4.6 Second-Harmonic Generation
101
4.6 Second-Harmonic Generation
From the previous sections, we understand that a crystal with sufficiently high nonlinear susceptibilities can convert a part of the incident fundamental frequency into the
second-harmonic frequency and that the efficiency of conversion depends, to a great
extent, on the length of the crystal. Armstrong et al. [183, 206] presented a beautiful mathematical analysis of the exchange of power between the two frequency as
they propagate through the crystal. They derived exact solutions to the coupled wave
equations for both the perfect (PPM) and the imperfect phase matching (IPM) cases
and represented the two field amplitudes as a function of distance. The mathematical
analysis by Armstrong et al. has been discussed below.
Firstly, let us consider a nonlinear crystal of length L, as shown in Fig. 4.11. Light
is assumed to be propagating along the z-direction. As light propagates through the
crystal and fundamental frequency converts into the second-harmonic frequency, the
amplitudes of the two waves vary as [183]
A 1 =
I
2n 1 0 c
1/2
u 1 e
iφ 1
(4.82)
A 2 =
I
2n 2 0 c
1/2
u 2 e
iφ 2
(4.83)
where u 1 and u 2 are the normalized field amplitudes of the fundamental and the
second-harmonic waves, respectively, I = I 1 + I 2 is the total intensity of the two
waves, and the rest of the symbols have their usual meaning. As a consequence of
the law of conservation of energy, I remains constant. Hence,
u
2
1 + u
2
2 = 1
(4.84)
In their mathematical analysis, instead of distance z, Armstrong et al. use a normalized distance parameter ζ given as
ζ = z/l
(4.85)
where
l =
2n
2
1 n 2
0 cI
1/2
c
2ω 1 d e f f
(4.86)
is the characteristic distance for the exchange of power between the two frequencies.
They also introduced a relative phase parameter θ given by
θ = 2φ 1 − φ 2 + kz
(4.87)
101
4.6 Second-Harmonic Generation
From the previous sections, we understand that a crystal with sufficiently high nonlinear susceptibilities can convert a part of the incident fundamental frequency into the
second-harmonic frequency and that the efficiency of conversion depends, to a great
extent, on the length of the crystal. Armstrong et al. [183, 206] presented a beautiful mathematical analysis of the exchange of power between the two frequency as
they propagate through the crystal. They derived exact solutions to the coupled wave
equations for both the perfect (PPM) and the imperfect phase matching (IPM) cases
and represented the two field amplitudes as a function of distance. The mathematical
analysis by Armstrong et al. has been discussed below.
Firstly, let us consider a nonlinear crystal of length L, as shown in Fig. 4.11. Light
is assumed to be propagating along the z-direction. As light propagates through the
crystal and fundamental frequency converts into the second-harmonic frequency, the
amplitudes of the two waves vary as [183]
A 1 =
I
2n 1 0 c
1/2
u 1 e
iφ 1
(4.82)
A 2 =
I
2n 2 0 c
1/2
u 2 e
iφ 2
(4.83)
where u 1 and u 2 are the normalized field amplitudes of the fundamental and the
second-harmonic waves, respectively, I = I 1 + I 2 is the total intensity of the two
waves, and the rest of the symbols have their usual meaning. As a consequence of
the law of conservation of energy, I remains constant. Hence,
u
2
1 + u
2
2 = 1
(4.84)
In their mathematical analysis, instead of distance z, Armstrong et al. use a normalized distance parameter ζ given as
ζ = z/l
(4.85)
where
l =
2n
2
1 n 2
0 cI
1/2
c
2ω 1 d e f f
(4.86)
is the characteristic distance for the exchange of power between the two frequencies.
They also introduced a relative phase parameter θ given by
θ = 2φ 1 − φ 2 + kz
(4.87)