Abstract

The class of causal negative index media is identified and analyzed through the development of a conceptual framework relying on polar paths and zero-pole placements of rational functions. In particular, necessary conditions on the refractive index function n(ω) toward the achievement of Ren<0 with Imn=0 are found. The requirement of causality necessitates either significant loss/gain in some frequency bands or steep variation in n immediately below or above the observation frequency. A particular system demonstrating negative refraction with arbitrarily small loss/gain for all frequencies through steep variation in n is analyzed.

© 2013 Optical Society of America

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  1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [CrossRef]
  2. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
  3. D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
    [CrossRef]
  4. J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
    [CrossRef]
  5. M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
    [CrossRef]
  6. T. G. Mackay and A. Lakhtakia, “Comment on “criterion for negative refraction with low optical losses from a fundamental principle of causality”,” Phys. Rev. Lett. 99, 189701 (2007).
    [CrossRef]
  7. P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101167401 (2008).
    [CrossRef]
  8. B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
    [CrossRef]
  9. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef]
  10. Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
    [CrossRef]
  11. A. Bernland, A. Luger, and M. Gustafsson, “Sum rules and constraints on passive systems,” J. Phys. A 44145205 (2011).
    [CrossRef]
  12. M. Gustafsson and D. Sjøberg, “Sum rules and physical bounds on passive metamaterials,” New J. Phys. 12, 043046 (2010).
    [CrossRef]
  13. P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. 112, 1488–1503 (1958).
    [CrossRef]
  14. R. J. Briggs, Electron-Stream Interactions with Plasmas(MIT, 1964).
  15. J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E 73, 026605 (2006).
    [CrossRef]
  16. J. Skaar, “On resolving the refractive index and the wave vector,” Opt. Lett. 31, 3372–3374 (2006).
    [CrossRef]
  17. 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]
  18. B. Nistad and J. Skaar, “Simulations and realizations of active right-handed metamaterials with negative refractive index,” Opt. Express 15, 10935–10946 (2007).
    [CrossRef]
  19. P. Kinsler, “How to be causal: time, spacetime and spectra,” Eur. J. Phys. 32, 1687–1700 (2011).
    [CrossRef]
  20. O. V. Dolgov, D. A. Kirzhnits, and E. G. Maksimov, “On an admissible sign of the static dielectric function of matter,” Rev. Mod. Phys. 53, 81–93 (1981).
    [CrossRef]
  21. P. Kinsler, “Active drains and causality,” Phys. Rev. A 82, 055804 (2010).
    [CrossRef]

2011 (2)

A. Bernland, A. Luger, and M. Gustafsson, “Sum rules and constraints on passive systems,” J. Phys. A 44145205 (2011).
[CrossRef]

P. Kinsler, “How to be causal: time, spacetime and spectra,” Eur. J. Phys. 32, 1687–1700 (2011).
[CrossRef]

2010 (2)

P. Kinsler, “Active drains and causality,” Phys. Rev. A 82, 055804 (2010).
[CrossRef]

M. Gustafsson and D. Sjøberg, “Sum rules and physical bounds on passive metamaterials,” New J. Phys. 12, 043046 (2010).
[CrossRef]

2009 (1)

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

2008 (2)

P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101167401 (2008).
[CrossRef]

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

2007 (3)

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

T. G. Mackay and A. Lakhtakia, “Comment on “criterion for negative refraction with low optical losses from a fundamental principle of causality”,” Phys. Rev. Lett. 99, 189701 (2007).
[CrossRef]

B. Nistad and J. Skaar, “Simulations and realizations of active right-handed metamaterials with negative refractive index,” Opt. Express 15, 10935–10946 (2007).
[CrossRef]

2006 (3)

J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E 73, 026605 (2006).
[CrossRef]

J. Skaar, “On resolving the refractive index and the wave vector,” Opt. Lett. 31, 3372–3374 (2006).
[CrossRef]

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
[CrossRef]

2005 (1)

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]

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]

2000 (2)

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

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

1981 (1)

O. V. Dolgov, D. A. Kirzhnits, and E. G. Maksimov, “On an admissible sign of the static dielectric function of matter,” Rev. Mod. Phys. 53, 81–93 (1981).
[CrossRef]

1958 (1)

P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. 112, 1488–1503 (1958).
[CrossRef]

Bernland, A.

A. Bernland, A. Luger, and M. Gustafsson, “Sum rules and constraints on passive systems,” J. Phys. A 44145205 (2011).
[CrossRef]

Briggs, R. J.

R. J. Briggs, Electron-Stream Interactions with Plasmas(MIT, 1964).

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]

Dolgov, O. V.

O. V. Dolgov, D. A. Kirzhnits, and E. G. Maksimov, “On an admissible sign of the static dielectric function of matter,” Rev. Mod. Phys. 53, 81–93 (1981).
[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]

Gustafsson, M.

A. Bernland, A. Luger, and M. Gustafsson, “Sum rules and constraints on passive systems,” J. Phys. A 44145205 (2011).
[CrossRef]

M. Gustafsson and D. Sjøberg, “Sum rules and physical bounds on passive metamaterials,” New J. Phys. 12, 043046 (2010).
[CrossRef]

Kinsler, P.

P. Kinsler, “How to be causal: time, spacetime and spectra,” Eur. J. Phys. 32, 1687–1700 (2011).
[CrossRef]

P. Kinsler, “Active drains and causality,” Phys. Rev. A 82, 055804 (2010).
[CrossRef]

P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101167401 (2008).
[CrossRef]

Kirzhnits, D. A.

O. V. Dolgov, D. A. Kirzhnits, and E. G. Maksimov, “On an admissible sign of the static dielectric function of matter,” Rev. Mod. Phys. 53, 81–93 (1981).
[CrossRef]

Kroll, N.

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

Lakhtakia, A.

T. G. Mackay and A. Lakhtakia, “Comment on “criterion for negative refraction with low optical losses from a fundamental principle of causality”,” Phys. Rev. Lett. 99, 189701 (2007).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Lind-Johansen, Ø.

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

Luger, A.

A. Bernland, A. Luger, and M. Gustafsson, “Sum rules and constraints on passive systems,” J. Phys. A 44145205 (2011).
[CrossRef]

Mackay, T. G.

T. G. Mackay and A. Lakhtakia, “Comment on “criterion for negative refraction with low optical losses from a fundamental principle of causality”,” Phys. Rev. Lett. 99, 189701 (2007).
[CrossRef]

Maksimov, E. G.

O. V. Dolgov, D. A. Kirzhnits, and E. G. Maksimov, “On an admissible sign of the static dielectric function of matter,” Rev. Mod. Phys. 53, 81–93 (1981).
[CrossRef]

McCall, M. W.

P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101167401 (2008).
[CrossRef]

Nistad, B.

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (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]

Seip, K.

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
[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]

Sjøberg, D.

M. Gustafsson and D. Sjøberg, “Sum rules and physical bounds on passive metamaterials,” New J. Phys. 12, 043046 (2010).
[CrossRef]

Skaar, J.

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

B. Nistad and J. Skaar, “Simulations and realizations of active right-handed metamaterials with negative refractive index,” Opt. Express 15, 10935–10946 (2007).
[CrossRef]

J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E 73, 026605 (2006).
[CrossRef]

J. Skaar, “On resolving the refractive index and the wave vector,” Opt. Lett. 31, 3372–3374 (2006).
[CrossRef]

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
[CrossRef]

Smith, D. R.

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

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

Stockman, M. I.

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

Sturrock, P. A.

P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. 112, 1488–1503 (1958).
[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]

Eur. J. Phys. (1)

P. Kinsler, “How to be causal: time, spacetime and spectra,” Eur. J. Phys. 32, 1687–1700 (2011).
[CrossRef]

J. Math. Phys. (1)

Ø. Lind-Johansen, K. Seip, and J. Skaar, “The perfect lens on a finite bandwidth,” J. Math. Phys. 50, 012908 (2009).
[CrossRef]

J. Phys. A (1)

A. Bernland, A. Luger, and M. Gustafsson, “Sum rules and constraints on passive systems,” J. Phys. A 44145205 (2011).
[CrossRef]

J. Phys. D (1)

J. Skaar and K. Seip, “Bounds for the refractive indices of metamaterials,” J. Phys. D 39, 1226–1229 (2006).
[CrossRef]

New J. Phys. (1)

M. Gustafsson and D. Sjøberg, “Sum rules and physical bounds on passive metamaterials,” New J. Phys. 12, 043046 (2010).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. (1)

P. A. Sturrock, “Kinematics of growing waves,” Phys. Rev. 112, 1488–1503 (1958).
[CrossRef]

Phys. Rev. A (1)

P. Kinsler, “Active drains and causality,” Phys. Rev. A 82, 055804 (2010).
[CrossRef]

Phys. Rev. E (2)

J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E 73, 026605 (2006).
[CrossRef]

B. Nistad and J. Skaar, “Causality and electromagnetic properties of active media,” Phys. Rev. E 78, 036603 (2008).
[CrossRef]

Phys. Rev. Lett. (6)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

T. G. Mackay and A. Lakhtakia, “Comment on “criterion for negative refraction with low optical losses from a fundamental principle of causality”,” Phys. Rev. Lett. 99, 189701 (2007).
[CrossRef]

P. Kinsler and M. W. McCall, “Causality-based criteria for a negative refractive index must be used with care,” Phys. Rev. Lett. 101167401 (2008).
[CrossRef]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[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]

Rev. Mod. Phys. (1)

O. V. Dolgov, D. A. Kirzhnits, and E. G. Maksimov, “On an admissible sign of the static dielectric function of matter,” Rev. Mod. Phys. 53, 81–93 (1981).
[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]

Other (2)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

R. J. Briggs, Electron-Stream Interactions with Plasmas(MIT, 1964).

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Figures (9)

Fig. 1.
Fig. 1.

Polar plots of electromagnetic response functions as the frequency is moved from along the real ω axis down to an observation frequency. (a) Polar path traversed by the product ϵμ of permittivity and permeability. (b) Polar path traversed by the corresponding refractive index n. The response ϵμ in (a) leads to Ren=1 with Imn=0 for a certain frequency in (b).

Fig. 2.
Fig. 2.

Two polar paths corresponding to two different functions ϵμ of permittivity and permeability as the frequency is moved from along the real ω axis toward an observation frequency. Despite their different appearance, both will lead to Ren=1 with Imn=0 at the frequency at which their polar path completes a loop.

Fig. 3.
Fig. 3.

(a) Permittivity ϵ1 plotted versus frequency. Notice that according to Eq. (3), this plot also represents the refractive index n1 plotted versus frequency. (b) ϵ1μ1 plotted in a polar representation for 0ω/ω02.

Fig. 4.
Fig. 4.

(a) Permittivity ϵ2 plotted versus frequency. (b) ϵ2μ2 plotted in a polar representation for 0ω/ω02.

Fig. 5.
Fig. 5.

(a) Zero-pole locations in the complex frequency plane, corresponding to both ϵ1μ1 from example 1 in Fig. 3 and ϵ2μ2 from example 2 in Fig. 4. The superscript signifies that there are two zeros and two poles placed on each respective dot and cross. (b) Response as results from the zero-pole positions in (a). The solid curve represents Reϵμ, while the dashed curve represents Imϵμ. This is the same response as displayed in Fig. 4(a), except here negative frequencies are also displayed.

Fig. 6.
Fig. 6.

Some various possibilities of zero-pole positions toward the goal of achieving a phase θϵμ±2π, in order to have Ren<0 with Imn0.

Fig. 7.
Fig. 7.

Some various possibilities of zero-pole positions.

Fig. 8.
Fig. 8.

Im(n) versus ω for an imagined refractive index response. For ω1<ω<ω2 one has Im(n)=0, and one desires that Ren(ωobs)<0 within this region.

Fig. 9.
Fig. 9.

Modification of the system displayed in Figs. 4 and 5. (a) Zero-pole positions in the complex frequency plane for positive frequencies. Note that there are two zeros and two poles on each respective cross and dot as indicated by the legend in Fig. 9(a). (b) Product ϵμ. The solid curves represent Re(ϵμ), while the stippled curves represent Im(ϵμ). (c) Polar plot of ϵμ for 0ω/ω010. (d) Refractive index n. The solid curves represent Re(n), while the stippled curves represent Im(n).

Equations (18)

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n(ω)=±|ϵ(ω)μ(ω)|exp[iθϵ(ω)+θμ(ω)2].
ϵ1(ω)=1+f(ω),
μ1(ω)=1+f(ω),
f(ω)=Fω02ω02ω2iωΓ,
n1(ω)=1+f(ω).
ϵ2(ω)=(1+f(ω))2,
μ2(ω)=1.
ϵ(ω)μ(ω)1forω±.
ϵ(ω)μ(ω)=ϵ*(ω)μ*(ω).
ϵ(ω)μ(ω)=(ωω01)(ωω02)(ωω0k)(ω+ω01*)(ω+ω02*)(ω+ω0k*)(ωωp1)(ωωp2)(ωωpk)(ω+ωp1*)(ω+ωp2*)(ω+ωpk*).
1+f(ω)=(ν(1.410.05i))(ν(1.410.05i))(ν(10.05i))(ν(10.05i)).
ϵ(ω)μ(ω)(ωω01)(ωω02)(ωω0k)(ωωp1)(ωωp2)(ωωpk).
θϵμ=θ0θp=θ.
θϵμ=2(θ0θp)=2θ.
Ren(ωobs)=12π0ω1Imn(ω)ωωobs2ω2dω+2πω2Imn(ω)ωω2ωobs2dω.
ρb(ω)=·P(ω)=·(D(ω)ϵ0E(ω))=(1/ϵ(ω)1)ρ(ω).
(ωn+bn1ωn1++b0)ρb(ω)=(an1ωn1++a0)ρ(ω),
k=0nikbktkρb(t)=k=0n1ikaktkρ(t),

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