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4 Nonlinear Optics with Zero-Index Metamaterials
where ˜
P
(1)
= 0 χ
(1) ˜
E(t) is the linear polarization and ˜
P
N L is the nonlinear polarization due to the rest of the terms. The net displacement field ˜
D is written as
˜
D(t) = 0 ˜
E(t) + ˜
P(t)
(4.3)
Normally, for relatively low electric fields, only the linear part ˜
P
(1) is significant and
the nonlinear part ˜
P
N L can be ignored due to small values. Then,
˜
D(t) = ˜
D
(1)
(t)
(4.4)
= 0 ˜
E(t) + ˜
P
(1)
(t)
(4.5)
= 0 ˜
E(t) + 0 χ
(1) ˜
E(t)
(4.6)
= 0 (1 + χ
(1)
) ˜
E(t)
(4.7)
= 0
(1) ˜
E(t)
(4.8)
where
(1) is the first-order relative permittivity. However, in the case of intense
electric fields, as in a laser, the nonlinear part begins to play a significant role and
gives rise to several interesting phenomena. This chapter is dedicated to the study of
those nonlinear phenomena and how useful the zero-index metamaterials can be for
them.
4.1.1 Wave Equation in a Nonlinear Medium
From the various texts on electromagnetics, we know that the wave equation in a
medium is written as
∇
2 ˜
E − μμ
∂
2 ˜
E
∂t 2 = 0
(4.9)
or
∇
2 ˜
E −
μ r r
c 2
∂
2 ˜
E
∂t 2 = 0
(4.10)
For a non-magnetic medium, i.e., μ r = 1,
∇
2 ˜
E −
r
c 2
∂
2 ˜
E
∂t 2 = 0
(4.11)
Multiplying and dividing the second term by 0 ,