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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 17 — Aug. 20, 2007
  • pp: 10935–10946
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Simulations and realizations of active right-handed metamaterials with negative refractive index

Bertil Nistad and Johannes Skaar  »View Author Affiliations


Optics Express, Vol. 15, Issue 17, pp. 10935-10946 (2007)
http://dx.doi.org/10.1364/OE.15.010935


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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

In 1968 Veselago predicted that isotropic materials with simultaneously negative permittivity 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]

]. Veselago referred to such materials as left-handed due to the fact that the electric field, the magnetic field, and the wave vector constitute a left-handed set of vectors. Thus, the phase velocity and Poynting vector point in opposite directions, in contrast to the situation in right-handed materials. Left-handed materials do not exist naturally; however, it is possible to make artificial metamaterials with negative 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

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

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

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]

].

Passive metamaterials often suffer from limited performance due to large losses. It has therefore been suggested to combine them with active gain media [6

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]

], or even create active metamaterials [7

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]

]. Nonlinear processes such as parametric amplification have also been suggested as a method for overcoming inherent losses in metamaterials [8

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]

]. As the interest in active metamaterials seem to be increasing [9

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

], it is crucial to understand fundamentally how they behave electromagnetically.

Of particular interest are the sign of the refractive index (direction of phase velocity), and the direction of the energy flow. The sign of the refractive index depends on the location of zeros and poles of 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]

]. In such right-handed materials the wave vector and Poynting vector both point towards the source [11

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

].

Recently, these results have been discussed in the literature [11

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

, 12

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]

, 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]

, 14

14. Y.-F. Chen, P. Fischer, and F.W. Wise, “Chen, Fischer, and Wise reply,” Phys. Rev. Lett. 96, 159702 (2006). [CrossRef]

, 15

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]

, 16

16. Y.-F. Chen, P. Fischer, and F.W. Wise, “Chen, Fischer, and Wise reply,” Phys. Rev. Lett. 98(5), 059702 (2007). [CrossRef]

]. While the conclusions differ, all authors seem to argue that causality must determine the sign of the refractive index. However, the interpretations of causality differ and in fact, give different results for certain active media [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

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

, 12

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]

, 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]

, 14

14. Y.-F. Chen, P. Fischer, and F.W. Wise, “Chen, Fischer, and Wise reply,” Phys. Rev. Lett. 96, 159702 (2006). [CrossRef]

, 15

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]

, 16

16. Y.-F. Chen, P. Fischer, and F.W. Wise, “Chen, Fischer, and Wise reply,” Phys. Rev. Lett. 98(5), 059702 (2007). [CrossRef]

, 17

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

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

, 19

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

]. To identify the correct interpretation and resolve this controversy, it is necessary to go back to first principles: Causality means that if an excitation starts at time t=0, no effect can be detected a distance z away before t=z/c. In other words, the front of an electromagnetic wave cannot travel faster than c, the vacuum velocity of light [20

20. L. Brillouin, Wave propagation and group velocity (Academic Press, New York and London, 1960).

]. Using this principle, it has been shown analytically how to determine the sign of the refractive index in active media [11

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

].

In this work, we will first present animations of numerical simulations, to demonstrate negative refractive index in right-handed, active media [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

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

]. The electromagnetic solution in a slab of thickness 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.

To concretize this class of media, we propose a 1-D transmission line model with lumped circuit elements, implementing a right-handed medium with negative refractive index. This model can be used as a starting point for analysis and possible implementation. Although we consider a one-dimensional structure, generalization to two dimensions is straightforward.

2. Determining the sign of the refractive index

When a plane wave is normally incident from vacuum to a semi-infinite medium, the reflection R and transmission S are given by the Fresnel equations:

R=η1η+1,
(1a)
S=2ηη+1exp(iωnzc).
(1b)

Here k=ωn/c, η=µ/n, and n2=εµ; ε and 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(ω)=Fω02ω02ω2iωΓ.
(2)

We take F=2 and Γ=0.005ω 0. This material is causal and realizable as ε is analytic in the upper half-plane of complex frequency and ε(ω)-1~-2 2 0/ω 2 as ω→∞ [21

21. L. D. Landau and E. M. Lifshits, Electrodynamics of continuous media, chap. IX.62 (Pergamon Press, 1960).

]. Using the method of Refs. [20

20. L. Brillouin, Wave propagation and group velocity (Academic Press, New York and London, 1960).

, 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

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

, 12

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

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

], we find the refractive index 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

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

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

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

], we find the refractive index 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=1.73ω 0, where n 2(ω) is discontinuous.

To resolve the above controversy, we first compute the time-domain solutions associated with the choices 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

E(z,t)=i2πiγiγ+ωSexp(iωt)ω12ω2dω.
(3)
Fig. 1. Refractive index vs. frequency for the two choices n 1(ω) and n 2(ω).

Here γ is a sufficiently large, positive number, such that Imω<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

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

, 22

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.

]. The excitation frequency is taken to be ω 1=1.4ω 0; at this frequency ε(ω 1)≈+1.13-0.32i. 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

20. L. Brillouin, Wave propagation and group velocity (Academic Press, New York and London, 1960).

]. 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.

Fig. 2. Numerical simulation of the time-domain electric field E(z,t) for a semi-infinite medium with refractive index n(ω)=n1(ω) and excitation frequency ω 1=1.4. The excitation, cos(ω1t), is initiated at t=0. After roughly a time 70/ω 0 the steady-state solution (monochromatic solution) has been built up. The excitation is turned off at t=100/ω 0. Frame grabbed at t=31.6/ω 0. [fig2.mov 2.5MB] [Media 1]
Fig. 3. Numerical simulation of E(z,t) of a semi-infinite medium with n(ω)=n 1(ω) and ω 1=1.4. Frame grabbed at t=0.3/ω 0. Note that fields exist in front of the red line z=ct. The figure does not have the same axes and time scale as for Fig. 2. [fig3.mov 1.5MB] [Media 2]

To see that 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

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

]

R=(η21)exp(ikd)(η21)exp(ikd)(η+1)2exp(ikd)(η1)2exp(ikd),
(4a)
S+=2η(η+1)(η+1)2(η1)2exp(2ikd),
(4b)
S=2η(η1)(η1)2(η+1)2exp(2ikd),
(4c)
T=4η(η+1)2exp(ikd)(η1)2exp(ikd).
(4d)

Note that 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

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

]. 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]

], by choosing a sufficiently large d<∞, it can be shown that there are no such poles. Then we can set γ=0+. We take the thickness to be ω0d/c=35, which in this case means that 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.

Fig. 4. Numerical simulation of E(z,t) for a medium with ε(ω)=(1+f(ω))2, µ(ω)=1, finite thickness ω0d/c=35, and ω 1=1.4ω 0. The excitation is initiated at t=0 and turned off at t=100/ω 0. The wave front z=ct is indicated with a vertical red line. Frame grabbed at t=31.6/ω 0. [fig4.mov 3MB] [Media 3]

We conclude that 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.

n=εμexp[i(φε+φμ)2],
(5)

where φε+φµ is the complex argument of εµ, unwrapped so that it is continuous and tends to zero as ω→∞.

More generally, 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

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

]), we find the refractive index from the analytic branch of εμ that tends to +1 as Imω→∞. The refractive index for real frequencies is the limit as Imω→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 φε+φµ →0 as ω→+∞.

The necessary phase unwrapping procedure means that the sign of the refractive index cannot be determined at a single frequency without knowing the global behavior of 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

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

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

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

], are incorrect in general.

Fig. 5. A general transmission line representation.

It may seem that we have used causality as an extra principle in addition to Maxwell’s equations, to calculate the field. However, causality is in fact built into the time-domain versions of the Maxwell equations: Provided 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

In 2002 Eleftheriades et. al. [5

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]

] proposed a negative index medium based on periodically L-C loaded transmission lines. As long as the periodicity of the structure remains less than roughly λ0/30, where λ0 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]

], the structure can be described electromagnetically by an effective dielectric permittivity 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:

dVdz=IZ,
(6a)
dIdz=VY.
(6b)

Combining Eqs. (6a) and (6b) yields

d2Vdz2+β2V=0,β2=ZY,
(7)

where β is the propagation constant. Mapping the voltage V to Ey and the current I to -Hx [5

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]

], Eqs. (6a) and (6b) can be rewritten as field equations:

dEydz=iωμμ0Hx,
(8a)
dHxdz=iωεμ0Ey,
(8b)

which yield the effective material parameters as

μ=Ziωμ0,
(9a)
ε=Yiωε0.
(9b)
Fig. 6. Transmission line with additional shunt admittance Ys.

The propagation constant satisfies

β2=ZY=ω2c2εμ.
(10)

It can easily be seen that interchanging the inductance and the capacitance of a conventional transmission line leads to negative 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

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]

, 25

25. 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.

]. An example circuit is given in Fig. 6, where the total admittance is

Y=iωCl1iωLniωLp1LpCpω2iωRpLp+Ra+RnRaRn1LaCaω2iω(Ra+Rn)Ca1LaCaω2iωRaCa.
(11)

Thus the effective relative permittivity and permeability become

ε(ω)=1ε0Λ[Cl1ω2Ln+1Lp1LpCpω2iωRpLpRa+RniωRaRn1LaCaω2iω(Ra+Rn)Ca1LaCaω2iωRaCa],
(12a)
μ(ω)=Llμ0Λ.
(12b)
Fig. 7. Refractive index vs. frequency for the transmission-line effective medium.

Here Λ, Cl, and Ll are the length, capacitance, and inductance for a section of conventional transmission line; Lp, Cp, and Rp are the inductance, capacitance and resistance of the passive resonator; La, Ca, and Ra are the inductance, capacitance and resistance of the parallel resonator part of the circuit; Rn is the negative resistor element which provides gain. Negative resistance can be realized in several ways [26

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

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]

]; transistors and Gunn-diodes are feasible alternatives. The parallel resonance has zero admittance at the resonance frequency. This acts together with the gain element to give negative admittance and negative imaginary part of the permittivity. The inductance Ln is included to remove a zero of ε from the positive imaginary frequency axis.

One must be careful when choosing the circuit parameters; it is not sufficient to have negative real part of the refractive index to obtain a medium similar to that in Section 2. It is important that 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

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

]).

We have found that the following parameters will give negative refractive index in a small frequency window: Λ=6.25cm, Ll=µ 0Λ, Cl=ε 0Λ, Rn=-300Ω, Ln=Ll, Rp=20Ω, Lp=40nH, Cp=2pF, Ra=310Ω, La=1.8nH, and Ca=10pF. Figure 7 shows the refractive index versus frequency calculated by Eq. (5). Figure 8 shows the reflection coefficient calculated by Eq. (4a) using 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=0.0063-0.23i, 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

The effective medium theory yields a useful design guide for realization of the effective parameters as a transmission line with discrete elements. To investigate such periodic structures exactly, we use the transmission matrix method. The transmission matrix of a transmission line can be written [28

28. D. M. Pozar, Microwave engineering, chap. 4, 2nd ed. (Wiley, 1998).

]

Ttl=[cos(β0Λ)isin(β0Λ)Z0isin(β0Λ)Z0cos(β0Λ)],
(13)

where Λ, β0=ωLlCl, and Z0=LlCl are the length, propagation constant, and characteristic impedance, respectively. For a general shunt admittance Ys, the transmission matrix is

Ts=[10Ys1].
(14)
Fig. 8. R vs. frequency for total length d=20Λ.

The transmission matrix for a unit cell is T uc=T tl T s, and the transmission matrix for N unit cells becomes

T=TucN=[ABCD].
(15)

Since detT uc=1, the Nth power of T uc can be calculated analytically with Chebyshev’s identity [29

29. M. Born and E. Wolf, Principles of Optics, chap. 1.6.5, pp. 66–67, 3rd ed. (Pergamon Press, 1965).

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

28. D. M. Pozar, Microwave engineering, chap. 4, 2nd ed. (Wiley, 1998).

]

S=[S11S12S21S22]=[A+BZ0CZ0DA+BZ0+CZ0+D2(ADBC)A+BZ0+CZ0+D2A+BZ0+CZ0+DA+BZ0CZ0+DA+BZ0+CZ0+D].
(16)

The scattering matrix element S 11 is the reflection coefficient and S 21 is the transmission coefficient. Note that S is symmetric due to reciprocity.

In the limit N→∞ while dNΛ and Ys/Λ are fixed, the transmission matrix model should correspond exactly to the effective medium theory. Indeed, we may expand T uc to first order in Λ to obtain T uc=I+γΛ+O(Λ2)=exp(γΛ)+O(Λ2), where I is the identity matrix and

ϒ=[0iβ0Z0iβ0Z0+YsΛ0].
(17)

Thus, in the above limit, T=exp(γd), 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).

[V2(ω)I2(ω)]=Tuc1[V1(ω)I1(ω)],
(18)

and so forth. (Since detT 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.

Fig. 9. |S 11-R| vs frequency for Λ=λ 0/8,λ0/16,λ0/30,λ0/60. λ0=25cm is the vacuum wavelength where |R| is maximal.

Figure 9 shows the difference between the scattering parameter 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 dN 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=λ0/30 and even better when Λ=λ0/50, 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]

]. Figure 10 shows the time-domain field when exciting the periodic structure with the frequency f 1=1.22GHz. At this frequency the effective medium has ε=0.063-0.23i. The excitation, cos(2πf 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.

The periodic circuit model is based on ideal passive components along with an ideal, constant negative resistor. For a more realistic model of the components it is possible to include resistive, capacitive and inductive parasitics into the components in use. Nevertheless, preliminary sensitivity studies show that the circuit is not particularly sensitive to perturbations. The implementation of the negative resistor element should yield a broad-banded response compared to the two resonances.

Fig. 10. Electrical field vs. distance and time, for a causal excitation [u(t)-u(t-20ns)]cos(2πf 1 t); f1=1.22GHz, d=150Λ, and Λ=0.83cm. Frame grabbed at t=3.1ns. [fig10.mov 3.4 MB] [Media 4]

5. Conclusion

We have presented theory and simulations which demonstrate how to determine the refractive index in active metamaterials. We have also proposed and analyzed a lumped circuit transmission line model for an active right-handed metamaterial with negative refractive index. This model demonstrates a backward propagating wave with phase velocity and Poynting’s vector towards the source, and can be used as a starting point for implementation.

References and links

1.

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

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]

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]

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]

14.

Y.-F. Chen, P. Fischer, and F.W. Wise, “Chen, Fischer, and Wise reply,” Phys. Rev. Lett. 96, 159702 (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]

16.

Y.-F. Chen, P. Fischer, and F.W. Wise, “Chen, Fischer, and Wise reply,” Phys. Rev. Lett. 98(5), 059702 (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]

19.

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

20.

L. Brillouin, Wave propagation and group velocity (Academic Press, New York and London, 1960).

21.

L. D. Landau and E. M. Lifshits, Electrodynamics of continuous media, chap. IX.62 (Pergamon Press, 1960).

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 71(24), 245105 (2005). [CrossRef]

25.

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.

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]

28.

D. M. Pozar, Microwave engineering, chap. 4, 2nd ed. (Wiley, 1998).

29.

M. Born and E. Wolf, Principles of Optics, chap. 1.6.5, pp. 66–67, 3rd ed. (Pergamon Press, 1965).

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


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References

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative ε and μ," Sov. Phys. Usp. 10(4), 509-514 (1968). [CrossRef]
  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, andW. 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 metamaterials 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]
  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]
  13. 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]
  14. Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 96, 159702 (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]
  16. Y.-F. Chen, P. Fischer, and F.W. Wise, "Chen, Fischer, and Wise reply," Phys. Rev. Lett. 98(5), 059702 (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]
  19. J. Skaar, "On resolving the refractive index and the wave vector," Opt. Lett. 31, 3372 (2006). [CrossRef] [PubMed]
  20. L. Brillouin, Wave propagation and group velocity (Academic Press, New York and London, 1960).
  21. L. D. Landau and E. M. Lifshits, Electrodynamics of continuous media, chap. IX.62 (Pergamon Press, 1960).
  22. 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.
  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 71(24), 245105 (2005). [CrossRef]
  25. 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.
  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]
  28. D. M. Pozar, Microwave engineering, chap. 4, 2nd ed. (Wiley, 1998).
  29. M. Born and E. Wolf, Principles of Optics, chap. 1.6.5, pp. 66-67, 3rd ed. (Pergamon Press, 1965).

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