## Simulations and realizations of active right-handed metamaterials with negative refractive index

Optics Express, Vol. 15, Issue 17, pp. 10935-10946 (2007)

http://dx.doi.org/10.1364/OE.15.010935

Acrobat PDF (176 KB)

### Abstract

The theory of determining the sign of the refractive index in active materials is discussed. Animations of numerical simulations are presented, supporting the claim that negative refractive index may occur in right-handed media. An example of such a medium, in the form of a lumped circuit model with active and passive resonances, is presented.

© 2007 Optical Society of America

## 1. Introduction

*ε*and permeability

*µ*may yield negative refraction [1

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative ε and µ,” Sov. Phys. Usp. **10(4)**, 509–514 (1968). [CrossRef]

*ε*and

*µ*in the same frequency range [2

2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys.: Condens. Matter **10(22)**, 4785–4809 (1998). [CrossRef]

3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. **47(11)**, 2075–2084 (1999). [CrossRef]

4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. **84(18)**, 4184–4187 (2000). [CrossRef]

5. 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(12)**, 2702–2712 (2002). [CrossRef]

6. S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B **67(20)**, 201101 (2003). [CrossRef]

7. M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. **31**, 3022 (2006). [CrossRef] [PubMed]

8. A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index meta-materials by optical parametric amplification,” Opt. Lett. **31**, 2169 (2006). [CrossRef] [PubMed]

9. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics **1**, 41–48 (2007). [CrossRef]

*εµ*in the complex

*ω*-plane. This indicates that it is possible to obtain negative refractive index without any magnetic resonances,

*µ*=1, but instead two electric resonances. Indeed, it has been suggested that certain nonmagnetic media, with active and passive dielectric resonances, can exhibit negative refraction [10

10. Y.-F. Chen, P. Fischer, and F.W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. **95(6)**, 067402 (2005). [CrossRef]

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

10. Y.-F. Chen, P. Fischer, and F.W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. **95(6)**, 067402 (2005). [CrossRef]

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

*d*can be found unambiguously as it only dependents on

*ε, µ*, and

*d*, and is independent of the refractive index. Moreover, by causality,

*the time-domain solution for t<d/c must be identical to that of a semi-infinite medium*when the slab is excited at one side, starting at

*t*=0. Letting

*d*be sufficiently large, the solutions can be compared in an arbitrarily large time window. The movies clearly support the claim that negative refractive index may be obtained in active, right-handed media.

## 2. Determining the sign of the refractive index

*R*and transmission

*S*are given by the Fresnel equations:

*k=ωn/c, η=µ/n*, and

*n*and

^{2}=εµ; ε*µ*are the permittivity and permeability of the semi-infinite medium. The interface between vacuum and the semi-infinite medium is the plane

*z*=0; the region

*z*>0 is the location of the semi-infinite medium. For convenience, the propagation factor exp(

*iωnz/c*) is included in

*S*. The sign of the refractive index

*n*must be identified to ensure causality; for passive materials this can be achieved simply by requiring that the Poynting vector point in +

*z*-direction, or that the wave decays in the +z-direction. As an example of an active medium, we consider a material with

*ε(ω)*=(1+

*f (ω)*)

^{2}and

*µ(ω)*=1, where

*F*=2 and Γ=0.005

*ω*

_{0}. This material is causal and realizable as

*ε*is analytic in the upper half-plane of complex frequency and

*ε(ω)*-1~-2

*Fω*

^{2}

_{0}/

*ω*

^{2}as

*ω*→∞ [21]. Using the method of Refs. [20, 10

10. Y.-F. Chen, P. Fischer, and F.W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. **95(6)**, 067402 (2005). [CrossRef]

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

12. Y.-F. Chen, P. Fischer, and F. W. Wise, “Sign of the refractive index in a gain medium with negative permittivity and permeability,” J. Opt. Soc. Am. B **23**, 45–50 (2006). [CrossRef]

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

*n*

_{1}(

*ω*)=1+

*f(ω)*. On the other hand, using the approaches in Refs. [13

13. T. G. Mackay and A. Lakhtakia, “Comment on “Negative refraction at optical frequencies in nonmagnetic two-component molecular media”,” Phys. Rev. Lett. **96(15)**, 159701 (2006). [CrossRef]

15. S. A. Ramakrishna, “Comment on “Negative refraction at optical frequencies in nonmagnetic two-component molecular media”,” Phys. Rev. Lett. **98(5)**, 059701 (2007). [CrossRef]

17. S. A. Ramakrishna and O. J. F. Martin, “Resolving the wave vector in negative refractive index media,” Opt. Lett. **30**, 2626 (2005). [CrossRef] [PubMed]

18. A. N. Grigorenko, “Negative refractive index in artificial metamaterials,” Opt. Lett. **31**, 2483 (2006). [CrossRef] [PubMed]

*n*

_{2}(

*ω*)=(1+

*f(ω)*)Re (1+

*f (ω)*)/|Re (1+

*f (ω)*)|. The real and imaginary parts of the refractive index for the two different solutions are plotted in Fig. 1. Note that there is a frequency band where Re

*n*

_{1}(

*ω*)<0; thus the first solution predicts that both phase velocity and steady-state energy flow may point towards the source. For the other solution, Re

*n*

_{2}(

*ω*)>0 for all frequencies; thus this solution means that the phase velocity and energy flow are directed away from the source. Note the two points,

*ω=ω*

_{0}and ω=1.73

*ω*

_{0}, where

*n*

_{2}(

*ω*) is discontinuous.

*n*

_{1}(

*ω*) and

*n*

_{2}(

*ω*). This is achieved with the inverse Laplace transform. We excite the medium by a plane wave, normally incident at

*z*=0-, starting at

*t*=0. For example, if the excitation at

*z*=0- is

*u(t)*cos(

*ω*

_{1}

*t*), where

*u(t)*is the unit step function and

*ω*

_{1}is the excitation frequency, the time-domain field in the material can be expressed as

*ω<γ*for all poles or nonanalytic points. For

*S*given by Eq. (1b), assuming that the refractive index has meaning for real frequencies, the integral can be evaluated just above the real frequency axis (

*γ*=0+) [11

**73**, 026605 (2006). [CrossRef]

22. In general, we can set γ=0+ provided the refractive index is analytic in the upper half-plane and the denominator in the Fresnel equations is nonzero in the upper half-plane. Excluding media with absolute instabilities, the refractive index can always be identified as an analytic function in the upper half-plane.

*ω*

_{1}=1.4

*ω*

_{0}; at this frequency

*ε(ω*

_{1})≈+1.13-0.32

*i*. Now we can examine the solutions associated with

*n(ω)*=

*n*

_{1}(

*ω*) (Fig. 2) and

*n(ω)*=

*n*

_{2}(

*ω*) (Fig. 3), and judge whether they satisfy causality. Note that causality should be interpreted in its most fundamental form; no field can arise at z before the time

*t=z/c*. We clearly see that the wave in Fig. 2 satisfies causality, while the wave in Fig. 3 does not. In fact, in Fig. 3 a nonzero field exists everywhere, even at

*t*=0. In Fig. 2, the wave front propagates at exactly the speed

*c*[20]. Similarly, when the excitation is turned off, the high-frequency components induced by the abrupt envelope propagate at c. Note that the field eventually dies out after the excitation has been turned off. Some ripple can be seen in front of the red line in Fig. 2. This is a numerical error, which relate to Gibbs’ effect, and can be reduced if a broader frequency representation is used.

*n*

_{1}(

*ω*) not only gives a causal solution, but also the correct one, we compare the solution in Fig. 2 to the electromagnetic field in a slab of finite thickness

*d*. Such a comparison makes sense since, by causality, the time-domain fields in a slab and in a semi-infinite medium must coincide for

*t<d/c*. In an active slab, the fields may blow up with time and thus Fourier transformed fields may not exist. A natural remedy is to use Laplace transformed fields, i.e., to introduce complex frequencies with Imω>0. For a normally incident electromagnetic field, the reflected field

*R*, the field in the slab

*S*=

*S*

^{+}exp(

*ikz*)+

*S*

^{-}exp(-

*ikz*), and the transmitted field

*T*can be found as [11

**73**, 026605 (2006). [CrossRef]

*R, S,*and

*T*are unchanged if

*n*→-

*n*; thus the sign of

*n*is irrelevant in Eq. (4). The time-domain field in the slab is computed by an inverse Laplace transform. The parameter

*γ*must be chosen such that the integration path is located above all poles of S in the upper half-plane [23]. For the present choice of medium [

*ε(ω)*=(1+

*f(ω)*)

^{2}and

*µ(ω)*=1

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative ε and µ,” Sov. Phys. Usp. **10(4)**, 509–514 (1968). [CrossRef]

*d*<∞, it can be shown that there are no such poles. Then we can set

*γ*=0

^{+}. We take the thickness to be

*ω*=35, which in this case means that

_{0}d/c*S*does not have poles in the upper half-plane. The resulting time-domain field is given in Fig. 4. We clearly see that the solution in Fig. 4 equals the solution in Fig. 2 when

*t<d/c*. Actually, the solutions are rather similar also for later times, which can be attributed to the fact that the field is small at

*z=d*. Again we note that the field dies out after the excitation has been turned off; thus the excitation of the amplified “backward” wave in Fig. 4 is fundamentally different to conventional lasing.

*n*

_{1}(

*ω*) is the correct refractive index function. The steady-state (monochromatic) solution, approached using the excitation

*u(t)*cos(

*ω*

_{1}

*t*) in the limit

*t*→∞, demonstrates that the phase velocity points towards the source. Since

*µ*=1, also the energy flow points towards the source. This backward wave draws energy from the active medium.

*φ*+

_{ε}*φ*is the complex argument of

_{µ}*εµ*, unwrapped so that it is continuous and tends to zero as ω→∞.

*ε*and

*µ*may not be continuous for real frequencies (the Kramers-Kronig relations only imply analyticity in the upper half-plane Imω>0, not for Imω=0). Moreover, εµ may contain odd-order zeros in the upper half-plane, which means that n cannot be identified as an analytic function there. Excluding the latter possibility (which corresponds to materials with so-called absolute instabilities [11

**73**, 026605 (2006). [CrossRef]

*ω*→∞. The refractive index for real frequencies is the limit as Im

*ω*→0

^{+}of this analytic branch. However, if

*εµ*contains only a single discontinuity or zero for real frequencies (at

*ω*=0), we can still use Eq. (5), unwrapping in the frequency interval (0,∞), and ensuring the limit

*φ*+

_{ε}*φ*→0 as

_{µ}*ω*→+∞.

*ε (ω)µ(ω)*(at all frequencies). In other words, two media with

*ε*=1-

*iα, α*>0 at a single frequency may have different refractive indices there. Thus the treatment of the example medium above does not contradict the fact that conventional gain media, with e.g. inverted Lorentzian responses, have positive real part of the refractive index. We note that any direct identification of

*n*from

*ε*and

*µ*at a single frequency, such as in Refs. [13

13. T. G. Mackay and A. Lakhtakia, “Comment on “Negative refraction at optical frequencies in nonmagnetic two-component molecular media”,” Phys. Rev. Lett. **96(15)**, 159701 (2006). [CrossRef]

15. S. A. Ramakrishna, “Comment on “Negative refraction at optical frequencies in nonmagnetic two-component molecular media”,” Phys. Rev. Lett. **98(5)**, 059701 (2007). [CrossRef]

17. S. A. Ramakrishna and O. J. F. Martin, “Resolving the wave vector in negative refractive index media,” Opt. Lett. **30**, 2626 (2005). [CrossRef] [PubMed]

18. A. N. Grigorenko, “Negative refractive index in artificial metamaterials,” Opt. Lett. **31**, 2483 (2006). [CrossRef] [PubMed]

*ε*and

*µ*describe a microscopically causal medium (i.e., the polarization and magnetization do not precede the electric and magnetic fields), an electromagnetic excitation cannot give any response before the time

*z/c*, where

*z*is the distance between the excitation point and the observation point. Thus, one can actually verify Fig. 2 by solving Maxwell’s equations in the time-domain, with no extra assumptions.

## 3. Effective medium based on a transmission line structure

**95(6)**, 067402 (2005). [CrossRef]

**73**, 026605 (2006). [CrossRef]

5. 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(12)**, 2702–2712 (2002). [CrossRef]

_{0}/30, where λ

_{0}is the vacuum wavelength [24

24. T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B **71(24)**, 245105 (2005). [CrossRef]

*εε*

_{0}[F/m] and an effective magnetic permeability

*µµ*

_{0}[H/m], where

*ε*

_{0}and

*µ*

_{0}are the vacuum permittivity and permeability. Consider a transmission line as shown in Fig. 5, with series impedance per length unit

*Z*, and shunt admittance per length unit

*Y*. The telegrapher’s equation for a general 1-D transmission line can be expressed as:

*β*is the propagation constant. Mapping the voltage

*V*to

*E*and the current

_{y}*I*to -

*H*[5

_{x}5. 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(12)**, 2702–2712 (2002). [CrossRef]

*ε*and

*µ*; thus the transmission line becomes left-handed. Since our goal is a right-handed transmission line medium with

*µ*=1, we will leave the impedance inductive,

*Z*=-

*iωµ*

_{0}, and rather change the admittance

*Y*so that

*ε*becomes a sum of one passive and one active resonance [10

**95(6)**, 067402 (2005). [CrossRef]

*C*, and

_{l}*L*are the length, capacitance, and inductance for a section of conventional transmission line;

_{l}*L*and

_{p}, C_{p},*R*are the inductance, capacitance and resistance of the passive resonator;

_{p}*L*and

_{a}, C_{a},*R*are the inductance, capacitance and resistance of the parallel resonator part of the circuit;

_{a}*R*is the negative resistor element which provides gain. Negative resistance can be realized in several ways [26

_{n}26. J.-S. Lee and Y.-S. Kwon, “Negative resistance circuit for monolithic resonators using gate-to-source resistive feedback,” Electron. Lett. **34(18)**, 1758–1760 (1998). [CrossRef]

27. L. O. Chua, J. Yu, and Y. Yu, “Bipolar-JFET-MOSFET negative resistance devices,” IEEE Trans. Circuits Syst. **31(1)**, 46–61 (1985). [CrossRef]

*L*is included to remove a zero of ε from the positive imaginary frequency axis.

_{n}*ε*does not have any odd-order zeros in the upper half plane, as this would create branch cuts in

*n*. When there are such branch cuts, the refractive index loses its usual interpretation for real frequencies; moreover, the media are electromagnetically unstable (that is, with absolute instabilities, see [11

**73**, 026605 (2006). [CrossRef]

*L*=

_{l}*µ*

_{0}Λ,

*C*=

_{l}*ε*

_{0}Λ,

*R*=-300Ω,

_{n}*L*=20Ω,

_{n}=L_{l}, R_{p}*L*=40nH,

_{p}*C*=2pF,

_{p}*R*=310Ω,

_{a}*L*=1.8nH, and

_{a}*C*=10pF. Figure 7 shows the refractive index versus frequency calculated by Eq. (5). Figure 8 shows the reflection coefficient calculated by Eq. (4a) using

_{a}*d*=20Λ. Negative real part of the refractive index exists in a small bandwidth from 1.11GHz to 1.30GHz, with the minimum value Re

*n*=-0.48 at 1.20GHz. A maximum 2.1 of the reflection coefficient is attained at a slightly larger frequency. At 1.22GHz the permittivity has positive real part,

*ε*=0.0063-0.23

*i*, corresponding to

*n*=-0.34+0.33i and |

*R*|=1.9. The bandwidth where Re

*n*<0 coincides with the bandwidth where |

*R*|>1.

## 4. Periodic structure

*Y*, the transmission matrix is

_{s}*T*

_{uc}=

*T*

_{tl}

*T*

_{s}, and the transmission matrix for

*N*unit cells becomes

*T*

_{uc}=1, the Nth power of

*T*

_{uc}can be calculated analytically with Chebyshev’s identity [29].

*T*can now be converted into a scattering matrix [28]

*S*

_{11}is the reflection coefficient and

*S*

_{21}is the transmission coefficient. Note that

*S*is symmetric due to reciprocity.

*N*→∞ while

*d*≡

*N*Λ and

*Y*/Λ are fixed, the transmission matrix model should correspond exactly to the effective medium theory. Indeed, we may expand

_{s}*T*

_{uc}to first order in Λ to obtain

*T*

_{uc}=I+γΛ+O(Λ

^{2})=exp(γΛ)+O(Λ

^{2}), where

*I*is the identity matrix and

*T*=exp(γ

*d*), and analytical expressions for the scattering parameters

*S*

_{11}and

*S*

_{21}can be found and verified to be equal to Eqs. (4a) and (4d).

*V*

^{+}(

*ω*). The reflected voltage is

*V*-(

*ω*)=

*V*

^{+}(

*ω*)

*S*

_{11}(

*ω*), and the voltage and current at the material interface are calculated as

*V*

_{1}(

*ω*)=

*V*

^{+}(

*ω*)+

*V*

^{-}(

*ω*) and

*I*

_{1}(

*ω*)=(

*V*

^{+}(

*ω*)-

*V*

^{-}(

*ω*))/

*Z*

_{0}, respectively. Then the voltage and current in the next unit cell are given by

*T*

_{uc}=1,

*T*

_{uc}is invertible.) By standard inverse Laplace transforms, we finally find the time-domain fields at each unit cell of the structure.

*S*

_{11}for the periodic structure and the reflection coefficient

*R*for the effective medium slab. The length scale Λ is varied while

*Y*/Λ and

*d*=Λ

*N*are fixed. To enable fair comparison, the reference plane of the periodic structure is shifted by an amountΛ/2 to the right, i.e., the periodic structure is symmetrized with respect to the point

*z=d*/2. We see that there is a good agreement between the two calculations for Λ=λ

_{0}/30 and even better when Λ=λ

_{0}/50, as predicted by [24

24. T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B **71(24)**, 245105 (2005). [CrossRef]

*f*

_{1}=1.22GHz. At this frequency the effective medium has

*ε*=0.063-0.23

*i*. The excitation, cos(2

*πf*

_{1}

*t*), is initiated at

*t*=0 and shut off at

*t*=20ns. We observe that the forerunner propagates straight through the material at the vacuum light velocity. The backward wave builds up before the reflection from the far end has returned to the region near

*z*=0

^{+}. After 20ns the excitation is turned off. Then the associated transient propagates at

*c*, and the backward wave dies out, demonstrating stability. This behavior is consistent with the example given in Sec. 2. We stress that we have analyzed a discrete lumped model with finite

*d*; the scattering parameters are calculated by the transmission matrix method so that the refractive index is not considered.

## 5. Conclusion

## References and links

1. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative ε and µ,” Sov. Phys. Usp. |

2. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys.: Condens. Matter |

3. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. |

4. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

5. | 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. |

6. | S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B |

7. | M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. |

8. | A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index meta-materials by optical parametric amplification,” Opt. Lett. |

9. | V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics |

10. | Y.-F. Chen, P. Fischer, and F.W. Wise, “Negative refraction at optical frequencies in nonmagnetic two-component molecular media,” Phys. Rev. Lett. |

11. | J. Skaar, “Fresnel equations and the refractive index of active media,” Phys. Rev. E |

12. | Y.-F. Chen, P. Fischer, and F. W. Wise, “Sign of the refractive index in a gain medium with negative permittivity and permeability,” J. Opt. Soc. Am. B |

13. | T. G. Mackay and A. Lakhtakia, “Comment on “Negative refraction at optical frequencies in nonmagnetic two-component molecular media”,” Phys. Rev. Lett. |

14. | Y.-F. Chen, P. Fischer, and F.W. Wise, “Chen, Fischer, and Wise reply,” Phys. Rev. Lett. |

15. | S. A. Ramakrishna, “Comment on “Negative refraction at optical frequencies in nonmagnetic two-component molecular media”,” Phys. Rev. Lett. |

16. | Y.-F. Chen, P. Fischer, and F.W. Wise, “Chen, Fischer, and Wise reply,” Phys. Rev. Lett. |

17. | S. A. Ramakrishna and O. J. F. Martin, “Resolving the wave vector in negative refractive index media,” Opt. Lett. |

18. | A. N. Grigorenko, “Negative refractive index in artificial metamaterials,” Opt. Lett. |

19. | J. Skaar, “On resolving the refractive index and the wave vector,” Opt. Lett. |

20. | L. Brillouin, |

21. | L. D. Landau and E. M. Lifshits, |

22. | In general, we can set γ=0+ provided the refractive index is analytic in the upper half-plane and the denominator in the Fresnel equations is nonzero in the upper half-plane. Excluding media with absolute instabilities, the refractive index can always be identified as an analytic function in the upper half-plane. |

23. | Poles in the upper half-plane mean that the slab will start lasing. |

24. | T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials,” Phys. Rev. B |

25. | B. Nistad and J. Skaar, in |

26. | J.-S. Lee and Y.-S. Kwon, “Negative resistance circuit for monolithic resonators using gate-to-source resistive feedback,” Electron. Lett. |

27. | L. O. Chua, J. Yu, and Y. Yu, “Bipolar-JFET-MOSFET negative resistance devices,” IEEE Trans. Circuits Syst. |

28. | D. M. Pozar, |

29. | M. Born and E. Wolf, |

**OCIS Codes**

(160.4670) Materials : Optical materials

(160.4760) Materials : Optical properties

(260.2030) Physical optics : Dispersion

(260.2110) Physical optics : Electromagnetic optics

(350.4010) Other areas of optics : Microwaves

(260.2065) Physical optics : Effective medium theory

**ToC Category:**

Metamaterials

**History**

Original Manuscript: July 9, 2007

Revised Manuscript: August 14, 2007

Manuscript Accepted: August 14, 2007

Published: August 15, 2007

**Citation**

Bertil Nistad and Johannes Skaar, "Simulations and realizations of active right-handed metamaterials with negative refractive index," Opt. Express **15**, 10935-10946 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-17-10935

Sort: Year | Journal | Reset

### References

- V. G. Veselago, "The electrodynamics of substances with simultaneously negative ε and μ," Sov. Phys. Usp. 10(4), 509-514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Low frequency plasmons in thin-wire structures," J. Phys.: Condens. Matter 10(22), 4785-4809 (1998). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, andW. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47(11), 2075-2084 (1999). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84(18), 4184-4187 (2000). [CrossRef]
- 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(12), 2702-2712 (2002). [CrossRef]
- S. A. Ramakrishna and J. B. Pendry, "Removal of absorption and increase in resolution in a near-field lens via optical gain," Phys. Rev. B 67(20), 201101 (2003). [CrossRef]
- M. A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C. E. Small, B. A. Ritzo, V. P. Drachev, and V. M. Shalaev, "Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium," Opt. Lett. 31, 3022 (2006). [CrossRef] [PubMed]
- A. K. Popov and V. M. Shalaev, "Compensating losses in negative-index metamaterials by optical parametric amplification," Opt. Lett. 31, 2169 (2006). [CrossRef] [PubMed]
- V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007). [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(6), 067402 (2005). [CrossRef]
- J. Skaar, "Fresnel equations and the refractive index of active media," Phys. Rev. E 73, 026605 (2006). [CrossRef]
- Y.-F. Chen, P. Fischer, and F. W. Wise, "Sign of the refractive index in a gain medium with negative permittivity and permeability," J. Opt. Soc. Am. B 23, 45-50 (2006). [CrossRef]
- T. G. Mackay and A. Lakhtakia, "Comment on "Negative refraction at optical frequencies in nonmagnetic twocomponent molecular media," Phys. Rev. Lett. 96(15), 159701 (2006). [CrossRef]
- Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 96, 159702 (2006). [CrossRef]
- S. A. Ramakrishna, "Comment on "Negative refraction at optical frequencies in nonmagnetic two-component molecular media"," Phys. Rev. Lett. 98(5), 059701 (2007). [CrossRef]
- Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 98(5), 059702 (2007). [CrossRef]
- S. A. Ramakrishna and O. J. F. Martin, "Resolving the wave vector in negative refractive index media," Opt. Lett. 30, 2626 (2005). [CrossRef] [PubMed]
- A. N. Grigorenko, "Negative refractive index in artificial metamaterials," Opt. Lett. 31, 2483 (2006). [CrossRef] [PubMed]
- J. Skaar, "On resolving the refractive index and the wave vector," Opt. Lett. 31, 3372 (2006). [CrossRef] [PubMed]
- L. Brillouin, Wave propagation and group velocity (Academic Press, New York and London, 1960).
- L. D. Landau and E. M. Lifshits, Electrodynamics of continuous media, chap. IX.62 (Pergamon Press, 1960).
- In general, we can set © =0+ provided the refractive index is analytic in the upper half-plane and the denominator in the Fresnel equations is nonzero in the upper half-plane. Excluding media with absolute instabilities, the refractive index can always be identified as an analytic function in the upper half-plane.
- Poles in the upper half-plane mean that the slab will start lasing.
- T. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, "Impact of inherent periodic structure on effective medium description of left-handed and related metamaterials," Phys. Rev. B 71(24), 245105 (2005). [CrossRef]
- B. Nistad and J. Skaar, in Photonic metamaterials: From random to periodic, V. M. Shalaev and A. Genack, eds. (OSA, 2006). ISBN:1-55752-808-X.
- J.-S. Lee and Y.-S. Kwon, "Negative resistance circuit for monolithic resonators using gate-to-source resistive feedback," Electron. Lett. 34(18), 1758-1760 (1998). [CrossRef]
- L. O. Chua, J. Yu, and Y. Yu, "Bipolar-JFET-MOSFET negative resistance devices," IEEE Trans. Circuits Syst. 31(1), 46-61 (1985). [CrossRef]
- D. M. Pozar, Microwave engineering, chap. 4, 2nd ed. (Wiley, 1998).
- M. Born and E. Wolf, Principles of Optics, chap. 1.6.5, pp. 66-67, 3rd ed. (Pergamon Press, 1965).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Multimedia

Multimedia Files | Recommended Software |

» Media 1: MOV (2577 KB) | |

» Media 2: MOV (1508 KB) | |

» Media 3: MOV (3073 KB) | |

» Media 4: MOV (3490 KB) |

« Previous Article | Next Article »

OSA is a member of CrossRef.