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Doubly negative metamaterials in the near infrared and visible regimes based on thin film nanocomposites

Vitaliy Lomakin, Yeshaiahu Fainman, Yaroslav Urzhumov, and Gennady Shvets

Open Access Open Access

Abstract

An optical metamaterial characterized simultaneously by negative permittivity and permeability, viz. doubly negative metamaterial (DNM), that comprises deeply subwavelength unit cells is introduced. The DNM can operate in the near infrared and visible spectra and can be manufactured using standard nanofabrication methods with compatible materials. The DNM”s unit cell comprise a continuous optically thin metal film sandwiched between two identical optically thin metal strips separated by a small distance form the film. The incorporation of the middle thin metal film avoids limitations of metamaterials comprised of arrays of paired wires/strips/patches to operate for large wavelength / unit cell ratios. A cavity model, which is a modification of the conventional patch antenna cavity model, is developed to elucidate the structure”s electromagnetic properties. A novel procedure for extracting the effective permittivity and permeability is developed for an arbitrary incident angle and those parameters were shown to be nearly angle-independent. Extensions of the presented two dimensional structure to three dimensions by using square patches are straightforward and will enable more isotropic DNMs.

©2006 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. Structure’s unit cell

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Fig. 2.
Fig. 2. Equivalent modified cavity used to model the cavity in Fig. 1.

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Fig. 3.
Fig. 3. Dependence of the structures scattering coefficients and effective properties on the middle film thickness. The structure’s parameters are chosen as Lx =100nm, w=50nm, ds =15nm for three values of the strip-symmetry plane separation and film thickness being h=7nm and df =0 (Lz =44.5nm), h=10.25 nm and df =6.5nm (Lz =50.5nm), as well as h=11.25 nm and df =8.5nm (Lz =52.5nm). (a): magnitude of the zeroth order transmission coefficient |T 0|; (b): effective permeability µ eff; (c) effective permittivity ε eff. For df =0, i.e. in the absence of the middle film, only a single magnetic (longer wavelength) and electric (lower wavelength) resonances are obtained; the resonances manifest themselves as minima in the transmission coefficient magnitude dependence. For d ≠ ⁠0, i.e. in the presence of the middle film, one magnetic are obtained in the wavelength range between two electric resonances. Associated with the magnetic resonance and longer wavelength electric resonance are bands of negative Re{µ eff} and Re{ε eff}, respectively. The bands are more separated for df =6.5nm and overlap for df =8.5nm thus resulting in a DNM.

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Fig. 4.
Fig. 4. Electric field distribution corresponding to (a) magnetic resonance and (b) electric resonances. The field distribution was obtained assuming static approximation and assuming that SiO2 (εd =2.25) occupies the entire space.

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Fig. 5.
Fig. 5. Comparison between the quasi-static dielectric permittivity ε qs ( ω ) = 1 − f 0 s − ∑ i f i ( s − s i ) (where s(ω)=(1-εm (ω)/εd )-1 and εm is the dielectric permittivity of gold) and the extracted from fully can electromagnetic simulations ε eff(ω) as described in Sec. 4.1.

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Fig. 6.
Fig. 6. Effective index of refraction neff for different sets of parameters for a single DNM layer.

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Fig. 7.
Fig. 7. The ratio Re{n eff}/Im{n eff} characterizing the losses in the system as a function of the gain (Im{εd }) in the dielectric layer for a single DNM layer. The structure parameters are chosen as Lx =100nm, Lz =51.5nm, w=50nm, ds =15nm, df =7.5nm, h=10.75nm. It is evident that the loss is not high without any gain and it further improves significantly by increasing the gain; the required values of gain correspond to practically achievable values.

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Fig. 8.
Fig. 8. The effective index of refraction for different number of layers ml . The structure parameters are chosen as Lx =100nm, Lz =102.5nm, w=50nm, ds =15nm, df =8.5nm, h=11.25nm. DNM operation with stable negative index in the range 640nm–680nm.

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Fig. 9.
Fig. 9. The real part of effective parameters µ eff and ε eff for different angles of incidence for a single DNM layer. The structure parameters are chosen as Lx =100nm, Lx =52.5nm, w=50nm, ds =15nm, df =8.5nm, h=11.25nm. DNM operation is obtained over a wide range of incident angles.

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Equations (12)

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Y s strip + Y z ( 1 − j cot k z h ) = 0 ,
f magn ≈ f p π q d s h ε d w 2 .
Y s strip + Y z + Y z Y s film + j 2 Y z tan k z h 2 Y z + j Y s film tan k z h = 0 .
μ eff , yy = μ 0 , eff , yy − f p , magn , yy f − f magn , ε eff , ii = ε 0 , eff , ii − f p , elect , ii ( 1 ) f − f elect ( 1 ) − f p , elect , ii ( 2 ) f − f elect ( 2 ) ,
T = ( cos ( k 0 n z , eff H ) + j 2 ( Z z , eff cos θ + cos θ Z z , eff ) sin ( k 0 n z , eff H ) ) − 1 ,
R = j 2 ( Z z , eff cos θ + cos θ Z z , eff ) sin ( k 0 n z , eff H ) T ,
n z , eff = ( cos − 1 ( 1 − ( R 2 − T 2 ) 2 T ) + 2 π l ) ( k 0 H ) − 1 ,
Z z , eff = cos θ ( 1 − R 2 ) − T 2 ( 1 − R 2 ) − T 2 .
n z , eff = ( μ eff , yy ε eff , xx − sin 2 θ ε eff , zz ) 1 2 ,
Z z , eff = n z , eff ε eff , xx ,
ε eff , xx = n z , eff Z z , eff ,
μ eff , yy = n z , eff Z z , eff + sin 2 θ ε eff , zz ,
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