## CNP optical metamaterials

Optics Express, Vol. 16, Issue 9, pp. 6692-6716 (2008)

http://dx.doi.org/10.1364/OE.16.006692

Acrobat PDF (945 KB)

### Abstract

Simulation results for optical metamaterials (MTMs) derived from active coated nano-particle (CNP) inclusions for operation in the visible range of the spectrum between 400nm and 700nm are presented. Several examples of optical MTMs designed with these inclusions are characterized, including two-dimensional (2D) CNP metafilms; three-dimensional (3D) periodic CNP arrays; and 3D random CNP distributions. The properties of these optical MTMs are explored using effective medium theories that are applicable to these inclusion configurations. The effective permittivities and refractive indexes of these optical MTMs are compared and contrasted to the scattering properties of their active CNP inclusions.

© 2008 Optical Society of America

## 1. Introduction

1. N. Engheta and R. W. Ziolkowski, “A positive future for double negative metamaterials,” IEEE Microwave Theory Tech. **53**, 1535–1556 (2005). [CrossRef]

1. N. Engheta and R. W. Ziolkowski, “A positive future for double negative metamaterials,” IEEE Microwave Theory Tech. **53**, 1535–1556 (2005). [CrossRef]

1. N. Engheta and R. W. Ziolkowski, “A positive future for double negative metamaterials,” IEEE Microwave Theory Tech. **53**, 1535–1556 (2005). [CrossRef]

4. R. W. Ziolkowski, “Metamaterial-based antennas: Research and developments,” IEICE Trans. Electron . **E89-C**, 1267–1275 (2006). [CrossRef]

5. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterials coatings,” Phys. Rev. E **72**, 016623 (2005). [CrossRef]

6. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (2006). [CrossRef] [PubMed]

10. R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B. **16**, 1824–1832 (1999). [CrossRef]

13. A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express **14**, 1557–1567 (2006). [CrossRef] [PubMed]

14. V. M. Shalaev and W. Cai, “Negative index of refraction in optical metamaterials,” Opt. Lett. **30**, 3356–3358 (2005). [CrossRef]

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

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

*and not a bulk material. Consequently, the claims that DNG or NIM optical MTMs have been achieved is a bit misleading. A better designation for these systems of inclusions would be a “metafilm”. In fact one of the major obstacles in achieving true optical MTMs is the alignment of multiple layers of these metafilms. Although there has been some very recent successes [17*

**film**17. N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Maters. **7**, 31–37 (2007). [CrossRef]

*MNG or DNG responses. Another issue that plagues MTMs at optical frequencies is the large losses inherent in the materials, typically metals, used to achieve the resonant electrically small inclusions. In this paper, both metafilms and bulk three dimensional optical MTMs comprised of spherically shaped, electrically small (highly sub-wavelength), resonant inclusions will be reported. The introduction of a gain medium into these spherical inclusions will be investigated as a means to combat the high losses that plague many current optical MTM realizations.*

**bulk**10. R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B. **16**, 1824–1832 (1999). [CrossRef]

10. R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B. **16**, 1824–1832 (1999). [CrossRef]

18. 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]

*active*dielectric sphere that is coated with a silver plasmonic shell and their use as inclusions to realize optical MTMs. The basic active CNP was studied extensively in [18

18. 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]

## 2. Optical properties of CNP inclusions

*μ*=1 for

_{i}*i*=1, 2. The plane wave is incident on the CNP from the surrounding free space region. The active CNP is represented by the layered sphere bounded by R

_{2}. Its core is defined by the outer radius R

_{1}and by the permittivity

*ε*

_{1}and permeability

*μ*

_{1}. The plasmonic shell surrounds the core and is defined by the permittivity

*ε*

_{2}and permeability

*μ*

_{2}and by the radii R

_{1}and R

_{2}of its inner and outer boundaries, respectively. The core permittivity in the absence of gain takes on the value for silica, SiO

_{2}, which is

*ε*

_{1}=2.05

*ε*

_{0}. In our investigation the simulations were performed assuming that the dimensions of the active CNP were R

_{1}=8nm and R

_{2}=10nm with a resonance peak positioned at the free space wavelength:

*λ*=491.2

_{res}*nm*.

*m⃑*(

*ρ,θ,ϕ*) and

*n⃑*(

*ρ,θ,ϕ*) [19], [20

20. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. **22**, 1242–1246 (1951). [CrossRef]

*a*and

_{n}*b*, is obtained and solved.

_{n}*I*, and can be expressed in terms of the scattered field coefficients as:

_{inc}*β*=2

*π*/

*λ*

_{0}, with

*λ*

_{0}is the free-space wavelength incident on the CNP. The corresponding scattering and absorption efficiencies are defined as the cross sections normalized by the geometric cross section of the particle.

18. 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]

*ω*and Γ are, respectively, the plasma and collision frequencies. The half width of the resonance

_{p}*A*~1. The term V

_{F}is the Fermi velocity.

**15**, 2622–2653 (2007). [CrossRef] [PubMed]

*n*and

*k*are, respectively, the real and imaginary parts of the refractive index. For optical gain the imaginary part of the refractive index

*k*takes on negative values, i.e., optical gain occurs when

*k*<0. In Figs. 2(a) and 2(b) the scattering and absorption resonances for the 10nm active silver CNP are shown near the SR wavelength,

*λ*=491.2

_{res}*nm*, where the gain term approaches its SR value:

*k*=-0.453. At

*λ*, this SR value of

_{res}*k*corresponds to a gain coefficient of

*α*=2

*πk*/

*λ*~-10

_{res}^{4}

*cm*

^{-1}. Recently, gain values as high as |

*α*|~10

^{5}

*cm*

^{-1}have been demonstrated in quantum dots [22

22. S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L.C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, “Self-Tuned quantum dot gain in photonic crystal lasers,” Phys. Rev. Lett. **96**, 127404 (2006). [CrossRef] [PubMed]

23. D. Bimberg, N. Kirstaedter, N. N. Ledentsov, Zh. I. Alferov, P. S. Kop’ev, and V. M. Ustinov, “InGaAs—GaAs quantum-dot lasers,” IEEE J. Sel. Top. Quantum Electron. **3**, 196–205 (1997). [CrossRef]

*a*and

_{n}*b*, respectively. Because of the spherical symmetry of the CNP, the scattering response is isotropic. Consequently, the electric and magnetic polarizabilities,

_{n}*α*and

_{E}*α*, of the CNP can be expressed as scalar quantities. The effective electric and magnetic dipole moments,

_{M}*p⃗*and

*m⃗*, of the CNP are then defined in terms of the corresponding polarizabilities and the corresponding local electric and magnetic fields,

*E⃗*and

_{loc}*H⃗*, which are acting on it, as:

_{loc}## 3. Two Dimensional CNP metafilms

24. E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans Antennas Propag. **51**, 2641–2651 (2003). [CrossRef]

*discrete*inclusion while properly taking into account the

*spatially*averaged electric and magnetic polarization densities of the sheet. For the quasi-static limit of the inclusions:

*βR*

_{2}→0, and for a square periodic array of period:

*d*, it has been shown by [25] that

*R*≃0.6956

*d*. After determining the macroscopic field for an array of inclusions embedded in free space, the GSTCs take the form,

*a⃗*,

_{x}*a⃗*,

_{y}*a⃗*are unit vectors, and the effective electric sheet polarizabilities per unit area,

_{z}*α*, are expressed in terms of the averaged particle polarizabilities, 〈

^{ij}_{ES}*α*〉, as

_{E,ij}*N*is the number of inclusions per unit area in the film. Similarly the effective sheet magnetic polarizabilities,

*α*, can be expressed in terms of the average particle magnetic polarizabilities,〈

^{ij}_{MS}*α*〉, as

_{M,ij}### 3. 1 Scattering from a CNP metafilm

*T*, and reflection, Γ, coefficients take the form:

*transmittance*,

*t*=|

*T*|

^{2}, and the

*reflectance*,

*r*=|Γ|

^{2}. Thus the amount of energy absorbed by the metafilm can be quantified by defining the

*absorptance*as,

*d*, between the CNPs in the square periodic metafilm, were varied. The 2D density plots in Figs. 4, 5, and 6 depict these gain variation results. The gain values were varied from the passive scenario:

*k*=0 to

*k*=-0.353, the latter being a significant value yet still below the SR value, through the SR value at

*k*=-0.453, to just above the SR value at

*k*=-0.463. The spacing values were varied over a range of 100nm, from 30 nm to 130nm.

*passive*CNPs becomes larger. One observes that the transmittance is reduced by as much as 30% near

*λ*≃491.2

_{res}*nm*, and the absorption increases. This increase in absorption can be attributed to the higher surface density of inclusions, each of which is inherently highly absorbing. As the CNPs are more densely packed in the passive metafilm, a larger fraction of the incident field is interacting with the inclusions. This increases the amount of energy that is removed from the incident field; and, consequently, the transmission is reduced and the absorption is increased.

*active*metafilm show significant changes as the gain and spacing are varied. As observed for the passive cases, there is little change in the relative transmittance and reflectance values as the separation distance is varied when the gain values are very small. However, as the gain is increased, a significant departure from the passive case behavior is observed. For particular values of the gain and the spacing between the inclusions, the transmittance decreases to zero near the CNP’s resonance wavelength

*λ*. On the other hand, the corresponding reflectance may attain values larger than those for the passive case and can even become larger than unity in some instances.

_{res}*d*≃51

*nm*, and the gain value

*k*=-0.463. In this configuration the transmittance is reduced completely to zero at

*λ*=495

*nm*, while the reflectance becomes unity. This behavior occurs just slightly past the CNP’s resonance wavelength. In Fig. 7 this behavior is compared explicitly to the passive case whose cross sections have been taken from the data presented in Figs. 4 and 5 for the separation distance

*d*≃51

*nm*. In this configuration the metafilm is essentially acting like a narrowband mirror, reflecting all incident radiation just slightly above the CNP’s resonance wavelength

*λ*≃491.2

_{res}*nm*. The active metafilm under these operating conditions can be thought of as acting like a plasmonic material with a plasma wavelength value that can be specifically designed via the geometry. In particular, the geometry of the CNP can be tuned to achieve localized plasmon resonances having different resonant wavelengths by selecting the appropriate inner and outer radii values of the CNP.

*λ*. In this wavelength region the metafilm behaves like a lossless plasmonic material and therefore takes on characteristics reminiscent of a lossless Drude material. In such a material the reflectance quickly approaches unity for operation above the bulk plasmon wavelength. From this point of view, the CNP resonance defines the “plasma wavelength” for the entire metafilm. When the plasmon oscillations of each CNP are excited coherently throughout the metafilm, the reflectance is maximized. Coupling between the particles moves the metafilm’s resonant wavelength to a value slightly larger than the individual CNP’s resonance wavelength

_{res}*λ*. Farther away from

_{res}*λ*, the field no longer strongly couples to the CNP plasmon modes; and, therefore, the characteristics of the metafilm no longer appear similar to a lossless plasmonic material but rather take on characteristics that can be attributed to the intrinsic permittivity of the silver used in the outer layer of the CNP and of its dielectric core.

_{res}*a*<0, for wavelengths below

*λ*and a net loss,

_{res}*a*>0, for wavelengths above

*λ*when the gain values are at, or above the SR gain value, i.e. when the metafilm acts like a lossless plasmonic material.

_{res}*λ*. This is shown explicitly in Fig. 8 at the SR gain value as,

_{res}*d*, is varied from 40nm to 70nm. At

*λ*the net loss and gain is zero,

_{res}*a*=0, however there is a net gain above and net loss below this wavelength. These features mimic the gain and loss behaviors of the individual active CNP inclusions at wavelengths above and below

*λ*[18

_{res}**15**, 2622–2653 (2007). [CrossRef] [PubMed]

*δ*≃25.7nm near

_{Ag}*λ*, is more than an order of magnitude larger than the 2

_{res}*nm*thickness of the layer of silver used in the outer coating of the CNP. Consequently, it was unexpected that the metafilm would act as a highly efficient mirror given that such a small amount of silver is used in the metafilm compared to a silver slab of the equivalent thickness

*t*=60

_{film}*nm*.

*k*=-0.463 is near

*d*≈80

*nm*, the scattering characteristics begin to resemble those of the individual inclusions, where the reflectance and transmittance profiles narrow to a few nanometers about

*λ*and attain values greater than unity. This strong enhancement can be explained by the fact that for large separation distances the evanescent coupling between the inclusions is extremely weak and the individual inclusions interact with the incident field as though they were isolated. The response to the incident field at

_{res}*λ*, therefore, approaches that of the individual CNP. Consequently, the metafilm radiates energy into the primary directions defined by the dipole radiation pattern lobes of the active CNP, thereby generating wave fronts (in accordance with Huygens principle) which are forward and counter propagating relative to the direction of propagation of the incident plane wave [18

_{res}**15**, 2622–2653 (2007). [CrossRef] [PubMed]

*d*, when the source is operating at the resonance wavelength,

*λ*, this configuration acts as a beam splitter which, in contrast to a traditional beam splitter that produces two beams whose individual amplitudes are less than that of the incident field, would create two beams whose amplitudes are larger than that of the incident field.

_{res}## 4. Three dimensional arrays of CNPs - CNP crystals

27. S.A Tretyakov and A. J. Viitanen, “Plane waves in regular arrays of dipole scatterers and effective-medium modeling,” J. Opt. Soc. Am. A **17**, 1791–1797 (2000). [CrossRef]

*d*, as well as the size of the inclusions,

*r*, was restricted to be much smaller than the excitation wavelength so that

*βd*≪1 and

*βr*≪1. The incident plane wave was assumed to propagate along the positive z-axis. With these assumptions the complex effective permittivity of the lattice takes the form,

*n*is the number density of the inclusions with units of [

*m*

^{-3}] and

*α*is the electric polarizability of the inclusion as defined above in (14). Expression (32) is similar to the familiar Clausius-Mossotti form of the effective permittivity of a random distribution of scatterers, but has been modified by the term,

_{E}*iβ*

^{3}/(6

*πε*

_{0}), added to the inclusion polarizability in the denominator. This extra term results from the periodicity of the inclusions in the lattice. It takes into account the coherent scattering interactions between inclusions, effectively canceling any losses arising from incoherent scattering effects that would otherwise arise in a random distribution of inclusions.

### 4.1 Effective Permittivity of the CNP Crystal

*k*=0 to

*k*=-0.353, through the SR value at

*k*=-0.453, to just above the SR value at

*k*=-0.463. The lattice spacing,

*d*, was varied over a range from 25 nm to 50nm, which maintained spacing values within the limits consistent for the validity of expression (32), i.e., to maintain

*βd*<1. The volume fraction of inclusions in the crystal is defined as

*f*=

*V*/

*d*

^{3}, where

*V*=4

*πr*

^{3}/3 is the volume of the CNP inclusion used. These values of

*d*correspond to a volume fraction range: 0.03<

*f*<0.26.

*λ*. This resonance wavelength coincides with the peak value of

_{res}*ε*″

*and will be denoted as*

_{lat}*λ*; it signifies a resonance associated with the entire volume of the crystal lattice. As can be seen from Figs. 9 and 10, the lattice resonance wavelength,

_{lattice}*λ*, exhibits tunability with the lattice spacing. It shifts to longer wavelengths as the lattice period is decreased and approaches the CNP resonance value,

_{lattice}*λ*, as the lattice spacing is increased. For the cases where the gain is below the SR gain value,

_{res}*ε*′

*takes on mostly positive values, but exhibits some negative values for lattice periods around 25nm as seen in Fig. 11. When the gain in the core of the CNP inclusions is increased above the SR gain value,*

_{lat}*ε*′

*takes on negative values for source wavelengths below*

_{lat}*λ*, and remains positive for those above

_{lattice}*λ*. These observations are consistent with those discussed above for the metafilms. In particular,, the coupling efficiency of the incident field to the localized plasmon of the CNP particle is maximized when the gain reaches the SR value at which point the absorption efficiency (7) of this inclusion becomes negative. The collective response of the inclusions then produces behavior indicative of a plasmonic material, i.e., the effective permittivity becomes negative. These features are emphasized in the more detailed cross sectional plots extracted from Figs. 8 and 9 at the lattice spacings of 25nm, 35nm, and 45nm given in Figs. 11, 12, and 13.

_{lattice}*ε*″

*may cross through zero and become negative in the region below*

_{lat}*λ*. The wavelength for which

_{lattice}*ε*″

*=0 depends on both the gain value in the CNP as well as the lattice period of the crystal. In the case with the gain value*

_{lat}*k*=-0.353 and lattice spacing

*d*=35

*nm*, this wavelength is

*λ*≃455.5

*nm*. At this zero crossing the effective permittivity is purely

*real*; i.e.,

*ε*=

_{lattice}*ε*′

*. Although only one zero crossing is observed for*

_{lat}*ε*″

*, the real part of the permittivity becomes zero at two wavelengths. One zero occurs at*

_{lat}*λ*, where

_{lattice}*ε*′

*transitions from positive values above*

_{lat}*λ*to negative values just below

_{lattice}*λ*. The second occurs at a wavelength that is below

_{lattice}*λ*, where

_{lattice}*ε*′

*changes sign again, transitioning from negative values to positive values as it re-passes through zero. At these two zero crossings the effective permittivity becomes purely*

_{lat}*imaginary*; i.e.,

*ε*=

_{lattice}*iε*″

*. The wavelength below*

_{lat}*λ*for which

_{lattice}*ε*′

*=0 similarly depends on both the gain value in the CNP core as well as the lattice period of the crystal. For example, in the case with the gain value*

_{lat}*k*=-0.353 and the lattice spacing

*d*=35

*nm*, this wavelength is

*λ*≃470.7

*nm*. As can be seen from these examples, the wavelengths where the complex permittivity becomes either purely real or purely imaginary in general do not coincide. These zero crossings have implications for wave propagation in the crystal lattice, as discussed next.

### 4.2 Effective Index of the CNP Crystal

*µ*=

_{eff}*µ*

_{0}. The effective complex refractive index of the lattice can then be defined as

*n*and

_{lat}*k*are, respectively, the real and imaginary parts of the effective complex index. The real part describes the effective phase velocity of a wave traveling through the CNP lattice; the imaginary part describes the decay or growth of a wave propagating through the CNP lattice. The real and imaginary parts of the effective complex index corresponding to the results shown in Figs. 11-13 are shown, respectively, in Figs. 14 and 15.

_{lat}*k*<0 and, loss

_{lat}*k*>0. The transition between gain or loss occurs at the wavelength,

_{lat}*λ*

_{n=0}, where the real part of the refractive index is identically zero, i.e., where

*n*=0, and, consequently, the imaginary part of the effective permittivity is also zero,

_{lat}*ε*″

*=0. It is important not to confuse the wavelength,*

_{lat}*λ*

_{n=0}, where the complex index is zero with the lattice resonance wavelength

*λ*. Therefore at

_{lattice}*λ*

_{n=0}, only evanescent waves having uniform phase throughout the crystal may exist.

*λ*

_{n=0}, the gain of the crystal approaches

*k*≃-1 while the gains in the core of the CNP inclusions in all cases presented in Fig. 14 all satisfy

_{lat}*k*>-0.5. In addition, for small lattice spacings in the range 25nm-30nm, an effective gain in the lattice is observed even when the gain in the core of the CNP inclusions is well below the SR value, i.e., when the individual CNP inclusions exhibit net absorption and their absorption efficiency is positive. This can be attributed to an enhancement in the plasmon resonance that was similarly responsible for the

*ε*′

*<0 values mentioned above when the gain in the core of the inclusions was below the SR value and the lattice spacing was near 25nm. To observe these features more closely, more detailed cross sectional plots taken from Figs. 14 and 15 at lattice spacings equal to 25nm, 35nm, and 45nm are presented in Figs. 16, 17, and 18 showing these features more closely.*

_{lat}*λ*<

*λ*, to greater than unity for

_{lattice}*λ*>

*λ*. The losses present in the crystal near

_{lattice}*λ*are the largest when the lattice spacing is small enough to ensure significant coupling between the inclusions. When the wavelength exceeds

_{lattice}*λ*, the losses drop off considerably. For large lattice spacings and, hence, the coupling between inclusions is weak, behaviors resembling the CNP inclusions themselves ensue. For example, the imaginary part of the index,

_{lattice}*k*, becomes very narrow and negative, and the crystal exhibits only gain. From these observations it is apparent that the strength and width of the absorption/gain resonance may be tuned by varying the lattice spacing,

_{lat}*d*, as is demonstrated in Figs. 15 and 16.

## 5. Random distribution of CNP inclusions

*random*distribution of active CNP inclusions was also investigated. The same gain values considered in the core of the CNP inclusions for the metafilm and the 3D periodic array were again used for this study. Recall that these gain values were:

*k*={0, -0.353, -0.453, -0.463}. The Clausius-Mossotti formula [28

28. A. Sihvola, *Electromagnetic Mixing Formulas and Applications* (Institute of Electrical Engineers, London, 1999). [CrossRef]

*n*remains the number density of the inclusions with units of [

*m*

^{-3}] and

*α*remains the electric polarizability of the inclusion as defined above in (14). This expression is a valid representation of the effective permittivity given that the size of the scatterers satisfies the quasi-static approximation, and the distance between the scatterers is large enough apart to make any close range coupling negligible. These conditions require that

_{E}*βr*≪1 and that the volume fraction of the particles,

*f*, satisfy the constraint:

*f*≪1. Therefore, the volume fractions used in the following simulations was limited to the range:

*f*≤0.01. Because the system of inclusions was randomly distributed, the volume fraction was used instead of the lattice period,

*d*, to parameterize the volume density of scatterers. Therefore, the number density that appears in expression (35) was defined in terms of the volume fraction,

*f*, and the volume of the inclusions,

*V*as

*n*=

*f*/

*V*. The random nature of this metamaterial essentially removes the coherent scattering effects that are present in the periodic case. Consequently, the term

*iβ*

^{3}/(6

*πε*

_{0}), which was present in the expression (32) for the periodic array of inclusions and was responsible for canceling the scattering losses from the individual inclusions, is absent in expression (35).

### 5.1 Effective Permittivity of the Random CNP Medium

*ε*′

*, and*

_{rand}*ε*″

*as the volume fraction is varied. In the periodic lattice it was observed that changing the lattice period and, therefore, the volume density had a noticeable affect on the position of the resonance. For the random CNP medium, there appears to be no such dependence on the resonance wavelength as the volume fraction is varied within the simulation range depicted in the Figs. 19, and 20, i.e., in the range 0.001*

_{rand}*f*<0.01. Also, as can be observed in Figs. 21 and 22, there is little effect on the width of the resonance for both

*ε*′

*and*

_{rand}*ε*″

*as*

_{rand}*f*is varied. Note that in Figs. 21 and 22 the profiles of

*ε*′

*and*

_{rand}*ε*″

*have been normalized to the range [-1,+1] to allow better observation of the relative width of the resonances. On the other hand, an enhancement in the strength of the resonances is observed as the volume fraction is increased, as was shown in Figs. 19 and 20.*

_{rand}*ε*″

*near the resonance may be affected by the volume fraction when the gain in the core of the inclusions is at the SR gain value,*

_{rand}*k*=-0.453. In particular, for very small volume fractions,

*f*<0.001, the imaginary part of the permittivity is negative, i.e.,

*ε*″

*<0, near the resonance, whereas for*

_{rand}*f*>0.001 it becomes positive, i.e.,

*ε*″

*>0. For gain values in the core of the inclusions above the SR gain value,*

_{rand}*k*=-0.453, the imaginary part of the permittivity becomes strictly positive, i.e.,

*ε*″

*>0, for all of the simulated volume fractions.*

_{rand}### 5.2 Effective Index of the Random CNP Medium

*µ*=

_{eff}*µ*

_{0}, the effective complex refractive index of the random CNP medium is defined as

*n*is the real part of the effective complex index and

_{rand}*k*is its imaginary part. Figures 23 and 24 show, respectively, the real and imaginary parts of the complex effective index for the same random CNP media used in the permittivity studies.

_{rand}*f*={4×10

^{-3},6×10

^{-3},8×10

^{-3},1×10

^{-3}}. This allows the trend in the enhancement of the effective gain with the volume fraction,

*f*, as well as with the gain value in the core of the CNP inclusions,

*k*, to be more easily seen.

*k*, goes from positive values when

_{rand}*k*is below the SR value, to negative values when

*k*is at or above the SR value. This behavior is shown in Figs. 25-28. It is interesting to note that the effective loss is enhanced compared to the passive case when there is gain present in the cores of the CNP inclusions and its value is below the SR value. This effect can be seen by comparing Fig. 27 to Fig. 28. Therefore, these results indicate that the random CNP medium exhibits either enhanced effective loss or gain in comparison to the passive case, depending on the amount of gain in the core of the inclusions.

## 6. Conclusions

*active*dielectric core surrounded by a 2nm thick silver layer, were presented. In particular, the effective material properties of these optical MTMs were explored using effective medium theories applicable to a variety of inclusion configurations. Two-dimensional (2D) mono-layer metafilms; three-dimensional (3D) square periodic array crystals; and 3D random distributions of these active CNPs were investigated.

## Acknowledgments

## References and Links

1. | N. Engheta and R. W. Ziolkowski, “A positive future for double negative metamaterials,” IEEE Microwave Theory Tech. |

2. | N. Engheta and R. W. Ziolkowski, eds., |

3. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of |

4. | R. W. Ziolkowski, “Metamaterial-based antennas: Research and developments,” IEICE Trans. Electron . |

5. | A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterials coatings,” Phys. Rev. E |

6. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

7. | 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 |

8. | U. Leonhardt, “Optical conformal mapping,” Science |

9. | G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. A |

10. | R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B. |

11. | A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double negative-positive metamaterials,” J. Appl. Phys. |

12. | R. W. Ziolkowski and A. Kipple, “Application of double negative metamaterials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propagat. |

13. | A. Alù, A. Salandrino, and N. Engheta, “Negative effective permeability and left-handed materials at optical frequencies,” Opt. Express |

14. | V. M. Shalaev and W. Cai, “Negative index of refraction in optical metamaterials,” Opt. Lett. |

15. | G. Dolling, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. |

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

17. | N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Maters. |

18. | J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express |

19. | J. A. Stratton, |

20. | A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. |

21. | U. Kreibig and M. Vollmer, |

22. | S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L.C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, “Self-Tuned quantum dot gain in photonic crystal lasers,” Phys. Rev. Lett. |

23. | D. Bimberg, N. Kirstaedter, N. N. Ledentsov, Zh. I. Alferov, P. S. Kop’ev, and V. M. Ustinov, “InGaAs—GaAs quantum-dot lasers,” IEEE J. Sel. Top. Quantum Electron. |

24. | E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans Antennas Propag. |

25. | S. I. Maslovski and S. A. Tretyakov, “Full-wave interaction field in two dimensional arrays of dipole scatterers,” Int. J. Electron. Comun., Arch. Elek. Übertragungstech. (AEÜ) |

26. | C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, “Reflection and transmission properties of a metafilm: with an application to a controllable surface composed of resonant particles,” IEEE Trans Antennas Propag. |

27. | S.A Tretyakov and A. J. Viitanen, “Plane waves in regular arrays of dipole scatterers and effective-medium modeling,” J. Opt. Soc. Am. A |

28. | A. Sihvola, |

**OCIS Codes**

(160.3918) Materials : Metamaterials

(160.4236) Materials : Nanomaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: February 26, 2008

Revised Manuscript: April 22, 2008

Manuscript Accepted: April 23, 2008

Published: April 25, 2008

**Citation**

Joshua A. Gordon and Richard W. Ziolkowski, "CNP optical metamaterials," Opt. Express **16**, 6692-6716 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6692

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### References

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- 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]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
- G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. A 462, 3027-3059 (2006). [CrossRef]
- R. D. Averitt, S. L. Westcott, and N. J. Halas, "Linear optical properties of gold nanoshells," J. Opt. Soc. Am. B. 16, 1824-1832 (1999). [CrossRef]
- A. Alù and N. Engheta, "Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and/or double negative-positive metamaterials," J. Appl. Phys. 97, 094310 (2005). [CrossRef]
- R. W. Ziolkowski and A. Kipple, "Application of double negative metamaterials to increase the power radiated by electrically small antennas," IEEE Trans. Antennas Propag. 51, 2626-2640 (2003) [CrossRef]
- A. Alù, A. Salandrino, and N. Engheta, "Negative effective permeability and left-handed materials at optical frequencies," Opt. Express 14, 1557-1567 (2006). [CrossRef] [PubMed]
- V. M. Shalaev and W. Cai, "Negative index of refraction in optical metamaterials," Opt. Lett. 30, 3356-3358 (2005). [CrossRef]
- G. Dolling, C. M. Soukoulis, and S. Linden, "Low-loss negative-index metamaterial at telecommunication wavelengths," Opt. Lett. 31, 1800-1802 (2006). [CrossRef] [PubMed]
- V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photon. 1, 41-48 (2007). [CrossRef]
- N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, "Three-dimensional photonic metamaterials at optical frequencies," Nat. Mater. 7, 31-37 (2007). [CrossRef]
- 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]
- J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
- A. L. Aden and M. Kerker, "Scattering of electromagnetic waves from two concentric spheres," J. Appl. Phys. 22, 1242-1246 (1951). [CrossRef]
- U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, New York, 1995).
- S. Strauf, K. Hennessy, M. T. Rakher, Y.-S. Choi, A. Badolato, L.C. Andreani, E. L. Hu, P. M. Petroff, and D. Bouwmeester, "Self-Tuned quantum dot gain in photonic crystal lasers," Phys. Rev. Lett. 96, 127404 (2006). [CrossRef] [PubMed]
- D. Bimberg, N. Kirstaedter, N. N. Ledentsov, Zh. I. Alferov, P. S. Kop??ev, and V. M. Ustinov, "InGaAs-GaAs quantum-dot lasers," IEEE J. Sel. Top. Quantum Electron. 3, 196-205 (1997). [CrossRef]
- E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, "Averaged transition conditions for electromagnetic fields at a metafilm," IEEE Trans Antennas Propag. 51, 2641-2651 (2003). [CrossRef]
- S. I. Maslovski and S. A. Tretyakov, "Full-wave interaction field in two dimensional arrays of dipole scatterers," Int. J. Electron. Comun., Arch. Elek. ?bertragungstech. (AE?) 53, 135-139 (1999).
- C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, "Reflection and transmission properties of a metafilm: with an application to a controllable surface composed of resonant particles," IEEE Trans Antennas Propag. 47, 853-865 (2005).
- S. A Tretyakov and A. J. Viitanen, "Plane waves in regular arrays of dipole scatterers and effective-medium modeling," J. Opt. Soc. Am. A 17, 1791-1797 (2000). [CrossRef]
- A. Sihvola, Electromagnetic Mixing Formulas and Applications (Institute of Electrical Engineers, London, 1999). [CrossRef]

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