106
4 Nonlinear Optics with Zero-Index Metamaterials
X
M
X
0
0.2
0.4
0.6
0.8
1
1.2
Frequency
a/2 c
Fig. 4.15 Second-harmonic generation with imperfect phase matching, according to
Armstrong et al.
1.3
1.4
1.5
1.6
1.7
-0.4
-0.2
0
0.2
0.4
(a)
0.74
0.76
0.78
0.8
0.82
-0.2
-0.1
0
0.1
0.2
(b)
Fig. 4.16 Refractive index of the metamaterial determined around a the fundamental and b the
second-harmonic frequencies. The markers indicate the actual values obtained from the graph and
the blue curved are the fourth-order polynomial fit
refractive index of silicon at λ 1 is n 1 = 3.477 and at λ 2 is n 2 = 3.714 (Palik 1999).
In the case of Si, we get = −1.921 × 10
6 m
−1 , which is more than 30 times than
that in the case of the zero-index metamaterial. Hence, we see that the zero-index
metamaterial allows drastically low phase mismatch compared to the homogeneous
medium of the same material.
4.6.1.1 Time-Domain Computation
Lowering the phase mismatch results in the enhancement of the second-harmonic
field. But before proceeding to the calculation of the field, one first needs to ensure