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WOLFRAM|DEMONSTRATIONS PROJECT

Phase Matching of SHG in Nonlinear Optics

​
view
standard vs. mismatched phases
amplitude sums
intensity sums
Δk
This Demonstration studies the phase matching condition of second harmonic generation (SHG) in nonlinear optics. The pump laser wave is given by
y=cos(kx)
. The wave of the second harmonic generation (SHG) is
y
standard
=cos(2kx)
if the phase is matched, and
y
m
=cos(2k'x+
φ
0
)=cos((2k+Δk)x+
φ
0
)
if the phase is mismatched, where
2k'=2k+Δk
.
If
x=X°
,
2kx=2k'x+
φ
0
, then
φ
0
=(2k-2k')X°=-ΔkX°
, and
y
m
=cos((2k+Δk)x+
φ
0
)=cos((2k+Δk)x-ΔkX°)
, where the SHG wave is generated at the position of
X°
. In this Demonstration,
k=1,X°=
π
12
,
2π
12
,
3π
12
,…,
12π
12
.
Click the first button to compare the standard phase (blue line) and the mismatched phase (other lines). Click the second and third buttons to see the summations of amplitudes and intensities of the mismatched waves, respectively. The intensity of optical wave is proportional to the square of the wave amplitude. It can be seen that when
Δk=0
, the sum of the intensites reaches the maximum value, which means that the phase is matched.
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