4.4 Phase Matching
93
Hence, the intensity now depends on the sinc part only and is maximum for k = 0,
as illustrated in Fig. 4.4b. All the above discussion throws light at the one and only
one fact, that is, k should be as low as possible because the smaller the mismatch the
greater the efficiency. This sufficiently explains the significance of phase matching
in nonlinear optics.
4.5 Achievement of Phase Matching
From the previous section, we know that the condition of phase matching is
k = k 1 + k 2 − k 3 = 0
(4.57)
Since k i = n i ω i /c, where n i is the refractive index of the medium for frequency ω i ,
Eq. 4.57 can be written as
n 1 ω 1
c
+
n 2 ω 2
c
=
n 3 ω 3
c
(4.58)
where ω 1 + ω 2 = ω 3 . In case of second-harmonic generation, ω 1 = ω 2 = ω and ω 3 =
2ω. Hence, for second-harmonic generation, Eq. 4.58 gives
n(ω) = n(2ω)
(4.59)
However, for sum-frequency generation we can rearrange Eq. 4.58 and get
n 3 =
n 1 ω 1 + n 2 ω 2
ω 3
(4.60)
n 3 − n 2 =
n 1 ω 1 + n 2 ω 2
ω 3
− n 2
(4.61)
=
n 1 ω 1 − n 2 (ω 3 − ω 2 )
ω 3
(4.62)
=
n 1 ω 1 − n 2 ω 1
ω 3
(4.63)
= (n 1 − n 2 )
ω 1
ω 3
(4.64)
Equation 4.64 is the condition of perfect phase matching (PPM), which cannot be
satisfied by normal dispersion where the refractive index exhibits increasing trend
with increasing frequency, i.e., if ω 3 > ω 2 > ω 1 , then n 3 > n 2 > n 1 . However, the
condition can be met in the abnormal dispersion region, albeit it is a region of
high absorption. Figure 4.5 shows the normal and anomalous dispersion regions in
the refractive index versus frequency plot of silica. The data for the plot has been
adapted from Palik [105]. The blue curve shows the real and the red curve shows
93
Hence, the intensity now depends on the sinc part only and is maximum for k = 0,
as illustrated in Fig. 4.4b. All the above discussion throws light at the one and only
one fact, that is, k should be as low as possible because the smaller the mismatch the
greater the efficiency. This sufficiently explains the significance of phase matching
in nonlinear optics.
4.5 Achievement of Phase Matching
From the previous section, we know that the condition of phase matching is
k = k 1 + k 2 − k 3 = 0
(4.57)
Since k i = n i ω i /c, where n i is the refractive index of the medium for frequency ω i ,
Eq. 4.57 can be written as
n 1 ω 1
c
+
n 2 ω 2
c
=
n 3 ω 3
c
(4.58)
where ω 1 + ω 2 = ω 3 . In case of second-harmonic generation, ω 1 = ω 2 = ω and ω 3 =
2ω. Hence, for second-harmonic generation, Eq. 4.58 gives
n(ω) = n(2ω)
(4.59)
However, for sum-frequency generation we can rearrange Eq. 4.58 and get
n 3 =
n 1 ω 1 + n 2 ω 2
ω 3
(4.60)
n 3 − n 2 =
n 1 ω 1 + n 2 ω 2
ω 3
− n 2
(4.61)
=
n 1 ω 1 − n 2 (ω 3 − ω 2 )
ω 3
(4.62)
=
n 1 ω 1 − n 2 ω 1
ω 3
(4.63)
= (n 1 − n 2 )
ω 1
ω 3
(4.64)
Equation 4.64 is the condition of perfect phase matching (PPM), which cannot be
satisfied by normal dispersion where the refractive index exhibits increasing trend
with increasing frequency, i.e., if ω 3 > ω 2 > ω 1 , then n 3 > n 2 > n 1 . However, the
condition can be met in the abnormal dispersion region, albeit it is a region of
high absorption. Figure 4.5 shows the normal and anomalous dispersion regions in
the refractive index versus frequency plot of silica. The data for the plot has been
adapted from Palik [105]. The blue curve shows the real and the red curve shows