1.3 Solutions to Problems
9
The term that includes exp(−i(ω k + ω l )t) in Eq. (1.9) indicates the oscillating
component at the frequency ω k + ω l . Therefore, this term corresponds to the
polarization of sum frequency,
P
(2)
p (ω k + ω l ) =
x−z
q,r
χ
(2)
pqr (ω k + ω l , ω k , ω l )E q (ω k )E r (ω l ).
(1.10)
This equation coincides with Eq. (1.4), by replacing ω k , ω l , ω k + ω l with ω 1 , ω 2 , ,
respectively. Using monochromatic lights of ω 1 and ω 2 , the sum frequency signal at
= ω 1 + ω 2 occurs by the second-order process.
In relation to the above derivation of the sum frequency polarization, we can also
derive the difference frequency generation (DFG), a related second-order nonlinear
process to SFG. We have learned in Sect. 1.1 that the light field of frequency
ω, E(ω) exp(−iωt) in Eq. (1.3), is accompanied with its complex conjugate,
E(ω) ∗ exp(iωt). Therefore, the combinations of light fields at ω 1 ( = 0) and ω 2
( = 0) actually give rise to four possible phase factors, exp (−i(±ω 1 ± ω 2 )t). All the
possible combinations of ω 1 and ω 2 are presented in analogous forms to Eq. (1.4):
P
(2)
p (ω 1 + ω 2 ) =
x−z
q,r
χ
(2)
pqr (ω 1 + ω 2 , ω 1 , ω 2 )E q (ω 1 )E r (ω 2 ),
(1.11)
P
(2)
p (ω 1 − ω 2 ) =
x−z
q,r
χ
(2)
pqr (ω 1 − ω 2 , ω 1 , −ω 2 )E q (ω 1 )E r (ω 2 )
∗ ,
(1.12)
P
(2)
p (−ω 1 + ω 2 ) =
x−z
q,r
χ
(2)
pqr (−ω 1 + ω 2 , −ω 1 , ω 2 )E q (ω 1 )
∗ E r (ω 2 ), (1.13)
P
(2)
p (−ω 1 −ω 2 ) =
x−z
q,r
χ
(2)
pqr (−ω 1 −ω 2 , −ω 1 , −ω 2 )E q (ω 1 )
∗ E r (ω 2 )
∗ . (1.14)
Among the above four terms, Eqs. (1.11) and (1.14) including ω 1 +ω 2 and −ω 1 −ω 2
correspond to SFG, whereas Eqs. (1.12) and (1.13) including ω 1 −ω 2 , and −ω 1 +ω 2
to DFG.
1.3.3 Red Shift of O-H Frequency
[Problem 1.3] Answer two frequencies of O-H stretching vibration of an isolated
water molecule (symmetric and anti-symmetric stretching). Where are these frequencies located in the spectra of Fig. 1.1?
9
The term that includes exp(−i(ω k + ω l )t) in Eq. (1.9) indicates the oscillating
component at the frequency ω k + ω l . Therefore, this term corresponds to the
polarization of sum frequency,
P
(2)
p (ω k + ω l ) =
x−z
q,r
χ
(2)
pqr (ω k + ω l , ω k , ω l )E q (ω k )E r (ω l ).
(1.10)
This equation coincides with Eq. (1.4), by replacing ω k , ω l , ω k + ω l with ω 1 , ω 2 , ,
respectively. Using monochromatic lights of ω 1 and ω 2 , the sum frequency signal at
= ω 1 + ω 2 occurs by the second-order process.
In relation to the above derivation of the sum frequency polarization, we can also
derive the difference frequency generation (DFG), a related second-order nonlinear
process to SFG. We have learned in Sect. 1.1 that the light field of frequency
ω, E(ω) exp(−iωt) in Eq. (1.3), is accompanied with its complex conjugate,
E(ω) ∗ exp(iωt). Therefore, the combinations of light fields at ω 1 ( = 0) and ω 2
( = 0) actually give rise to four possible phase factors, exp (−i(±ω 1 ± ω 2 )t). All the
possible combinations of ω 1 and ω 2 are presented in analogous forms to Eq. (1.4):
P
(2)
p (ω 1 + ω 2 ) =
x−z
q,r
χ
(2)
pqr (ω 1 + ω 2 , ω 1 , ω 2 )E q (ω 1 )E r (ω 2 ),
(1.11)
P
(2)
p (ω 1 − ω 2 ) =
x−z
q,r
χ
(2)
pqr (ω 1 − ω 2 , ω 1 , −ω 2 )E q (ω 1 )E r (ω 2 )
∗ ,
(1.12)
P
(2)
p (−ω 1 + ω 2 ) =
x−z
q,r
χ
(2)
pqr (−ω 1 + ω 2 , −ω 1 , ω 2 )E q (ω 1 )
∗ E r (ω 2 ), (1.13)
P
(2)
p (−ω 1 −ω 2 ) =
x−z
q,r
χ
(2)
pqr (−ω 1 −ω 2 , −ω 1 , −ω 2 )E q (ω 1 )
∗ E r (ω 2 )
∗ . (1.14)
Among the above four terms, Eqs. (1.11) and (1.14) including ω 1 +ω 2 and −ω 1 −ω 2
correspond to SFG, whereas Eqs. (1.12) and (1.13) including ω 1 −ω 2 , and −ω 1 +ω 2
to DFG.
1.3.3 Red Shift of O-H Frequency
[Problem 1.3] Answer two frequencies of O-H stretching vibration of an isolated
water molecule (symmetric and anti-symmetric stretching). Where are these frequencies located in the spectra of Fig. 1.1?