92
4 Nonlinear Optics with Zero-Index Metamaterials
Fig. 4.4 Significance of
phase matching in nonlinear
phenomena
0
π/Δk=L
coh
2π/Δk
3 π/Δk
4 π/Δk
5 π/Δk
6 π/Δk
0
0.2
0.4
0.6
0.8
1
z
I
3
(z)
Δk=0
Δk≠ 0
(a)
-3 π
-2 π
-π
0
π
2π
3π
0
0.2
0.4
0.6
0.8
1
ΔkL/2
sinc
2
(ΔkL/2)
(b)
For a perfectly phase-matched situation ( = 0), the variation of intensity has
been shown by the red curve. For = 0, the sinc function becomes unity (Fig. 4.4b)
and only the z
2 part survives, and hence the nature of the curve is parabolic. It can be
observed that the intensity increases sharply with z in case of zero mismatch, since
the coherence is always maintained. For crystal of a particular length L, the intensity
I 3 only depends on k as
I 3 (z) =
8d
2
e f f ω
2
3 I 1 I 2
n 1 n 2 n 3 0 c 2 L
2 sinc
2
k L
2
(4.56)
4 Nonlinear Optics with Zero-Index Metamaterials
Fig. 4.4 Significance of
phase matching in nonlinear
phenomena
0
π/Δk=L
coh
2π/Δk
3 π/Δk
4 π/Δk
5 π/Δk
6 π/Δk
0
0.2
0.4
0.6
0.8
1
z
I
3
(z)
Δk=0
Δk≠ 0
(a)
-3 π
-2 π
-π
0
π
2π
3π
0
0.2
0.4
0.6
0.8
1
ΔkL/2
sinc
2
(ΔkL/2)
(b)
For a perfectly phase-matched situation ( = 0), the variation of intensity has
been shown by the red curve. For = 0, the sinc function becomes unity (Fig. 4.4b)
and only the z
2 part survives, and hence the nature of the curve is parabolic. It can be
observed that the intensity increases sharply with z in case of zero mismatch, since
the coherence is always maintained. For crystal of a particular length L, the intensity
I 3 only depends on k as
I 3 (z) =
8d
2
e f f ω
2
3 I 1 I 2
n 1 n 2 n 3 0 c 2 L
2 sinc
2
k L
2
(4.56)