4.2 Nonlinear Phenomena
89
P(2ω 1 ) = 0 χ
(2) E
2
1 = second harmonic o f ω 1
(4.33)
P(2ω 2 ) = 0 χ
(2) E
2
2 = second harmonic o f ω 2
(4.34)
P(ω 1 + ω 2 ) = 2 0 χ
(2) E 1 E 2 = sum f requency term
(4.35)
P(ω 1 − ω 2 ) = 2 0 χ
(2) E 1 E
∗
2 = di f f erence f requency term
(4.36)
P(0) = 2 0 χ
(2)
(E 1 E
∗
2 + E
∗
1 E 2 ) = zero f requency term
(4.37)
Though mathematically, all the above-obtained frequency components are probable
to exist in the output, the one which actually exists depends on values of the input
frequencies, which further depends on the energy band structure of the nonlinear
medium employed.
4.3 Coupled Wave Equations
In the previous section, for the sum-frequency generation, we represented the timevarying electric fields for the two input waves as ˜
E 1 (t) = E 1 e
−iω 1 t and ˜
E 2 (t) =
E 2 e
−iω 2 t . On similar lines, the field of the sum-frequency ω 3 = ω 1 + ω 2 can be
written as ˜
E 3 (t) = E 3 e
−iω 3 t . In general,
˜
E i (t) = E i e
−iω i t
, i = 1, 2, 3
(4.38)
where i = 1, 2 is meant for input frequencies and i = 3 is for sum-frequency component in the output. In this type of notation, it should be noted that the term e
−iω i t
represents the time-dependent part and the remaining E i is independent of time,
but intrinsically contains the harmonic variation of the field w.r.t. space, which is a
property of plane waves. The spatial term can be further expanded as
E i = A i e
ik i z
(4.39)
Hence, Eq. 4.38 becomes
˜
E i (t) = A i e
i(k i z+ω i t)
, i = 1, 2, 3
(4.40)
where k i = n i ω i /c is the wave vector for the ith frequency and n
2
i =
(1)
(ω i ) is the
permittivity of the nonlinear medium for the ith frequency. Using this in Eq. 4.35,
the amplitude of nonlinear polarization of the sum-frequency component is written
as
P 3 = P(ω 3 ) = 2 0 χ
(2) A 1 A 2 e
i(k 1 +k 2 )z
(4.41)
= 4 0 d e f f A 1 A 2 e
i(k 1 +k 2 )z
(4.42)
89
P(2ω 1 ) = 0 χ
(2) E
2
1 = second harmonic o f ω 1
(4.33)
P(2ω 2 ) = 0 χ
(2) E
2
2 = second harmonic o f ω 2
(4.34)
P(ω 1 + ω 2 ) = 2 0 χ
(2) E 1 E 2 = sum f requency term
(4.35)
P(ω 1 − ω 2 ) = 2 0 χ
(2) E 1 E
∗
2 = di f f erence f requency term
(4.36)
P(0) = 2 0 χ
(2)
(E 1 E
∗
2 + E
∗
1 E 2 ) = zero f requency term
(4.37)
Though mathematically, all the above-obtained frequency components are probable
to exist in the output, the one which actually exists depends on values of the input
frequencies, which further depends on the energy band structure of the nonlinear
medium employed.
4.3 Coupled Wave Equations
In the previous section, for the sum-frequency generation, we represented the timevarying electric fields for the two input waves as ˜
E 1 (t) = E 1 e
−iω 1 t and ˜
E 2 (t) =
E 2 e
−iω 2 t . On similar lines, the field of the sum-frequency ω 3 = ω 1 + ω 2 can be
written as ˜
E 3 (t) = E 3 e
−iω 3 t . In general,
˜
E i (t) = E i e
−iω i t
, i = 1, 2, 3
(4.38)
where i = 1, 2 is meant for input frequencies and i = 3 is for sum-frequency component in the output. In this type of notation, it should be noted that the term e
−iω i t
represents the time-dependent part and the remaining E i is independent of time,
but intrinsically contains the harmonic variation of the field w.r.t. space, which is a
property of plane waves. The spatial term can be further expanded as
E i = A i e
ik i z
(4.39)
Hence, Eq. 4.38 becomes
˜
E i (t) = A i e
i(k i z+ω i t)
, i = 1, 2, 3
(4.40)
where k i = n i ω i /c is the wave vector for the ith frequency and n
2
i =
(1)
(ω i ) is the
permittivity of the nonlinear medium for the ith frequency. Using this in Eq. 4.35,
the amplitude of nonlinear polarization of the sum-frequency component is written
as
P 3 = P(ω 3 ) = 2 0 χ
(2) A 1 A 2 e
i(k 1 +k 2 )z
(4.41)
= 4 0 d e f f A 1 A 2 e
i(k 1 +k 2 )z
(4.42)