34
2 Zero-Index Metamaterials
e f f (ω) = 1 −
ω
2
ep
ω 2 + iωγ
(2.12)
μ e f f (ω) = 1 −
ω
2
mp
ω 2 + iωγ
(2.13)
where ω ep and ω mp are the electric and magnetic plasma frequencies, and γ
(<< ω ep , ω mp ) is the loss factor. According to the above equations, when ω → ω ep ,
e f f (ω) → 0 and when ω → ω mp , μ e f f (ω) → 0. Now, if somehow, the magnetic
plasma frequency happens to be equal to electric plasma frequency, i.e., ω mp = ω ep =
ω D and γ = 0, then for ω = ω D , e f f and μ e f f both become zero. As a result, at the
Dirac point frequency ω = ω D , the effective refractive index n e f f =
√
e f f
√
μ e f f
also becomes zero. Figure 2.6 depicts this type of a system, where ω mp = ω ep =
ω D = 1 × 10
9 rad/s and γ = 10
−5
ω ep . Figure 2.6a shows its dispersion diagram in
which the Dirac cones and the Dirac point can be observed and Fig. 2.6b depicts
the resultant effective refractive index which is zero at ω D . Figure 2.7 illustrates
the electric field distribution inside an NZPI medium at 0.5 × 10
9 rad/s (ω < ω D ),
1.0 × 10
9 rad/s (ω = ω D ) and 1.5 × 10
9 rad/s (ω > ω D ). It shows that as a wave
initially propagating in air enters the NZPI medium, its wavelength is altered according to the refractive index of the medium (see Fig. 2.7). According to Eqs. 2.12 and
2.13, at ω = 0.5 × 10
9 rad/s, the medium becomes a negative-index medium with
n e f f =
√ e f f
√ μ e f f = −3.0, thus the wavelength inside the medium becomes onethird of that in air (Fig. 2.7b). Whereas at ω = 1.0 × 10
9 rad/s (i.e., ω = ω D ), the
medium becomes a zero-index medium with n e f f = 0, and hence the wavelength
inside the medium becomes infinite and the electric field attains a quasi-static state,
i.e., the electric field remains constant throughout the medium, as shown in Fig. 2.7c.
And finally, at ω = 0.5 × 10
9 rad/s, the medium acquires the positive-index character with n e f f =
√
e f f
√
μ e f f = 0.56, accordingly the wavelength becomes 1.78
times of that in air (Fig. 2.7d).
Please note that since these plots represent the steady-state fields obtained by
frequency-domain computation, the effect of negative refractive index is not visible.
If a time-domain computation method such as FDTD is employed [9, 44, 46], the
wave propagation inside the medium will certainly exhibit the negative refraction.
2.5 Accidental-Degeneracy-Induced Dirac Cones
in Photonic Crystals
Wang et al. presented a beautiful heuristic model for ZIM systems, but the real design
of zero-index media exhibiting Dirac dispersion (or conical dispersion) was proposed
by Huang et al. [27] in 2011. Huang reported that a square lattice of dielectric rods in
air (see Fig. 2.8) of a particular radius and periodicity exhibits Dirac cones and linear
dispersion, and hence exhibits zero refractive index at the Dirac point frequency.
2 Zero-Index Metamaterials
e f f (ω) = 1 −
ω
2
ep
ω 2 + iωγ
(2.12)
μ e f f (ω) = 1 −
ω
2
mp
ω 2 + iωγ
(2.13)
where ω ep and ω mp are the electric and magnetic plasma frequencies, and γ
(<< ω ep , ω mp ) is the loss factor. According to the above equations, when ω → ω ep ,
e f f (ω) → 0 and when ω → ω mp , μ e f f (ω) → 0. Now, if somehow, the magnetic
plasma frequency happens to be equal to electric plasma frequency, i.e., ω mp = ω ep =
ω D and γ = 0, then for ω = ω D , e f f and μ e f f both become zero. As a result, at the
Dirac point frequency ω = ω D , the effective refractive index n e f f =
√
e f f
√
μ e f f
also becomes zero. Figure 2.6 depicts this type of a system, where ω mp = ω ep =
ω D = 1 × 10
9 rad/s and γ = 10
−5
ω ep . Figure 2.6a shows its dispersion diagram in
which the Dirac cones and the Dirac point can be observed and Fig. 2.6b depicts
the resultant effective refractive index which is zero at ω D . Figure 2.7 illustrates
the electric field distribution inside an NZPI medium at 0.5 × 10
9 rad/s (ω < ω D ),
1.0 × 10
9 rad/s (ω = ω D ) and 1.5 × 10
9 rad/s (ω > ω D ). It shows that as a wave
initially propagating in air enters the NZPI medium, its wavelength is altered according to the refractive index of the medium (see Fig. 2.7). According to Eqs. 2.12 and
2.13, at ω = 0.5 × 10
9 rad/s, the medium becomes a negative-index medium with
n e f f =
√ e f f
√ μ e f f = −3.0, thus the wavelength inside the medium becomes onethird of that in air (Fig. 2.7b). Whereas at ω = 1.0 × 10
9 rad/s (i.e., ω = ω D ), the
medium becomes a zero-index medium with n e f f = 0, and hence the wavelength
inside the medium becomes infinite and the electric field attains a quasi-static state,
i.e., the electric field remains constant throughout the medium, as shown in Fig. 2.7c.
And finally, at ω = 0.5 × 10
9 rad/s, the medium acquires the positive-index character with n e f f =
√
e f f
√
μ e f f = 0.56, accordingly the wavelength becomes 1.78
times of that in air (Fig. 2.7d).
Please note that since these plots represent the steady-state fields obtained by
frequency-domain computation, the effect of negative refractive index is not visible.
If a time-domain computation method such as FDTD is employed [9, 44, 46], the
wave propagation inside the medium will certainly exhibit the negative refraction.
2.5 Accidental-Degeneracy-Induced Dirac Cones
in Photonic Crystals
Wang et al. presented a beautiful heuristic model for ZIM systems, but the real design
of zero-index media exhibiting Dirac dispersion (or conical dispersion) was proposed
by Huang et al. [27] in 2011. Huang reported that a square lattice of dielectric rods in
air (see Fig. 2.8) of a particular radius and periodicity exhibits Dirac cones and linear
dispersion, and hence exhibits zero refractive index at the Dirac point frequency.