4.6 Second-Harmonic Generation
105
4.6.1 SHG in Zero-Index Medium
From the previous sections, we know that the momentum mismatch or phase mismatch k in the second-harmonic generation is given by
k = 2k 1 − k 2
(4.96)
= 2n 1
ω
c
− n 2
2ω
c
(4.97)
= 2(n 1 − n 2 )
ω
c
(4.98)
Now, there are two ways of reducing k. One is to have n 1 ≈ n 2 , which is prevented
by normal dispersion. The other is by reducing both n 1 and n 2 , close to zero, which
can be achieved by employing zero-index metamaterials.
To showcase the efficiency of zero-index metamaterials in enhancing the secondharmonic field, we present below a comparative study between a homogeneous silicon medium and a silicon-based zero-index metamaterial. Let us first analyze the
case of a zero-index medium. We have chosen a rods-in-air-type metamaterial whose
zero-index behavior is a manifestation of accidental degeneracy-induced Dirac cone.
It is similar to the one proposed by Huang et al. [27], whose geometry has already
been shown in Fig. 2.7 Sect. 2.5 Chap. 2. The rods are made up of silicon and the
radius r of each rod is 0.2a, where a is the periodicity of the square lattice. The
band structure has been computed by plane wave expansion method (PWEM) [42],
in which a constant value of relative permittivity ( r = 12.5) has been chosen for the
silicon domains and the surrounding medium is air. The band diagram with ten bands
has been shown in Fig. 4.15. The zero-index character of this metamaterial around
the Dirac cone has already been explained in Chap. 2. Now, how to study secondharmonic generation in this zero-index metamaterial?. Firstly, the Dirac frequency
ωa/2πc = 0.542 = a/λ, marked by the red dot, has been chosen as the fundamental
frequency, whose corresponding second-harmonic frequency has been located in the
band diagram at ωa/2πc = 1.089 = a/λ, indicated by the blue dot. It can be noticed
that both these frequencies correspond to zero wave vector, being at the center of
the Brillouin zone ( point). Depending upon the desired fundamental frequency
(or wavelength) periodicity a is decided. We choose periodicity a = 840 nm, so
that the two frequencies translate into wavelengths λ 1 = 1550 nm (fundamental)
and λ 2 = 771 nm ≈ λ 1 /2 (second harmonic). Using the band structure with some
simple mathematics, the phase velocities v p = ω/k have been calculated, followed
by the calculation of refractive index as n = c/v p , at different frequencies (or equivalently at different free-space wavelengths). The refractive index obtained as the
function of wavelength, around both the fundamental and the second-harmonic frequencies, has been shown in Fig. 4.16. According to these two graphs, the refractive
index at λ 1 is n 1 = −0.0036, and at λ 2 is n 2 = 0.0040. Using these in Eq. 4.98,
we get k = −6.162 × 10
4 m
−1 . Now, to know if this value is substantially and
sufficiently low, it needs to be compared with a homogeneous silicon medium. The
105
4.6.1 SHG in Zero-Index Medium
From the previous sections, we know that the momentum mismatch or phase mismatch k in the second-harmonic generation is given by
k = 2k 1 − k 2
(4.96)
= 2n 1
ω
c
− n 2
2ω
c
(4.97)
= 2(n 1 − n 2 )
ω
c
(4.98)
Now, there are two ways of reducing k. One is to have n 1 ≈ n 2 , which is prevented
by normal dispersion. The other is by reducing both n 1 and n 2 , close to zero, which
can be achieved by employing zero-index metamaterials.
To showcase the efficiency of zero-index metamaterials in enhancing the secondharmonic field, we present below a comparative study between a homogeneous silicon medium and a silicon-based zero-index metamaterial. Let us first analyze the
case of a zero-index medium. We have chosen a rods-in-air-type metamaterial whose
zero-index behavior is a manifestation of accidental degeneracy-induced Dirac cone.
It is similar to the one proposed by Huang et al. [27], whose geometry has already
been shown in Fig. 2.7 Sect. 2.5 Chap. 2. The rods are made up of silicon and the
radius r of each rod is 0.2a, where a is the periodicity of the square lattice. The
band structure has been computed by plane wave expansion method (PWEM) [42],
in which a constant value of relative permittivity ( r = 12.5) has been chosen for the
silicon domains and the surrounding medium is air. The band diagram with ten bands
has been shown in Fig. 4.15. The zero-index character of this metamaterial around
the Dirac cone has already been explained in Chap. 2. Now, how to study secondharmonic generation in this zero-index metamaterial?. Firstly, the Dirac frequency
ωa/2πc = 0.542 = a/λ, marked by the red dot, has been chosen as the fundamental
frequency, whose corresponding second-harmonic frequency has been located in the
band diagram at ωa/2πc = 1.089 = a/λ, indicated by the blue dot. It can be noticed
that both these frequencies correspond to zero wave vector, being at the center of
the Brillouin zone ( point). Depending upon the desired fundamental frequency
(or wavelength) periodicity a is decided. We choose periodicity a = 840 nm, so
that the two frequencies translate into wavelengths λ 1 = 1550 nm (fundamental)
and λ 2 = 771 nm ≈ λ 1 /2 (second harmonic). Using the band structure with some
simple mathematics, the phase velocities v p = ω/k have been calculated, followed
by the calculation of refractive index as n = c/v p , at different frequencies (or equivalently at different free-space wavelengths). The refractive index obtained as the
function of wavelength, around both the fundamental and the second-harmonic frequencies, has been shown in Fig. 4.16. According to these two graphs, the refractive
index at λ 1 is n 1 = −0.0036, and at λ 2 is n 2 = 0.0040. Using these in Eq. 4.98,
we get k = −6.162 × 10
4 m
−1 . Now, to know if this value is substantially and
sufficiently low, it needs to be compared with a homogeneous silicon medium. The