am. B 73, 1739-1743 (1983).
    [CrossRef]
  24. When the denominator in Eqs. -->0, then r‖ and t‖-->∞. To determine the angle of incidence thetai at which this occurs, note that the denominator can be expressed as [(n1sinthetai)/n2][1−(n12/n22)sin2thetai]1/2−(µ1/∣µ2∣)sinthetaicosthetai. For the example considered in the text where µ2=−µ1 and µ1>0, the denominator vanishes when (n1/n2)[1−(n12/n22)sin2thetai]1/2=costhetai. This yields [(n12−n22)/n22]sin2thetai=[(n12−n22)/n12]cos2thetai, and the singularity occurs when tanthetai=(n2/n1). It then follows that the inclusion of the imaginary part of ϵ2 (which yields a complex n2) removes the singularity, since this relation can no longer be satisfied for a real angle of incidence thetai.

2005 (2)

V. Yannopapas and A. Moroz, "Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges," J. Phys.: Condens. Matter 17, 3717-3734 (2005).
[CrossRef]

Y.-F. Chen, P. Fischer, and F. W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

2004 (2)

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
[CrossRef]

R. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

2003 (2)

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, "Abnormal wave propagation in passive media," IEEE J. Sel. Top. Quantum Electron. 9, 30-39 (2003).
[CrossRef]

S. A. Cummer, "Dynamics of causal beam refraction in negative refractive index materials," Appl. Phys. Lett. 82, 2008-2010 (2003).
[CrossRef]

2002 (2)

P. W. Milonni, "Controlling the speed of light pulses," J. Phys. B 35, R31-R56 (2002).
[CrossRef]

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

2001 (1)

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

2000 (1)

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

1996 (1)

G. Diener, "Superluminal group velocities and information transfer," Phys. Lett. A 223, 327-331 (1996).
[CrossRef]

1983 (2)

R. F. Cybulski and M. P. Silverman, "Investigation of light amplification by enhanced internal reflection. I. Theoretical reflectance and transmittance of an exponentially nonuniform gain region," J. Opt. Soc. Am. B 73, 1732-1738 (1983).
[CrossRef]

M. P. Silverman and R. F. Cybulski, "Investigation of light amplification by enhanced internal reflection. II. Experimental determination of the single-pass reflectance of an optically pumped gain region," J. Opt. Soc. Am. B 73, 1739-1743 (1983).
[CrossRef]

1968 (1)

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

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, 1937).

Brillouin, L.

L. Brillouin and A. Sommerfeld, Wave Propagation and Group Velocity (Academic, 1960).

Chen, Y.-F.

Y.-F. Chen, P. Fischer, and F. W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

Chiao, R. Y.

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, "Abnormal wave propagation in passive media," IEEE J. Sel. Top. Quantum Electron. 9, 30-39 (2003).
[CrossRef]

R. Y. Chiao and A. M. Steinberg, "Tunneling times and superluminality," in Progress in Optics, E.Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 347-406.
[CrossRef]

Cummer, S. A.

S. A. Cummer, "Dynamics of causal beam refraction in negative refractive index materials," Appl. Phys. Lett. 82, 2008-2010 (2003).
[CrossRef]

Cybulski, R. F.

R. F. Cybulski and M. P. Silverman, "Investigation of light amplification by enhanced internal reflection. I. Theoretical reflectance and transmittance of an exponentially nonuniform gain region," J. Opt. Soc. Am. B 73, 1732-1738 (1983).
[CrossRef]

M. P. Silverman and R. F. Cybulski, "Investigation of light amplification by enhanced internal reflection. II. Experimental determination of the single-pass reflectance of an optically pumped gain region," J. Opt. Soc. Am. B 73, 1739-1743 (1983).
[CrossRef]

Depine, R.

R. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

Diener, G.

G. Diener, "Superluminal group velocities and information transfer," Phys. Lett. A 223, 327-331 (1996).
[CrossRef]

Eleftheriades, G. V.

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, "Abnormal wave propagation in passive media," IEEE J. Sel. Top. Quantum Electron. 9, 30-39 (2003).
[CrossRef]

Fischer, P.

Y.-F. Chen, P. Fischer, and F. W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

Heyman, E.

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

Huang, K. C.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

Joannopoulos, J. D.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
[CrossRef]

Kroll, N.

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

Lakhtakia, A.

R. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

Malloy, K. J.

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, "Abnormal wave propagation in passive media," IEEE J. Sel. Top. Quantum Electron. 9, 30-39 (2003).
[CrossRef]

McCall, M. W.

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

Milonni, P. W.

P. W. Milonni, "Controlling the speed of light pulses," J. Phys. B 35, R31-R56 (2002).
[CrossRef]

Mojahedi, M.

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, "Abnormal wave propagation in passive media," IEEE J. Sel. Top. Quantum Electron. 9, 30-39 (2003).
[CrossRef]

Moroz, A.

V. Yannopapas and A. Moroz, "Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges," J. Phys.: Condens. Matter 17, 3717-3734 (2005).
[CrossRef]

Povinelli, M. L.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
[CrossRef]

Silverman, M. P.

R. F. Cybulski and M. P. Silverman, "Investigation of light amplification by enhanced internal reflection. I. Theoretical reflectance and transmittance of an exponentially nonuniform gain region," J. Opt. Soc. Am. B 73, 1732-1738 (1983).
[CrossRef]

M. P. Silverman and R. F. Cybulski, "Investigation of light amplification by enhanced internal reflection. II. Experimental determination of the single-pass reflectance of an optically pumped gain region," J. Opt. Soc. Am. B 73, 1739-1743 (1983).
[CrossRef]

Smith, D. R.

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

Sommerfeld, A.

L. Brillouin and A. Sommerfeld, Wave Propagation and Group Velocity (Academic, 1960).

Steinberg, A. M.

R. Y. Chiao and A. M. Steinberg, "Tunneling times and superluminality," in Progress in Optics, E.Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 347-406.
[CrossRef]

Titchmarsh, E. C.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford U. Press, 1937).

Veselago, V. G.

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

Weiglhofer, W. S.

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

Wise, F. W.

Y.-F. Chen, P. Fischer, and F. W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

Woodley, J.

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, "Abnormal wave propagation in passive media," IEEE J. Sel. Top. Quantum Electron. 9, 30-39 (2003).
[CrossRef]

Yannopapas, V.

V. Yannopapas and A. Moroz, "Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges," J. Phys.: Condens. Matter 17, 3717-3734 (2005).
[CrossRef]

Ziolkowski, R. W.

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

Appl. Phys. Lett. (2)

S. A. Cummer, "Dynamics of causal beam refraction in negative refractive index materials," Appl. Phys. Lett. 82, 2008-2010 (2003).
[CrossRef]

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004).
[CrossRef]

Eur. J. Phys. (1)

M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, "The negative index of refraction demystified," Eur. J. Phys. 23, 353-359 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

M. Mojahedi, K. J. Malloy, G. V. Eleftheriades, J. Woodley, and R. Y. Chiao, "Abnormal wave propagation in passive media," IEEE J. Sel. Top. Quantum Electron. 9, 30-39 (2003).
[CrossRef]

J. Opt. Soc. Am. B (2)

R. F. Cybulski and M. P. Silverman, "Investigation of light amplification by enhanced internal reflection. I. Theoretical reflectance and transmittance of an exponentially nonuniform gain region," J. Opt. Soc. Am. B 73, 1732-1738 (1983).
[CrossRef]

M. P. Silverman and R. F. Cybulski, "Investigation of light amplification by enhanced internal reflection. II. Experimental determination of the single-pass reflectance of an optically pumped gain region," J. Opt. Soc. Am. B 73, 1739-1743 (1983).
[CrossRef]

J. Phys. B (1)

P. W. Milonni, "Controlling the speed of light pulses," J. Phys. B 35, R31-R56 (2002).
[CrossRef]

J. Phys.: Condens. Matter (1)

V. Yannopapas and A. Moroz, "Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges," J. Phys.: Condens. Matter 17, 3717-3734 (2005).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

R. Depine and A. Lakhtakia, "A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity," Microwave Opt. Technol. Lett. 41, 315-316 (2004).
[CrossRef]

Phys. Lett. A (1)

G. Diener, "Superluminal group velocities and information transfer," Phys. Lett. A 223, 327-331 (1996).
[CrossRef]

Phys. Rev. E (1)

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. (2)

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

Y.-F. Chen, P. Fischer, and F. W. Wise, "Negative refraction at optical frequencies in nonmagnetic two-component molecular media," Phys. Rev. Lett. 95, 067402 (2005).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

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

Other (10)

R. Y. Chiao and A. M. Steinberg, "Tunneling times and superluminality," in Progress in Optics, E.Wolf, ed. (Elsevier, 1997), Vol. 37, pp. 347-406.
[CrossRef]

The field psi(z,t) due to an arbitrary source s(t) located at z=0 is given by psi(z,t)=(1/2pi)∫−∞∞s(t′)g(z,t−t′)dt′. Given that g(z,t)=0 for t<(z/c), the field psi(z,t) depends only on the source s(t) for a time later than t>(z/c), such that the onset of the disturbance (see Ref. ) propagates at exactly the speed c.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

To simplify the presentation in Figs. , we have shown only the zero-pole pair for positive frequencies. More generally, phivϵ and phivµ are determined by both the positive and the negative frequency zero-pole pairs. However, since the zero-pole structure is always symmetric with respect to Re[omega]=0, the contribution to phivϵ and phivµ from the zero-pole pair on the negative frequency side is always smaller in magnitude and is of opposite sign when omega>0. The sign of phivϵ and phivµ is thus fully determined by the zero-pole pair on the positive frequency side. Furthermore, the magnitude of the contribution to phivϵ and phivµ from the zero-pole pair on the negative frequency side is ∣tan−1[Gamma/(omegapole+omega)]−tan−1[Gamma/(omegazero+omega)]∣ and it is negligibly small if omegazero⪢Gamma and omegapole⪢Gamma. In this case, phivϵ and phivµ are entirely determined by the zero-pole pair on the positive frequency side. Our results do not rely on this particular simplification and they apply to the general case where the contribution from the negative frequency zero-pole pairs is included.

In the present context left-handed and right-handed are unrelated to chirality.

L. Brillouin and A. Sommerfeld, Wave Propagation and Group Velocity (Academic, 1960).

G. Arfken, Mathematical Methods for Physicists (Academic, 1937).

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford U. Press, 1937).

Because of a material's finite bandwidth, ϵ̃(omega)-->ϵ0 and µ̃(omega)-->µ0 as ∣omega∣-->∞.

When the denominator in Eqs. -->0, then r‖ and t‖-->∞. To determine the angle of incidence thetai at which this occurs, note that the denominator can be expressed as [(n1sinthetai)/n2][1−(n12/n22)sin2thetai]1/2−(µ1/∣µ2∣)sinthetaicosthetai. For the example considered in the text where µ2=−µ1 and µ1>0, the denominator vanishes when (n1/n2)[1−(n12/n22)sin2thetai]1/2=costhetai. This yields [(n12−n22)/n22]sin2thetai=[(n12−n22)/n12]cos2thetai, and the singularity occurs when tanthetai=(n2/n1). It then follows that the inclusion of the imaginary part of ϵ2 (which yields a complex n2) removes the singularity, since this relation can no longer be satisfied for a real angle of incidence thetai.

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Figures (5)

Fig. 1
Fig. 1

Zero-pole pairs generated by (a) a noninverted Lorentz oscillator ( F > 0 ) and (b) an inverted Lorentz oscillator ( F < 0 ) . Only the poles and zeros for Re [ ω ] > 0 are indicated, as the structures are symmetric with respect to Re [ ω ] = 0 .

Fig. 2
Fig. 2

Three of the possible four different types of zero-pole pairs are shown for LHM: (a) F > 0 and G > 0 , (b) F < 0 and G > 0 , (c) F < 0 and G < 0 . The one not shown corresponds to (b) with ϵ and μ exchanged.

Fig. 3
Fig. 3

Boundary conditions for an interface between an ordinary material ( ϵ 1 , μ 1 > 0 ) and a type II or type III LHM ( ϵ 2 , μ 2 < 0 ) . Two conditions have to be met: (i) the transverse components of the wave vectors must be the same across the boundary ( k i x = k a x = k b x ) , and (ii) the ratio of the magnitudes of the transmitted wave vector to that of the incidence wave vector must satisfy ( k a k i ) 2 = ( k b k i ) 2 = ( ϵ 1 μ 1 ϵ 2 μ 2 ) .

Fig. 4
Fig. 4

Schematic of the energy flux across an interface between an ordinary material and a (type II or type III) LHM with a positive refractive index. Energy is conserved and the refraction is positive. The net energy derives from the inverted LHM.

Fig. 5
Fig. 5

Reflectance and transmission curves for electric field polarization perpendicular to the plane of incidence (top panel) and electric field polarization parallel to the plane of incidence (bottom panel).

Tables (1)

Tables Icon

Table 1 Four Types of LHM a

Equations (22)

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{ 2 E ̃ ( r , ω ) = ω 2 ϵ ̃ ( ω ) μ ̃ ( ω ) E ̃ ( r , ω ) 2 B ̃ ( r , ω ) = ω 2 ϵ ̃ ( ω ) μ ̃ ( ω ) B ̃ ( r , ω ) } .
f ̃ ( ω ) = 1 2 π f ( t ) exp ( i ω t ) d t .
2 g ̃ ( z , ω ) z 2 = ω 2 ϵ ̃ ( ω ) μ ̃ ( ω ) g ̃ ( z , ω ) ,
g ( z , t ) = 1 2 π g ̃ ( z , ω ) exp ( i ω t ) d ω = 1 2 π exp [ i ( k ̃ ( ω ) z ω t ) ] d ω .
α ̃ = 2 ω m g g p m 2 3 [ ω m g 2 ( ω + i Γ ) 2 ] ,
ϵ ̃ = ϵ 0 ( 1 + N α ̃ ϵ 0 ) ,
ϵ ̃ = ϵ 0 { 1 + 2 ω m g N g p m 2 3 ϵ 0 [ ω m g 2 ( ω + i Γ ) 2 ] } ,
F = 2 ω m g N g p m 2 3 ϵ 0 .
ϵ ̃ ( ω ) = ϵ 0 [ 1 + F ω pole ϵ 2 ( ω + i Γ ) 2 ] = ϵ 0 ( ω + i Γ ) 2 ω zero ϵ 2 ( ω + i Γ ) 2 ω pole ϵ 2
μ ̃ ( ω ) = μ 0 [ 1 + G ω pole μ 2 ( ω + i Γ ) 2 ] = μ 0 ( ω + i Γ ) 2 ω zero μ 2 ( ω + i Γ ) 2 ω pole μ 2 ,
n 2 = + ϵ 2 μ 2 ϵ 0 μ 0 > 0 ,
r ( E r E i ) = sin θ t cos θ i + μ 1 μ 2 sin θ i cos θ t sin θ t cos θ i μ 1 μ 2 sin θ i cos θ t ,
t ( E t E i ) = 2 sin θ t cos θ i sin θ t cos θ i μ 1 μ 2 sin θ i cos θ t .
r ( E r E i ) = μ 1 μ 2 sin θ i cos θ i sin θ t cos θ t sin θ t cos θ t μ 1 μ 2 sin θ i cos θ i
t ( E t E i ) = 2 sin θ t cos θ i sin θ t cos θ t μ 1 μ 2 sin θ i cos θ i ,
R = r 2
T = ( μ 1 μ 2 ) ( n 2 n 1 ) ( cos θ t cos θ i ) t 2 ,
R = r 2 ,
T = ( μ 1 μ 2 ) ( n 2 n 1 ) ( cos θ t cos θ i ) t 2 ,
n 1 = + ϵ 1 μ 1 ϵ 0 μ 0 > 0 .
R + T = R + T = 1 .
u t + S = 0 ,

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