Molecular Fluorescence

128 5 Fluorescence polarization. Emission anisotropy

Fig. 5.3. Relations between the fluorescence intensity components resulting from the Curie symmetry principle. The fluorescent sample is placed at the origin of the system of coordinates.

cause from which it results) leads to relations between intensity components, as shown in Figure 5.3. It should be noted that the symmetry principle is strictly valid only for a point source of light, which cannot be rigorously achieved in practice. Moreover, only homogeneous diluted solutions will be considered in this chapter, so that artefacts due for instance to inner filter e ects are avoided.

The various cases of excitation are now examined (Figure 5.3).

5.1 Characterization of the polarization state of fluorescence (polarization ratio, emission anisotropy) 129

5.1.1

Excitation by polarized light

5.1.1.1 Vertically polarized excitation

When the incident light is vertically polarized, the vertical axis Oz is an axis of symmetry for the emission of fluorescence according to the Curie principle, i.e. Ix ¼ Iy. The fluorescence observed in the direction of this axis is thus unpolarized.

The components of the fluorescence intensity that are parallel and perpendicular to the electric vector of the incident beam are usually denoted as Ik and I?, respectively. For vertically polarized incident light, Iz ¼ Ik and Ix ¼ Iy ¼ I?.

The intensity component Iz corresponding to oscillations of the electric field along the Oz axis cannot be detected by the eye or by a detector placed along this axis. The fluorescence intensity observed in the direction of this axis is thus

Ix þ Iy ¼ 2I?.

On the contrary, the Ox and Oy axes are not axes of symmetry for the emission of fluorescence. When fluorescence is observed through a polarizer along the Ox axis (or the Oy axis), the intensity measured is Iz ¼ Ik for the vertical position of the polarizer, and Iy ¼ I? (or Ix ¼ I?) for the horizontal position. Without polarizer, the measured intensity is Iz þ Iy ¼ Ik þ I? in the Ox direction and Iz þ Ix ¼ Ik þ I? in the Oy direction.

In most cases, fluorescence is observed in a horizontal plane at 90 to the propagation direction of the incident beam, i.e. in direction Ox (Figure 5.4). The fluorescence intensity components Ik and I? are measured by a photomultiplier in

Fig. 5.4. Usual configuration for measuring fluorescence polarization.

5.2 Instantaneous and steady-state anisotropy 131

population of fluorophores are additive. Upon observation at 90 in a horizontal plane, the incident vertical component has the same e ect as previously described, whereas the incident horizontal component leads to unpolarized fluorescence emission in the direction of observation Ox, which is an axis of symmetry: Iz ¼ Ix (Figure 5.3).

The components IV and IH, vertically and horizontally polarized respectively, are such that Iz ¼ IV ¼ Ix, Iy ¼ IH (Figure 5.3). The total fluorescence intensity is then 2IV þ IH. The polarization ratio and the emission anisotropy are given by

where the subscript n refers to natural exciting light. These two quantities are linked by the following relation:

It is easy to show that rn ¼ r=2. Therefore, the emission anisotropy observed upon excitation by natural light is half that upon excitation by vertically polarized light. In view of the di culty of producing perfectly natural light (i.e. totally unpolarized), vertically polarized light is always used in practice. Consequently, only excitation by polarized light will be considered in the rest of this chapter.

5.2

Instantaneous and steady-state anisotropy

5.2.1

Instantaneous anisotropy

Following an infinitely short pulse of light, the total fluorescence intensity at time t is IðtÞ ¼ IkðtÞ þ 2I?ðtÞ, and the instantaneous emission anisotropy at that time is

After recording IkðtÞ and I?ðtÞ, the emission anisotropy can be calculated by means

132 5 Fluorescence polarization. Emission anisotropy

of Eq. (5.6), provided that the light pulse is very short with respect to the fluorescence decay. Otherwise, we should take into account the fact that the measured polarized components are the convolution products of the d-pulse responses (5.7) and (5.8) by the instrument response (see Chapter 6).

5.2.2

Steady-state anisotropy

On continuous illumination (i.e. when the incident light intensity is constant), the measured anisotropy is called steady-state anisotropy r. Using the general definition of an averaged quantity, with the total normalized fluorescence intensity as the probability law, we obtain

In the case of a single exponential decay with time constant t (excited-state lifetime), the steady-state anisotropy is given by

5.3

Additivity law of anisotropy

When the sample contains a mixture of fluorophores, each has its own emission anisotropy ri:

From a practical point of view, we measure the components Ik and I?, which are now the sum of all individual components:

XX

and usually the same definition of the total emission anisotropy is kept because, in practice, we measure the components Ik and I?

134 5 Fluorescence polarization. Emission anisotropy

Fig. 5.5. System of coordinates for characterizing the orientation of the emission transition moments.

5.4

Relation between emission anisotropy and angular distribution of the emission transition moments

Let us consider a population of N molecules randomly oriented and excited at time 0 by an infinitely short pulse of light polarized along Oz. At time t, the emission transition moments ME of the excited molecules have a certain angular distribution. The orientation of these transition moments is characterized by yE, the angle with respect to the Oz axis, and by f (azimuth), the angle with respect to the Oz axis (Figure 5.5). The final expression of emission anisotropy should be independent of f because Oz is an axis of symmetry.

For a particular molecule i, the components of the emission transition moments

along the three axes Ox, Oy, Oz are MEaiðtÞ, MEbiðtÞ and MEgiðtÞ, where aiðtÞ, biðtÞ and giðtÞ are the cosines of the angles formed by the emission transition moment

with the three axes (such that ai2bi2gi2 ¼ 1) and ME is the modulus of the vector transition moment.

The total fluorescence intensity at time t is obtained by summing over all molecules emitting at that time. Because there is no phase relation between the elementary emissions, the contributions of each molecule to the intensity components along Ox, Oy and Oz are proportional to the square of its transition moment components along each axis. Summation over all molecules leads to the following expressions for the fluorescence intensity components:

XN

IzðtÞ ¼ KME2 gi2ðtÞ ¼ KME2Ng2ðtÞ

i¼1