Coherent Raman Scattering Microscopy - Emerging Raman Applications and Techniques in Biomedical and Pharmaceutical Fields - page 113

[32-35]. Accordingly, for a given Raman-active resonance r, the amplitudes
of χ (3)r

ijkl can be expressed in terms of the isotropy and symmetric anisotropy
invariants of the corresponding spontaneous Raman scattering tensor, α 2 and
γ s 2 , respectively. In the case of frequency-degenerate CRS considered here,
the two relevant independent tensor components assume the following form
[33, 34]:

4

α 2 +

45 γ s 2
δ r −

A r , 1111
δ r −

χ (3)r

1111 =

= CN Γ r (6.5)

iΓ r iΓ r

15 γ s 2
δ r −

1

A r , 1221
δ r −

χ (3)r

1221 =

= CN Γ r

iΓ r .

(6.6)

iΓ r

δ r =Ω r − ω S ) is the detuning from the Raman resonance at frequency

Ω r with a half width at half maximum (HWHM) of Γ r . N is the number
density of Raman-active scatterers and C is a proportionality constant.
The ratio between the susceptibility components, χ (3)

( ω p −

1221 / χ (3)

1111 , defines the

resonant and nonresonant nonlinear depolarization ratios:

ρ r = χ (3)r ρ nr = χ (3)nr

3 γ s 2
45 α 2 +4 γ s 2

= 1
3

1221
χ (3)r
1111

1221
χ (3)nr
1111

= and (6.7)

respectively. ρ r is the direct CRS analogue for the spontaneous Raman depo-
larization ratio of the r th Raman mode, with values ranging between 0 for an
isotropic and 0.75 for an anisotropic vibrational mode symmetry. In contrast,
ρ nr has no spontaneous Raman analogue. For the nonresonant background
susceptibility tensor components, Kleinman's symmetry conjecture holds [36],
χ (3)nr
1111

=3 χ (3)nr

1221 ,and ρ nr assumes the value of 1 / 3.
The spontaneous Raman spectrum is then described by the sum of the
imaginary parts of the complex resonant susceptibility,

r Im χ (3)r

I Raman ∝ (6.8)

which is linearly dependent on the number density of Raman-active scat-
terers.

6.2.3 Excitation Pulse Schemes with High Vibrational Selectivity

Because of the nonlinearity of CRS, conventional wisdom states that high
input field peak amplitudes, which are readily provided by ultrashort laser
pulses, are required to induce a strong nonlinear polarization (cf. (6.1)-
(6.3)). The use of transform-limited femtosecond pulses, however, results in
a broad distribution of Raman shifts, ω p − ω S , that by far exceeds the line

width of a typical Raman resonance ∼ 10 cm − 1 in condensed-phase sam-

ples at room temperatures. This is schematically depicted in Fig. 6.2A where
the frequency-time dependences of a transform-limited pump and Stokes