16.4 Two-Photon Spectroscopy
Ordinary fluorescence polarization spectroscopy is often a very valuable
supplement to LD spectroscopy. As we have seen, LD spectroscopy may
provide the numerical value of angles between a given molecular axis (the
orientation axis) and transition moments for individual transitions. Information
about M may help determine the signs of these angles (see above).
The description of ordinary luminescence polarization spectroscopy is simple
since it in reality consists of the (photo)selection of an aligned subset, followed
by an emission spectrum from this subset. In order to describe two-photon
experiments performed on aligned samples a considerably more extensive
treatment is required. In addition to the cosine square averages, the K, used in
LD spectroscopy, also averaged over the fourth power of the directional cosines,
are required; e.g.,
L
stuv
= ¢cos s cos t cos u cos v²
where the labels s, t, u, and v correspond to any selection of the molecular axes,
x, y, and z.
The number of different experiments that may be performed with linearly
polarized light is also larger in the case of two-photon spectroscopy applied to
aligned samples. For a uniaxial, aligned sample (sample axis U) the following
five experiments are (at least in the general case) different:
,
,
,
,
UUV
WV
WV W
U
VW
U
U
VW
W
V
Here the laboratory axes U, V, W indicate the polarization of (photon
1/photon 2). However, for practical reasons it is not always possible to record all
five independent spectra. Another variable is again the angle between the
directions of photon 1 and photon 2; the choice depends on a number of
experimental conditions. It is important that the observed intensities are
corrected for the actual angle used, according to Gallivan, Brinen, and Koren
[39].
The observed intensity E
PQ
(i,j) in a two-photon experiment, involving the
process (i, j), with the polarization direction of the two photons along laboratory
axes P and Q (= U, V, W) may now be written
E
PQ
(i, j) = 6 [F
PQ
]
stuv
[G(i, j)]
stuv
where the sum goes over s, t, u, and v = x, y, z. The matrix element [F
PQ
]
stuv
is
expressed in terms of the K and L and describe the orientational effects,
including the polarizer positions P and Q, while [G(i,j)]
stuv
describes the
molecular properties and the specific process involved. For example, in a Raman
experiment, with simultaneous photon processes, G is the Raman scattering
tensor, while in a luminescence experiment, with successive photon processes, G
is simply the product of the transition moments for the two transitions involved,
M
i
and M
j
. For more details, including the general, complete expressions for the
tensors F
PQ
, see [2].
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