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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 24 — Nov. 24, 2008
  • pp: 19785–19798
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Toy model for plasmonic metamaterial resonances coupled to two-level system gain

Martin Wegener, Juan Luis García-Pomar, Costas M. Soukoulis, Nina Meinzer, Matthias Ruther, and Stefan Linden  »View Author Affiliations


Optics Express, Vol. 16, Issue 24, pp. 19785-19798 (2008)
http://dx.doi.org/10.1364/OE.16.019785


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Abstract

We propose, solve, and discuss a simple model for a metamaterial incorporating optical gain: A single bosonic resonance is coupled to a fermionic (inverted) two-level-system resonance via local-field interactions. For given steady-state inversion, this model can be solved analytically, revealing a rich variety of (Fano) absorption/gain lineshapes. We also give an analytic expression for the fixed inversion resulting from gain pinning under steady-state conditions. Furthermore, the dynamic response of the “lasing SPASER”, i.e., its relaxation oscillations, can be obtained by simple numerical calculations within the same model. As a result, this toy model can be viewed as the near-field-optical counterpart of the usual LASER rate equations.

© 2008 Optical Society of America

1. Introduction

Reducing or compensating the large intrinsic loss of metal-based metamaterials operating at optical frequencies is one of the major challenges in the emerging field of photonic metamaterials [1

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

3

3. K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007). [CrossRef]

]. If this potential show-stopper could be eliminated, many of the envisioned applications such as perfect lenses [4

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

] or cloaking [5

5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

,6

6. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. 1, 224–227 (2008). [CrossRef]

] at optical frequencies might actually come into reach. For example, the best fabricated negative-index metamaterial structures operating at around 1.4-µm wavelength [7

7. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “A low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006). [CrossRef] [PubMed]

] have shown a figure of merit of FOM=3 (modulus of real to imaginary part of the refractive index). This experimental result can be translated into an absolute absorption coefficient of α=3×104 cm-1=3 µm-1 — which is even larger than the band-to-band absorption of typical direct-gap semiconductors such as, e.g., GaAs (there, α≈104 cm-1). At first sight, this level of absorption looks quite depressing as room-temperature steady-state gain coefficients g=-α of this magnitude are not easily achieved at all.

Fig. 1. (a) Illustration of a possible geometry for bringing the optical gain from a thin semiconductor film (bulk, quantum wells, or quantum dots) close to an array of, e.g., plasmonic split-ring resonators. This geometry aims at taking advantage of local-field-enhancement effects. The electric-field vector of the incident light lies in the layer plane. (b) Schematic illustration of the toy model for a plasmonic (bosonic) metamaterial resonance coupled to a (fermionic) two-level-system gain resonance via local-field interactions. This interaction is described by the phenomenological (Lorentz) parameter L.

However, an interesting recent theoretical publication by Zheludev et al. [8

8. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Coherent metamaterials and the lasing spaser,” Nature Photon. 2, 351–354 (2008). [CrossRef]

] — based on the concept of the SPASER (surface plasmon amplification by stimulated emission of radiation) introduced by Stockman et al. [9

9. D. J. Bergman and M. I. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett. 90, 027402:1–4 (2003). [CrossRef]

,10

10. M. I. Stockman, “Spasers explained,” Nature Photon. 2, 327–329 (2008). [CrossRef]

] in 2003 — has essentially shown that it is not the bulk gain coefficient that matters but rather the effective gain coefficient of the combined system. Due to pronounced local-field enhancement effects in the spatial vicinity of the periodic metallic nanostructure, the effective gain coefficient can be substantially larger than its bulk counterpart. As an extended planar periodic arrangement of identical plasmonic structures that oscillate in phase due to mutual coupling and that exhibit a sub-wavelength period clearly leads to coherent plane-wave emission of light normal to the plane (in close analogy to phased antenna arrays), this two-dimensional active metamaterial structure has been named the “lasing SPASER” [8

8. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Coherent metamaterials and the lasing spaser,” Nature Photon. 2, 351–354 (2008). [CrossRef]

]. We briefly mention that further theoretical investigations on particular SPASER [11

11. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007). [CrossRef] [PubMed]

,12

12. S.-W. Chang, C.-Y.A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008). [CrossRef] [PubMed]

] structures, on metamaterials including gain materials [13

13. T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-Index Metamaterials: Going Optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006). [CrossRef]

,14

14. A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007). [CrossRef]

], and on lasing SPASER structures [15

15. J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16, 6692–6716 (2008). [CrossRef] [PubMed]

] have recently been published. Other work regarding bringing gain to surface plasmons includes, e.g., [16

16. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12, 4072–4079 (2004). [CrossRef] [PubMed]

19

19. 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, 3022–3024 (2006). [CrossRef] [PubMed]

]. Furthermore, recently published experiments on active metal-dielectric nanocavities [20

20. M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2008). [CrossRef]

,21

21. C. Manolatou and F. Rana, “Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers,” IEEE J. Quantum Electron. 44, 435–447 (2008). [CrossRef]

] already come fairly close to the original SPASER [9

9. D. J. Bergman and M. I. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett. 90, 027402:1–4 (2003). [CrossRef]

] idea.

In the microscopic calculations by Zheludev et al. [8

8. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Coherent metamaterials and the lasing spaser,” Nature Photon. 2, 351–354 (2008). [CrossRef]

] based on numerical solutions of the three-dimensional vector Maxwell equations for specific (two-slit) split-ring-resonator nanostructures, the gain medium has been described by a constant (i.e., frequency-independent and emission-independent) negative imaginary part of the gain medium dielectric function. Clearly, it is interesting to investigate modifications of this result

• due to a more realistic frequency-dependent gain,

• due to gain pinning, and

• due to dynamic effects.

Complete microscopic self-consistent numerical calculations of this sort have not been published so far to the best of our knowledge. They may well be feasible, but are certainly rather demanding and will have to deal with the specifics of a particular design.

In this Letter, we rather propose a simple and general toy model based on a fermionic two-level-system resonance (representing the gain medium, e.g., an ensemble of densely spaced semiconductor quantum dots) and a single bosonic resonance (representing the plasmonic resonance of the metamaterial) — connected to each other via a local-field coupling analogous to the local-field Lorentz factor. Interestingly, our toy model can be solved analytically for (quasi-) steady-state conditions. Numerical solutions for the time-dependent case are also discussed.

2. Definition of the model

The physics of our toy model is graphically illustrated in Fig. 1. The key to the model are the local field (or evanescent-field) interactions, i.e., the local electric field of the two-level system (2LS) is given by the external electric field E

E(t)=E˜0cos(ωt)=E˜02(exp(iωt)+c.c.)
(1)

of the light plus a phenomenological constant, L, times the polarization of the bosonic mode, P pl, i.e.,

EE+LPpl.
(2)

Correspondingly, the local electric field of the bosonic mode is given by the external electric field plus constant L times the polarization of the 2LS, P 2LS, i.e.,

EE+LP2LS.
(3)

As the spatial separation between the gain film and the plasmonic nanostructures [see Fig. 1(a)] would be increased along the direction normal to the plane of the gain film, the coupling via the plasmonic evanescent field decreases and L is expected to approach zero. Note that we have omitted the self-interactions, which can generally also occur via the local fields. In linear optics, however, self-interaction merely renormalizes the effective resonance center frequency.

Mathematically, the equations of motion for a fermionic two-level system are the famous and well known optical Bloch equations [22

22. W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer, New York, 2002).

]. They can be arranged into the form

p˙2LS+(iΩ2LS+γ2LS)p2LS=i1d2LSE(12f),
(4)
f˙+Γ2LSf=i1(p2LS*d2LSEp2LSd2LS*E*).
(5)

Here p 2LS denotes the (dimensionless) complex transition amplitude, f=f 2LS the occupation probability of the upper level, d 2LS the dipole matrix element, E the total electric field, Ω 2LS the transition frequency, γ 2LS the damping (or transverse relaxation) rate, and Γ 2LS the population (or longitudinal) relaxation rate. The individual electric dipole moment is given by the product d 2LS p 2LS. Multiplying with the volume density of two-level systems, N 2LS, leads to the macroscopic 2LS polarization P 2LS, i.e., to P 2LS=N 2LS d 2LS p 2LS+c.c. The equations of motion for a corresponding single bosonic mode are strictly identical to Eq. (4) and Eq. (5), except that the factor (1-2f) in Eq. (4) has to be replaced by unity, i.e.,

p˙pl+(iΩpl+γpl)ppl=i1dplE.
(6)

Hence, the equation for the occupation f pl of the bosonic mode becomes irrelevant. In analogy to the 2LS, the macroscopic plasmonic polarization is given by P pl=N pl d pl p pl+c.c.

As discussed above, we assume that the coupling between the two resonances is governed by local-field effects. Under these conditions, we simply have to replace the electric fields E in Eq. (4) and Eq. (5) by the term given in Eq. (2) and the electric field E in Eq. (6) by Eq. (3).

We envision that the 2LS is pumped via additional energy levels of the system that are not explicitly accounted for in the 2LS model, resulting in an additional effective pump rate, Γ pump, of the 2LS on the right-hand side of Eq. (5).

To avoid confusion: Generally, plasmonic nanostructures can have both an electric-dipole and a magnetic-dipole response [1

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

3

3. K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007). [CrossRef]

]. Here, we have focused on the electric-dipole response (leading to the polarization P) as only that can couple directly to readily available gain media at optical frequencies. Thus, our model is not applicable directly to magnetic-dipole plasmonic resonances or to negative-index metamaterials. Yet, it may provide qualitative trends. Our model is applicable directly to, e.g., the electric-dipole (Mie) resonances of the “V” structures envisioned in Ref. 9, but not to the “dark” modes there.

Next, we discuss analytical and numerical solutions of the model defined so far.

3. Linear optical response for fixed occupation

First, we consider the case of a given (i.e., fixed) occupation f of the 2LS and calculate the linear optical response with respect to some probe light with electric field E. As long as the system does not exhibit effective stimulated emission, this assumption of fixed f is completely unproblematic. This situation can occur under steady-state pumping conditions indeed. In contrast, as soon as f is such that the system does reveal effective stimulated emission, the situation becomes unstable, hence, it cannot occur under steady-state conditions (also see sections 4 and 5). Still, such values of f can be meaningful under transient conditions, for example, for a probe pulse with electric field E that follows, with some delay, the excitation by a pump pulse at larger photon energies (via the pump rate Γ pump described above).

Under these conditions of fixed f, it is straightforward to solve the above coupled equations. Using the usual rotating-wave approximation (RWA) [17

17. I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B 70, 155416:1–6 (2004).

], we derive the transition amplitudes p=p̃ exp(-iωt)

p˜2LS=(12f)(1d2LSE˜02+VLS1dplE˜02(Ωplω)iγpl)(Ω2LSω)iγ2LS(12f)VplV2LS(Ωplω)iγpl,
(7)

and

p˜pl=1dplE˜02+Vplp˜2LS(Ωplω)iγpl.
(8)

Here, we have introduced the two resulting coupling frequencies

V2LS=1d2LSLNpldpl,
Vpl=1dplLN2LSd2LS,
(9)

describing the effective back-action of the plasmonic mode onto the 2LS and vice versa, respectively. Note that the numerator of the 2LS transition amplitude in Eq. (7) shows the anticipated resonant enhancement of its “oscillator strength”, hence of its absorption/gain, via the plasmonic resonance. Strictly speaking, this resonance indexed by “2LS” is a mixed mode composed of the 2LS and the plasmonic resonance. Only for zero coupling, this resonance becomes that of the 2LS. The denominator of Eq. (7) also contains a resonance. This aspect can be interpreted as an eigenfrequency and a damping that are effectively frequency dependent, hence, this aspect gives rise to non-trivial lineshapes. As p 2LS according to Eq. (7) also enters in the numerator of Eq. (8), the spectral dependence of p pl is quite complex as well. From this point on, one has two different options to arrive at optical spectra: via a transfer-matrix approach or via a Maxwell-Garnett effective-medium approach.

3.1. Transfer-Matrix approach

The system shown in Fig. 1 can be treated as composed of two films with thicknesses l 2LS and l pl for the 2LS gain and the plasmonic layer, respectively. The two corresponding dielectric functions and refractive indices result immediately from the two macroscopic polarizations P 2LS and P pl given above. Normal-incidence intensity transmittance T and reflectance R spectra can be computed from the well known transfer-matrix approach [26

26. E. Hecht, Optics (Addison Wesley, 1987)

] for layered films. In order to test the relevance of our toy model and in order to adjust the toy model parameters, especially the local-field parameter L, we compare such calculations with microscopic numerical calculations on the basis of the complete three-dimensional vector Maxwell equations. These calculations based on the commercially available finite-difference time-domain software package lumerical [27] are rather similar to those in Ref. 8, however, with a frequency-dependent two-level-system gain resonance (same parameters as for the toy model) and for plasmonic split-ring resonators (SRR) roughly similar to the ones discussed in Ref. 24. The SRR are l pl=20-nm thick and are arranged on a quadratic lattice with lattice constant a=300 nm. The SRR lateral dimensions can be seen from Fig. 1(a). The silver, the SRR are assumed to be composed of, is described by the Drude free-electron model with standard parameters, i.e., plasma frequency ω pl=1.32×1016 s-1 and collision frequency ω col=1.2×1014 s-1.

3.2. Maxwell-Garnett effective-medium approach

P=l2LSl2LS+lplP2LS+lpll2LS+lplPpl=ε0χE˜02exp(iωt)+c.c.
(10)

Equation (10) allows for calculating all relevant linear optical properties via the effective optical susceptibility χ, the effective electric permittivity ε=1+χ, and the effective complex refractive index nε 1/2. The sign of the root has to be chosen such that Re(n)≥0. As µ=1 here, we can equivalently say that the sign of the root has to be chosen such that the sign of Im(n) is the same as the sign of Im(χ). The formulas for the normal-incidence intensity transmittance T and reflectance R of a homogeneous slab with complex refractive index n (hence, as µ=1, complex impedance Z=Z0/n, with the vacuum impedance Z0) and thickness l=l 2LS+l pl can, e.g., easily be obtained from Eq. (5.119) and Eq. (5.120) of the review article [3

3. K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007). [CrossRef]

] or from optics textbooks (e.g., [26

26. E. Hecht, Optics (Addison Wesley, 1987)

]).

Fig. 2. Normal-incidence intensity transmittance T (black) and reflectance R (red) spectra. From top to bottom row (see schemes on the left-hand side): split-ring resonators only, gain film for f=0 only, both combined for f=0, and both combined for f=1. (a) Complete numerical finite-difference time-domain solutions of the three-dimensional vector Maxwell equations for the geometry depicted in Fig. 1. (b) Same for the transfer-matrix treatment of the toy model. (c) Same for the Maxwell-Garnett treatment of the toy model. Model parameters are: Ω 2LS=2π×200 THz, Ω pl=2π×200 THz, γ 2LS=7.53×1012 s-1, γ pl=29.5×1012 s-1, d 2LS=6.5×10-29 Cm, d pl=6.2×10-26 Cm, N 2LS=5.05×1023 m-3, N pl=5.56×1020 m-3, l 2LS=50 nm, l pl=20 nm, and L=3.3416×1010 m/F such that V 2LS=7.1×1011 s-1 and V pl=6.449×1014 s-1 result.
Fig. 3. (a) Toy model complex linear susceptibility χ, (b) complex refractive index n, and (c) absorption coefficient α within the Maxwell-Garnett treatment. The solid (dashed) curves are the results with (without) local field coupling L. Real (imaginary) parts of complex quantities are red (green). The two-level system occupation f increases from top to bottom row as indicated. Model parameters are identical to those in Fig. 2.

Examples for linear optical spectra according to the Maxwell-Garnett approach are given in column (c) of Fig. 2. It can be seen that the qualitative agreement with the transfer-matrix calculations shown in column (b) is very good. This is especially true for the lower two rows. The upper two rows of columns (c) and (b) are trivially identical (as they refer to only a single layer each). Furthermore, column (c) in Fig. 2 also nicely qualitatively agrees with the complete numerical calculations shown in column (a), although small quantitative deviations do arise. We can conclude that not only the local-field approximation but also the Maxwell-Garnett effective-medium approximation is well justified under these conditions.

Fig. 4. Readers interested in the numerous parameter combinations other than the few selected special cases shown in Figs. 2 and 3 may download a software free of cost [25], allowing to calculate all relevant quantities of our toy model within the Maxwell-Garnett approach. The surface of this software is depicted here. All ten model parameters can be adjusted and the spectra change in real time. Default parameters are those of Figs. 2 and 3 and f=1. All frequencies and dampings are normalized with respect to the fixed 2LS transition frequency Ω 2LS=2π×200 THz (corresponding to about 1.5-µm wavelength).

The corresponding Maxwell-Garnett effective parameters are depicted in Fig. 3. Here, the solid curves are the complete results. For the dashed curves, the local-field parameter is artificially set to L=0. The comparison of solid and dashed curves, respectively, allows for assessing the influence of the local-field coupling L. Generally, the absorption lineshape is Fano-like — an aspect that we have previously pointed out for coupled classical harmonic oscillators [23

23. M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, “Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs,” Phys. Rev. B 72, 115113:1–12 (2005).

]. The Fano character becomes especially obvious for f=0 and if one of the two resonances exhibits a much larger damping than the other one (not shown here, see Fig. 4). More importantly, Fig. 3 also shows the evolution of the absorption spectrum for increasing two-level-system upper-state occupation f=0, 0.50, 0.7426, and 1. As transparency of the 2LS at f=0.5 is approached, the avoided crossing obviously collapses. Increasing f further, a sharp feature develops that becomes infinitely sharp if the gain exactly compensates the loss. At this critical point (f=0.7426 for the parameters chosen in Figs. 2 and 3), we expect lasing (or “spasing” [9

9. D. J. Bergman and M. I. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett. 90, 027402:1–4 (2003). [CrossRef]

] or “lasing spasing” [8

8. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Coherent metamaterials and the lasing spaser,” Nature Photon. 2, 351–354 (2008). [CrossRef]

]). We will further investigate this critical value for f in the following section. Further increase of f again broadens the spectral features because the gain over-compensates the loss. It is important to note that no gain is observed at all in Fig. 3 in case that the local-field coupling is artificially switched off (dashed curves for L=0). This behavior is expected on the basis of the intuitive reasoning in section 1 in terms of the effective gain rather than the 2LS gain itself that matters. The discontinuities of Im(n) occur at zeroes of Re(n), where the sign of the root of the electric permittivity has to change in order to fulfill the condition Re(n)≥0.

Intuitively, one might have expected that the intensity transmittance T will exceed unity as soon as effective gain occurs (see above). This can be the case, but Figs. 2 and 3 show that this is not necessarily the case. At and around the critical point, the real part of the refractive index becomes very large compared to unity or close to zero, either of which leads to a large impedance mismatch with respect to vacuum that gives rise to a large intensity reflectance R. As a result, the transmittance T is well below unity and shows no obvious signs of gain at all. We can conclude that transmittance alone is not necessarily a good experimental observable at this point.

In Fig. 2(c) and Fig. 3, we have intentionally only shown few selected aspects of the rather rich behavior of the model. In order to allow for a broad overview, we provide the reader with a program free of cost [25

25. This software suitable for modern Windows compatible personal computers can be downloaded via www.aph.uni-karlsruhe.de/wegener/de/publikationen and then “Publications 2008”.

] that allows for playing with all parameters of the toy model in real time (see Fig. 4). Obviously, each of the two uncoupled resonances has four parameters (center frequency Ω, damping γ, dipole moment d, and density N). In addition, the important phenomenological local-field coupling parameter L and the two-level-system occupation f can be freely adjusted — altogether 10 parameters.

4. Steady-state gain and occupation pinning

While any value of the 2LS occupation f∈[0,1

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

] can occur under transient conditions, this is not true under steady-state conditions: If the coupled system exhibits a gain that exceeds the loss, stimulated emission will eventually reduce the two-level system upper-state occupation probability f until some steady-state value is reached — a phenomenon, which is well known as gain pinning in the context of a LASER [22

22. W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer, New York, 2002).

,28

28. W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser Physics (Springer, New York, 1994). [CrossRef]

]. Thus, values of f above that steady-state value must be treated with a grain of salt. Also see Ref. 29 for a corresponding discussion regarding constraints for negative-index metamaterials due to causality.

Mathematically, polarization envelopes which neither grow (“too much gain”) nor decay (“not enough gain”) obviously need to have constant envelope, i.e., the imaginary part of the corresponding eigenfrequency ω of the coupled system has to be strictly zero. Without external light field, i.e., for E=0 on the RHS of Eq. (2) and Eq. (3), and for constant pump rate Γ pump, the two complex-valued eigenfrequencies of the above coupled Eqs. (2)(6) are given by

ω=Ω2LS+Ωpl2iγ2LS+γpl2±(Ω2LSΩpl2iγ2LSγpl2)2+V2LSVpl(12f).
(11)

The behavior of ω in the complex frequency plane versus f (running from 0 to 1) according to Eq. (11) is illustrated in Fig. 5. Figure 5(a) corresponds to the degenerate case, i.e., to Ω 2LS=Ω pl. The other parameters are as in Figs. 2 and 3. In Fig. 5(b), the degeneracy is slightly lifted. In Figs. 5(c) and (d), the local-field coupling L is successively decreased. As a result, an eigenfrequency with zero imaginary part no longer occurs in (d) [see open circles in (a)–(c)]. Let us explicitly evaluate the occupation f corresponding to Im(ω)=0.

Zero imaginary part of Eq. (11) immediately translates into the general condition for the 2LS occupation f

γ2LS+γpl2=±Im((Ω2LSΩpl2iγ2LSγpl2)2+V2LSVpl(12f)).
(12)

If Eq. (12) is fulfilled, the denominator of Eq. (7) becomes strictly zero for the real frequency ω resulting from Eq. (11), hence the linear optical response according to Eq. (10) diverges at this point.

Case (a): Equation (12) has, e.g., a simple transparent special solution for the important degenerate case, i.e., for Ω 2LS=Ω pl. We obtain

f=12(1+γplγ2LSVplV2LS)12(1+γ2V2)[0,1].
(13)

For example, for the parameters of Fig. 2, Eq. (13) leads to f=0.7426 — which is why we have depicted this particular value in the third row of Fig. 3.

Recall that, according to the Pauli exclusion principle, the occupation f in Eq. (13) needs to be in the interval [0,1

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

]. Thus, an effective coupling frequency V (= geometric mean of the two couplings) smaller than the effective damping γ (= geometric mean of the two dampings) leads to values of f exceeding unity. In other words: We have found a critical threshold value for the strength of the effective local-field interaction V. For values below that critical value, no lasing (spasing) action can occur under steady-state conditions. From Eq. (11), the (real-valued) lasing SPASER frequency results as ω=Ω 2LS=Ω pl.

Case (b): Another simple special case of Eq. (12) results for identical damping rates, i.e., for γ=γ 2LS=γ pl, but finite detuning ΔΩ=Ω 2LS-Ω pl≠0. We derive

f=12(1+γ2+(ΔΩ2)2V2)[0,1].
(14)

Clearly, increasing the detuning ΔΩ increases the necessary 2LS occupation f, until eventually no physical solution f∈[0,1

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

] occurs any more for γ 2+(ΔΩ/2)2>V 2. The detuning simply reduces the optical gain that is accessible for the plasmonic resonance from the two-level system (and so does the damping). From Eq. (11), the (real-valued) lasing SPASER frequency results as ω=(Ω 2LS+Ω pl)/2.

Note that — in the entire reasoning of this section — we have deliberately and tacitly neglected any optical feedback due to back-reflections from sample interfaces or even from mirrors due to propagating electromagnetic waves (i.e., we have set E=0 on the RHS of Eq. (2) and Eq. (3)) because our aim has been to discuss the case of pure evanescent-field (or local-field) optical feedback. The same holds true for the following section. Experimentally, this ideal case could, in principle, be realized by appropriate anti-reflection coatings of the structure at the lasing wavelength. It should be clear, however, that there will be a continuous transition from an ideal SPASER to a usual LASER if the relative strength of the far-field optical feedback with respect to the near-field (local-field) optical feedback is increased continuously. Studying this transition is not subject of the present letter.

Fig. 5. (a) The two complex eigenfrequencies ω according to Eq. (11). The occupation factor f runs from 0 (no pumping) to 1 (complete inversion) along the direction indicated by the arrows. Model parameters are chosen as in Figs. 2 and 3. At the crossing with the real frequency axis, the imaginary part of ω becomes zero, corresponding to the only possible non-trivial stationary solution of the lasing SPASER (see open circle). The occupation f is pinned via this condition Eq. (12). The two eigenfrequencies without coupling, i.e., for L=0, are shown by the green filled circles. Note that these uncoupled complex eigenfrequencies do not depend on f at all. It is instructive to compare the complex eigenfrequencies shown here with the linear optical spectra shown in Figs. 2 and 3. (b) Same as (a) but Ω pl→ 0.99×Ω pl, (c) as (b) but LL/2, and (d) as (b) but LL/5. For the latter, stationary SPASER action is obviously no longer possible.

5. Lasing SPASER relaxation oscillations

Next, we discuss an example for the time-dependent behavior of our toy model [see Eqs. (2)(6)]. We consider a pump rate Γ pump=Γ 0(1-f). The Pauli blocking factor (1-f) acknowledges that the upper state of each two-level system cannot be occupied with more than one electron, i.e., it guarantees f∈[0,1

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

]. Γ 0 shall be zero until time t=0. Then it is switched on to a constant value. Also, we seed the polarizations with a tiny but finite value. This is necessary because our toy model does not contain any spontaneous emission whatsoever. As a result, the transition amplitudes would otherwise be strictly zero — no matter how much gain the system develops. Alternatively, one could use a weak external seed pulse of light E(t). None of these details is really important, the initial conditions just need to be non-zero.

Under these conditions, by inserting the local field according to Eq. (2) for E=0 on the RHS of Eq. (2), Eq. (5) in RWA becomes

f˙+Γ2LSf=i(p2LS*V2LSpplc.c.)+Γpump=Γstim+Γpump.
(15)

Here, we have abbreviated the effective rate of stimulated emission Γ stim. For example, for the degenerate case, the steady-state lasing threshold (where Γ stim=0) is reached for the pump rate Γ pump=Γ 2LS f, with the pinned 2LS occupation f according to Eq. (12). As usual, above this threshold pump rate, the effective rate of stimulated emission increases linearly with Γ pump, i.e., Γ stim=Γ pump-Γ 2LS f.

The square modulus of the effective polarization according to Eq. (10) shows a behavior closely similar to that of the effective rate of stimulated emission in Fig. 6 (hence it is not depicted here).

The damped oscillation scenario is just the counterpart of the relaxation oscillations that are well known from the usual LASER rate equations [22

22. W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer, New York, 2002).

,28

28. W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser Physics (Springer, New York, 1994). [CrossRef]

]. As usual, the relaxation oscillation frequency increases with increasing pumping level [22

22. W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer, New York, 2002).

,28

28. W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser Physics (Springer, New York, 1994). [CrossRef]

] — a trend that is also clearly visible in Fig. 6. This observation emphasizes that our toy model and the usual LASER rate equations share certain similarities. Section 3, however, has also shown aspects that do not occur in the LASER rate equations at all. Recall that the LASER rate equations address the time-dependent emission intensity but they do not address the spectral dependence of the optical response at all.

Fig. 6. Switch-on of the lasing SPASER within our toy model leading to pronounced rapid relaxation oscillations of the two-level system occupation f and the effective rate of stimulated emission Γ stim. The pump rate Γ pump=Γ 0(1-f) with constant Γ 0 after time t=0 is parameter. The model parameters are identical to those of Figs. 2 and 3, Γ 2LS=1010 s-1. The lasing SPASER frequency results as ω=Ω 2LS=Ω pl. (a) Γ 0=4×1010 s-1 (just slightly above threshold) and (b) Γ 0=6×1010 s-1.

6. Conclusions

In conclusion, we have proposed and analyzed a simple and intuitive model for a (bosonic) metamaterial resonance coupled to a (fermionic) two-level-system gain resonance via local-field (evanescent-field) interactions. Especially the presented analytic solutions might help experimentalists in getting a feeling for designing actual SPASER structures. The model contains one purely phenomenological parameter L that describes the strength of the local-field interaction. This parameter needs to be chosen through comparison with numerical calculations of the plasmonic nanostructure via the three-dimensional vector Maxwell equations. The choice of all other model parameters for a given particular configuration is straightforward. To allow the reader for playing with the altogether ten model parameters, we provide a corresponding software [25

25. This software suitable for modern Windows compatible personal computers can be downloaded via www.aph.uni-karlsruhe.de/wegener/de/publikationen and then “Publications 2008”.

]. Next, analytic results for steady-state gain/occupation pinning have been derived. Furthermore, we have presented numerical solutions for the time-dependent problem that exhibits the usual laser relaxation oscillations. The latter aspect shows that our toy model for the SPASER shares similarities with the well established LASER rate equations.

Clearly, our simple modeling leaves plenty of room for future improvements. For example, proper treatment of the semiconductor will require accounting for Coulomb interaction effects via the semiconductor Bloch equations [22

22. W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer, New York, 2002).

,28

28. W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser Physics (Springer, New York, 1994). [CrossRef]

]. Furthermore, self-consistent solutions of the material and the Maxwell equations should include a spatially inhomogeneous response of the gain material in the vicinity of the plasmonic nanostructure. Finally, if the two-dimensional array of coupled plasmonic nanostructures starts lasing from spontaneous emission (rather than being driven homogeneously and coherently as in our toy model or in Ref. 8), it is not clear under which conditions a homogeneous coherent solution of this complex nonlinear system will actually be stable. Regarding applications of the lasing SPASER it is obviously not desirable that the system breaks up into domains.

Broadly speaking, we have seen that one must not assume that gain can be added to a metamaterial just to reduce the losses and leave the metamaterial properties (e.g., magnetic permeability or negative refractive index) unaltered otherwise. Strong coupling to a gain resonance inherently and unavoidably changes the system, resulting in a new effective system with new effective properties that need to be evaluated.

Acknowledgments

We thank Hyatt Gibbs, Galina Khitrova, and Wolfgang Stolz for discussions. We acknowledge financial support provided by the Deutsche Forschungsgemeinschaft (DFG) and the State of Baden-Württemberg through the DFG-Center for Functional Nanostructures (CFN) within subproject A1.5. The project PHOME acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number 213390. Also, we acknowledge funding through the METAMAT project by the Bundesministerium für Bildung und Forschung (BMBF). The research of S.L. is further supported through a “Helmholtz-Hochschul-Nachwuchsgruppe” (VH-NG-232), the PhD education of N.M. and M.R. through the Karlsruhe School of Optics & Photonics (KSOP). J.L.G.-P. acknowledges support by the I3P-CSIC grant program.

References and links

1.

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

2.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007). [CrossRef] [PubMed]

3.

K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, “Periodic nanostructures for photonics,” Phys. Rep. 444, 101–202 (2007). [CrossRef]

4.

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

5.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

6.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nature Photon. 1, 224–227 (2008). [CrossRef]

7.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “A low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006). [CrossRef] [PubMed]

8.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Coherent metamaterials and the lasing spaser,” Nature Photon. 2, 351–354 (2008). [CrossRef]

9.

D. J. Bergman and M. I. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett. 90, 027402:1–4 (2003). [CrossRef]

10.

M. I. Stockman, “Spasers explained,” Nature Photon. 2, 327–329 (2008). [CrossRef]

11.

J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007). [CrossRef] [PubMed]

12.

S.-W. Chang, C.-Y.A. Ni, and S. L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 10580–10595 (2008). [CrossRef] [PubMed]

13.

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-Index Metamaterials: Going Optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006). [CrossRef]

14.

A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, “Active metamaterials: Sign of refractive index and gain-assisted dispersion management,” Appl. Phys. Lett. 91, 191103 (2007). [CrossRef]

15.

J. A. Gordon and R. W. Ziolkowski, “CNP optical metamaterials,” Opt. Express 16, 6692–6716 (2008). [CrossRef] [PubMed]

16.

P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12, 4072–4079 (2004). [CrossRef] [PubMed]

17.

I. Avrutsky, “Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain,” Phys. Rev. B 70, 155416:1–6 (2004).

18.

J. Seidel, S. Grafstroem, and L. Eng, “Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution,” Phys. Rev. Lett. 94, 177401:1–4 (2005). [CrossRef]

19.

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, 3022–3024 (2006). [CrossRef] [PubMed]

20.

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nature Photon. 1, 589–594 (2008). [CrossRef]

21.

C. Manolatou and F. Rana, “Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers,” IEEE J. Quantum Electron. 44, 435–447 (2008). [CrossRef]

22.

W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer, New York, 2002).

23.

M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, “Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs,” Phys. Rev. B 72, 115113:1–12 (2005).

24.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901:1–4 (2005). [CrossRef]

25.

This software suitable for modern Windows compatible personal computers can be downloaded via www.aph.uni-karlsruhe.de/wegener/de/publikationen and then “Publications 2008”.

26.

E. Hecht, Optics (Addison Wesley, 1987)

27.

www.lumerical.com

28.

W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser Physics (Springer, New York, 1994). [CrossRef]

29.

M. I. Stockman, “Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality,” Phys. Rev. Lett. 98, 177404:1–4 (2007). [CrossRef]

OCIS Codes
(160.4670) Materials : Optical materials
(260.5740) Physical optics : Resonance

ToC Category:
Metamaterials

History
Original Manuscript: September 2, 2008
Revised Manuscript: October 28, 2008
Manuscript Accepted: October 28, 2008
Published: November 14, 2008

Citation
Martin Wegener, Juan Luis García-Pomar, Costas M. Soukoulis, Nina Meinzer, Matthias Ruther, and Stefan Linden, "Toy model for plasmonic metamaterial resonances coupled to two-level system gain," Opt. Express 16, 19785-19798 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19785


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References

  1. V. M. Shalaev, "Optical negative-index metamaterials," Nature Photon. 1,41-48 (2007). [CrossRef]
  2. C. M. Soukoulis, S. Linden, and M. Wegener, "Negative refractive index at optical wavelengths," Science 315, 47-49 (2007). [CrossRef] [PubMed]
  3. K. Busch, G. von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007). [CrossRef]
  4. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314, 977-980 (2006). [CrossRef] [PubMed]
  6. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nature Photon. 1, 224-227 (2008). [CrossRef]
  7. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, "A low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006). [CrossRef] [PubMed]
  8. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, "Coherent metamaterials and the lasing spaser," Nature Photon. 2, 351-354 (2008). [CrossRef]
  9. D. J. Bergman and M. I. Stockman, "Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems," Phys. Rev. Lett. 90, 027402:1-4 (2003). [CrossRef]
  10. M. I. Stockman, "Spasers explained," Nature Photon. 2, 327-329 (2008). [CrossRef]
  11. J. A. Gordon and R. W. Ziolkowski, "The design and simulated performance of a coated nano-particle laser," Opt. Express 15, 2622-2653 (2007). [CrossRef] [PubMed]
  12. S.-W. Chang, C.-Y.A. Ni, and S. L. Chuang, "Theory for bowtie plasmonic nanolasers," Opt. Express 16, 10580-10595 (2008). [CrossRef] [PubMed]
  13. T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, "Negative-Index Metamaterials: Going Optical," IEEE J. Sel. Top. Quantum Electron. 12,1106-1115 (2006). [CrossRef]
  14. A. A. Govyadinov, V. A. Podolskiy, and M. A. Noginov, "Active metamaterials: Sign of refractive index and gain-assisted dispersion management," Appl. Phys. Lett. 91, 191103 (2007). [CrossRef]
  15. J. A. Gordon and R. W. Ziolkowski, "CNP optical metamaterials," Opt. Express 16, 6692-6716 (2008). [CrossRef] [PubMed]
  16. P. Nezhad, K. Tetz, and Y. Fainman, "Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides," Opt. Express 12, 4072-4079 (2004). [CrossRef] [PubMed]
  17. I. Avrutsky, "Surface plasmons at nanoscale relief gratings between a metal and a dielectric medium with optical gain," Phys. Rev. B 70, 155416:1-6 (2004).
  18. J. Seidel, S. Grafstroem, and L. Eng, "Stimulated Emission of Surface Plasmons at the Interface between a Silver Film and an Optically Pumped Dye Solution," Phys. Rev. Lett. 94, 177401:1-4 (2005). [CrossRef]
  19. 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, 3022-3024 (2006). [CrossRef] [PubMed]
  20. M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, "Lasing in metallic-coated nanocavities," Nature Photon. 1, 589-594 (2008). [CrossRef]
  21. C. Manolatou and F. Rana, "Subwavelength Nanopatch Cavities for Semiconductor Plasmon Lasers," IEEE J. Quantum Electron. 44, 435-447 (2008). [CrossRef]
  22. W. Schäfer and M. Wegener, Semiconductor Optics and Transport Phenomena (Springer, New York, 2002).
  23. M. W. Klein, T. Tritschler, M. Wegener, and S. Linden, "Lineshape of harmonic generation on metal nanoparticles and metallic Photonic Crystal slabs," Phys. Rev. B 72, 115113:1-12 (2005).
  24. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901:1-4 (2005). [CrossRef]
  25. This software suitable for modern Windows compatible personal computers can be downloaded via www.aph.uni-karlsruhe.de/wegener/de/publikationen and then "Publications 2008".
  26. E. Hecht, Optics (Addison Wesley, 1987)
  27. www.lumerical.com
  28. W. W. Chow, S. W. Koch, and M. SargentIII, Semiconductor-Laser Physics (Springer, New York, 1994). [CrossRef]
  29. M. I. Stockman, "Criterion for Negative Refraction with Low Optical Losses from a Fundamental Principle of Causality," Phys. Rev. Lett. 98,177404:1-4 (2007). [CrossRef]

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