**2.1 Basic theory of fluorescence**

This section provides a basic tutorial on specific topic in luminescence, namely fluorescence, and fluorescent instrumentations. To be able to understand the basic theoretical principles of luminescence spectroscopy, which include the electronic transitions, one should have a basic background on quantum mechanics and atomic orbitals, which was developed by Schrödinger in 1926. A tutorial review of Schrödinger's wave equation is out of the scope of this chapter, but briefly the most fruitful outcome of solving Schrödinger's wave equation is a set of wave functions (*called orbitals*), and their corresponding energies. An orbital is described by a set of three quantum numbers (Principal (n), Angular momentum (*l*), and magnetic (ml) quantum numbers). Later a fourth quantum number, that describes the magnetic field generated due to the spinning of an electron on its own axis, was discovered and named as spin magnetic quantum number (ms). The spin quantum number has only two allowed values: +1/2 and −1/2. According to Pauli Exclusion Principle, no two electrons in the same atom can have identical sets of four quantum numbers, *n, l, ml, and ms.* Thus any two electrons in same orbital (*n, l, ml*) must have different spins either ms=+1/2 or ms= -1/2. The total spin is defined by S = Σms and the Multiplicity (M) is defined as M = 2S+1(M=1, 2, 3… Singlet, Doublet, and Triplet, respectively,..) as shown in Figure 5.

The photophysical processes that occur from absorption to emission are often shown in a socalled Jabłoński diagram. Of course all possible energy routes cannot be encompassed in single figure, and different forms of the diagram can be found in different contexts. Figure 7 is a simple version of Jabłoński diagram, where absorption (S0→S1 or S0→S2 ), fluorescence (emission not involving spin change, S1→S0), phosphorescence (emission involving spin change, T1→S0), intersystem crossing (non-radiative transition with spin change, as an example from singlet to triplet states; S1→T1), internal conversion (non-radiative transition either to lower state when vibrational energy levels "match" or to lower state by collisional deactivation, S2→S1) ), vibration relaxation (within the vibrational levels in any excited electronic state) as well as intermolecular processes (radiationless relaxations) are shown.

parameters as absorbance—defined by the Beer-Lambert law as the product of the molar extinction coefficient, optical path-length, and concentration—as well as on the fluorescence quantum yield, the intensity of the excitation source, and the efficiency of the instrument

The summary points of this introduction to fluorescence are: 1. Fluorophores are molecules that, upon absorbing light energy, can reach an excited state, and then emit light energy. 2. The three-stage process of excitation, excited lifetime, and emission is called fluorescence. 3. Fluorophores absorb a range of wavelengths of light energy, and also emit a range of wavelengths. Within these ranges are the excitation maximum and the emission maximum. Because the excitation and emission wavelengths are different, the absorbed and emitted

The purpose of this chapter is to review the articles on the interior cited aspects published since 2000 about various aspects of application of fluorescence spectrophotometry in

This section provides a basic tutorial on specific topic in luminescence, namely fluorescence, and fluorescent instrumentations. To be able to understand the basic theoretical principles of luminescence spectroscopy, which include the electronic transitions, one should have a basic background on quantum mechanics and atomic orbitals, which was developed by Schrödinger in 1926. A tutorial review of Schrödinger's wave equation is out of the scope of this chapter, but briefly the most fruitful outcome of solving Schrödinger's wave equation is a set of wave functions (*called orbitals*), and their corresponding energies. An orbital is described by a set of three quantum numbers (Principal (n), Angular momentum (*l*), and magnetic (ml) quantum numbers). Later a fourth quantum number, that describes the magnetic field generated due to the spinning of an electron on its own axis, was discovered and named as spin magnetic quantum number (ms). The spin quantum number has only two allowed values: +1/2 and −1/2. According to Pauli Exclusion Principle, no two electrons in the same atom can have identical sets of four quantum numbers, *n, l, ml, and ms.* Thus any two electrons in same orbital (*n, l, ml*) must have different spins either ms=+1/2 or ms= -1/2. The total spin is defined by S = Σms and the Multiplicity (M) is defined as M =

2S+1(M=1, 2, 3… Singlet, Doublet, and Triplet, respectively,..) as shown in Figure 5.

The photophysical processes that occur from absorption to emission are often shown in a socalled Jabłoński diagram. Of course all possible energy routes cannot be encompassed in single figure, and different forms of the diagram can be found in different contexts. Figure 7 is a simple version of Jabłoński diagram, where absorption (S0→S1 or S0→S2 ), fluorescence (emission not involving spin change, S1→S0), phosphorescence (emission involving spin change, T1→S0), intersystem crossing (non-radiative transition with spin change, as an example from singlet to triplet states; S1→T1), internal conversion (non-radiative transition either to lower state when vibrational energy levels "match" or to lower state by collisional deactivation, S2→S1) ), vibration relaxation (within the vibrational levels in any excited electronic state) as well as intermolecular processes (radiationless relaxations) are shown.

and, in dilute solutions, is linearly proportional to these parameters.

lights are detectable as different colors or areas on the visible spectrum.

**2. Theoretical and instrumental aspects** 

**2.1 Basic theory of fluorescence** 

chemical analysis.

Other intermolecular processes (e.g. quenching, energy transfer, solvent interaction etc.) are omitted (Rendell, 1987; Lakowicz, 2006).

Fig. 5. Possible energy states according to their spin multiplicity

Once a molecule is excited by absorption of light it can return to the ground state with emission of fluorescence, but many other pathways for de-excitation are also possible, these are summarized in Figure 6.

Fig. 6. All possible pathways for de-excitation processes

Jablonski diagram (Fig. 7) explains the mechanism of light emission in most organic and inorganic luminophores. Upon absorption of the light by a molecule, the electron promoted from ground electronic state (S0) to an excited state that should possess the same spin multiplicity (such as, S1, S2,….) this process usually occurs within ∼10-15 s. This excludes the triplet excited state as the final state of electronic absorption because transitions between states of different multiplicities are improbable "*forbidden*" (e.g. T→S or S→T). According to the quantum mechanical selection rules for electronic transitions, spin state should be maintained upon excitation because it is harder for an electron to go from a singlet state to

Current Achievement and Future Potential of Fluorescence Spectroscopy 217

exceptions to the rule. The transitions that will predominate can be justified by Franck-Condon principle. The principle states that since the electronic absorption of light occurs in extremely short time (∼10-15s), thus during the time scale of absorption the nuclei are assumed to be frozen, that is that the transitions between various electronic levels are

The typical molecular photoluminescence relaxation processes that illustrated by Jablonski diagram (Fig. 7) can be classified to two main type of transition, these are radiative and nonradiative transitions. The non-radiative relaxation processes are vibrational relaxation (a rapid relaxation of excited molecules from higher to lowest vibrational level, occur within ∼10-14-10-12 s), internal conversion (a rapid relaxation of excited molecules to the lowest energy singlet excited state (S1) from higher energy excited singlet state (such as S2 in Fig. 7), occur within a time scale of 10-12 s), and intersystem crossing (relaxation between excited states of different spin multiplicity, such as S1→T1 in Fig. 7, occur within a time scale of 10-8 s). Intersystem crossing occurs more slowly than internal conversion since it's a less

probable process than internal conversion due to spin multiplicity is not conserved.

crossing and consequently make the spin forbidden transitions more probable.

The radiative processes are fluorescence and phosphorescence (Figure 9) (Lakowicz, 2006). The fluorescence refers to the emission of light associated with a radiative transition from an excited electronic state that has the same spin multiplicity as the ground electronic state (S1→S0, Fig. 7). Fluorescence transitions are spin allowed, thus they occur very rapidly and the average lifetimes of the excited states responsible for fluorescence are typically 10-9-10-5 s. Phosphorescence refers to the emission of light associated with a radiative transition from an excited electronic state that has a different spin multiplicity from that of the ground electronic state (T1→S0, Fig. 7). Phosphorescence transitions are spin forbidden, thus they are less probable than spin allowed transitions and the average lifetimes of the excited states responsible for phosphorescence are typically 10-3 s. However, spin forbidden transitions become more probable when a significant interaction between the spin angular moment and the orbital angular momentum is observed (spin-orbit coupling increases), this can be observed in the presence of heavy atoms. Furthermore, in solutions, the presences of paramagnetic species such as molecular oxygen increase the probability of intersystem

vertical (Rendell, 1987; Lakowicz, 2006).

Fig. 8. Typical absorption and emission spectra.

triplet state since the spin has to be flipped (i.e. *change in spin during the electronic transitions is not allowed*). Therefore, to go from a singlet to a triplet state (ΔM = 1) is so-called forbidden transition and occurs with a small rate constant and typically too weak to be of much importance. At room temperature the higher vibrational energy levels are in general not populated (less than 1% according to Boltzmann statistics). The magnitude of the absorbed energy decides which vibrational level of S1 (or S2) becomes populated. This process is very fast and happens within 10-15 s. In the next 10-12 s the molecule relaxes to the lowest vibrational level of S1, a process called internal conversion. Since emission typically occurs after 10-9 s the molecule is fully relaxed at the time of emission, hence, as a rule, emission occurs from the lowest vibrational level of S1 (Kasha.s rule) and the fluorescence spectrum is generally independent of the excitation wavelength. After emission the molecule returns to the ground state, possibly after vibrational relaxation. This completes the simplest case of fluorescence: excitation, internal conversion, emission and relaxation. The energy lost to the surroundings, due to vibrational relaxation and internal conversion is the reason why a Stokes' shift (*defined as the energy difference between the emission and absorption peak maxima for the same electronic transition*) is observed (Figure 8). The Stokes shift is due to the fact that some of the energy of the excited fluorophore is lost through molecular vibrations that occur during the brief lifetime of the molecule's excited state. This energy is dissipated as heat to surrounding solvent molecules as they collide with the excited fluorophore. The magnitude of the Stokes shift is determined by the electronic structure of the fluorophore, and is a characteristic of the fluorophore molecule (Rendell, 1987; Lakowicz, 2006).

Fig. 7. The Jablonski diagram. Four electronic levels are depicted along with four vibrational energy levels.

Since the energy spacing between the vibrational levels in S0 or S1 is of the same size, there often exist mirror image symmetry between the emission spectrum and the S0 → S1 absorption spectrum (not the S0 → S2 absorption) (Fig. 8), needless to say there are plenty of

triplet state since the spin has to be flipped (i.e. *change in spin during the electronic transitions is not allowed*). Therefore, to go from a singlet to a triplet state (ΔM = 1) is so-called forbidden transition and occurs with a small rate constant and typically too weak to be of much importance. At room temperature the higher vibrational energy levels are in general not populated (less than 1% according to Boltzmann statistics). The magnitude of the absorbed energy decides which vibrational level of S1 (or S2) becomes populated. This process is very fast and happens within 10-15 s. In the next 10-12 s the molecule relaxes to the lowest vibrational level of S1, a process called internal conversion. Since emission typically occurs after 10-9 s the molecule is fully relaxed at the time of emission, hence, as a rule, emission occurs from the lowest vibrational level of S1 (Kasha.s rule) and the fluorescence spectrum is generally independent of the excitation wavelength. After emission the molecule returns to the ground state, possibly after vibrational relaxation. This completes the simplest case of fluorescence: excitation, internal conversion, emission and relaxation. The energy lost to the surroundings, due to vibrational relaxation and internal conversion is the reason why a Stokes' shift (*defined as the energy difference between the emission and absorption peak maxima for the same electronic transition*) is observed (Figure 8). The Stokes shift is due to the fact that some of the energy of the excited fluorophore is lost through molecular vibrations that occur during the brief lifetime of the molecule's excited state. This energy is dissipated as heat to surrounding solvent molecules as they collide with the excited fluorophore. The magnitude of the Stokes shift is determined by the electronic structure of the fluorophore, and is a

characteristic of the fluorophore molecule (Rendell, 1987; Lakowicz, 2006).

Fig. 7. The Jablonski diagram. Four electronic levels are depicted along with four vibrational

Since the energy spacing between the vibrational levels in S0 or S1 is of the same size, there often exist mirror image symmetry between the emission spectrum and the S0 → S1 absorption spectrum (not the S0 → S2 absorption) (Fig. 8), needless to say there are plenty of

energy levels.

exceptions to the rule. The transitions that will predominate can be justified by Franck-Condon principle. The principle states that since the electronic absorption of light occurs in extremely short time (∼10-15s), thus during the time scale of absorption the nuclei are assumed to be frozen, that is that the transitions between various electronic levels are vertical (Rendell, 1987; Lakowicz, 2006).

Fig. 8. Typical absorption and emission spectra.

The typical molecular photoluminescence relaxation processes that illustrated by Jablonski diagram (Fig. 7) can be classified to two main type of transition, these are radiative and nonradiative transitions. The non-radiative relaxation processes are vibrational relaxation (a rapid relaxation of excited molecules from higher to lowest vibrational level, occur within ∼10-14-10-12 s), internal conversion (a rapid relaxation of excited molecules to the lowest energy singlet excited state (S1) from higher energy excited singlet state (such as S2 in Fig. 7), occur within a time scale of 10-12 s), and intersystem crossing (relaxation between excited states of different spin multiplicity, such as S1→T1 in Fig. 7, occur within a time scale of 10-8 s). Intersystem crossing occurs more slowly than internal conversion since it's a less probable process than internal conversion due to spin multiplicity is not conserved.

The radiative processes are fluorescence and phosphorescence (Figure 9) (Lakowicz, 2006). The fluorescence refers to the emission of light associated with a radiative transition from an excited electronic state that has the same spin multiplicity as the ground electronic state (S1→S0, Fig. 7). Fluorescence transitions are spin allowed, thus they occur very rapidly and the average lifetimes of the excited states responsible for fluorescence are typically 10-9-10-5 s. Phosphorescence refers to the emission of light associated with a radiative transition from an excited electronic state that has a different spin multiplicity from that of the ground electronic state (T1→S0, Fig. 7). Phosphorescence transitions are spin forbidden, thus they are less probable than spin allowed transitions and the average lifetimes of the excited states responsible for phosphorescence are typically 10-3 s. However, spin forbidden transitions become more probable when a significant interaction between the spin angular moment and the orbital angular momentum is observed (spin-orbit coupling increases), this can be observed in the presence of heavy atoms. Furthermore, in solutions, the presences of paramagnetic species such as molecular oxygen increase the probability of intersystem crossing and consequently make the spin forbidden transitions more probable.

Current Achievement and Future Potential of Fluorescence Spectroscopy 219

transitions; (2) lifetime of the excited state, the transition probability measured by the molar absorbitivity (ε), large ε implies short lifetime, thus largest fluorescence are observed from short lifetime and high ε state, as an example π\* → π > π\*→ n (10-9-10-7 s > 10-7-10-5 s); (3) structure of the molecule, emission of light is favored in aromatic molecules (conjugated systems) involving n → π\* and π → π\* transitions, and fluorescence increased by number of fused rings and substitution on or in the ring, such as pyridine, pyrrole, quinoline and indole; (4) rigidity of the structure, fluorescence quantum yield increases with increasing the rigidity of the molecules specially with chelation, such as fluorine and biphenyl; and (5) the fluorescence quantum yield is highly dependence on the temperature, pH and solvent

= Φ 2.303

Where *Io* is the incident radiant power, the term εcb is deduced from the well known Beers' law expression for the absorption (ε is the molar absorptivity, c is the molar concentration of the fluorescent substance, and b is the path length of the cell). The Beers' law is valid only

When an emission spectrum is obtained, data are typically collected for more than 0.1 sec. at each wavelength increment (typically 1nm), but since fluorescence lifetimes typically is measured in nanoseconds, it follows that the obtained spectrum is a time-average of a many events. The time averaging loses much information, and time-resolved experiments are often the more interesting when a system is investigated. The fluorescent lifetime of the excited state, τF, is the average time a molecule stays in the excited state before returning to ground state. Thus τF can be expressed as the inverse of the total depopulation rate as in

> τ <sup>=</sup> <sup>+</sup> 1

*F*

Where kF and knr are the rate constant of fluorescence and non-radiative processes,

Typically fluorescence lifetime values are in the 5-15 ns range. The expression in Eq. 3 is related to the expression for ΦF, in that way that they have a common denominator. Actually an approximation of τF can be obtained by measuring ΦF in aired and degassed solutions.

In the absence of non-radiative relaxation (*k*nr= 0), the lifetime becomes the inverse of *k*r and is often called the natural lifetime, denoted τN. For many compounds the natural lifetime can be calculated from the measured lifetime τ and the measured quantum yield ΦF, Equation

τ

It is important to notice that the fluorescent lifetime is what is experimentally obtained, and

*F*

τ <sup>=</sup> <sup>Φ</sup> *F*

*N*

ε

*F o I I cb* (2)

*F nr k k* (3)

(4)

(Rendell, 1987; Lakowicz, 2006).

The total fluorescence intensity (*I*) is given by Equation 2:

for diluted solutions (εbc< 0.05) (Guilbault, 1990)

Equation 3 (Rendell, 1987; Lakowicz, 2006).

(4) (Rendell, 1987; Lakowicz, 2006).

the natural lifetime can be calculated.

respectively.

Fig. 9. Typical excitation (E), fluorescence (F), and phosphorescence (P) spectra of phenanthrene (Lakowicz, 2006).

The relative intensity of fluorescence peak is controlled by the Frank-Condon principle, but also the total fluorescence peak intensity (*I*) is related to the fluorescence quantum yield (ΦF), which defined as the ratio of number of photons emitted to number of photons absorbed. Furthermore, the fluorescence quantum yield (ΦF) can be expressed as the rate of photons emitted divided by the total rate of depopulation of the excited state (Equation 1) (Rendell, 1987; Lakowicz, 2006).

$$
\Phi\_F = \frac{k\_F}{k\_F + \Sigma k\_{nr}} \tag{1}
$$

Where kF and knr are the rate constant of fluorescence and non-radiative processes, respectively. The fluorescence quantum yield (ΦF) value in the range of 0.0 to 1.0. If the nonradiative relaxation is fast compared to fluorescence (*k*nr > *k*r), Φ will be small, that is the compound will fluoresce very little or not at all. Often different non-radiative events are limited in the solid phase, and long-lived luminescence (e.g. phosphorescence) is often studied in frozen solution or other solid phases. Quenchers make non-radiative relaxation routes more favorable and often there is a simple relation between Φ and the quencher concentration. The best-known quencher is probably O2, which quench almost all fluorophores; other quenchers only quench a limited range of fluorophores. If a molecule is subject to intramolecular quenching, Φ may yield information about the structure.

The factors that affect the fluorescence quantum yield (ΦF) are: (1) the excitation wavelength (λex), the short wavelengths' break bonds and increase the rate constant of dissociation processes, the most common excitation wavelength are involving the n → π\* and π → π\*

Fig. 9. Typical excitation (E), fluorescence (F), and phosphorescence (P) spectra of

Φ =

*F*

subject to intramolecular quenching, Φ may yield information about the structure.

The factors that affect the fluorescence quantum yield (ΦF) are: (1) the excitation wavelength (λex), the short wavelengths' break bonds and increase the rate constant of dissociation processes, the most common excitation wavelength are involving the n → π\* and π → π\*

The relative intensity of fluorescence peak is controlled by the Frank-Condon principle, but also the total fluorescence peak intensity (*I*) is related to the fluorescence quantum yield (ΦF), which defined as the ratio of number of photons emitted to number of photons absorbed. Furthermore, the fluorescence quantum yield (ΦF) can be expressed as the rate of photons emitted divided by the total rate of depopulation of the excited state (Equation 1)

> + Σ *F*

Where kF and knr are the rate constant of fluorescence and non-radiative processes, respectively. The fluorescence quantum yield (ΦF) value in the range of 0.0 to 1.0. If the nonradiative relaxation is fast compared to fluorescence (*k*nr > *k*r), Φ will be small, that is the compound will fluoresce very little or not at all. Often different non-radiative events are limited in the solid phase, and long-lived luminescence (e.g. phosphorescence) is often studied in frozen solution or other solid phases. Quenchers make non-radiative relaxation routes more favorable and often there is a simple relation between Φ and the quencher concentration. The best-known quencher is probably O2, which quench almost all fluorophores; other quenchers only quench a limited range of fluorophores. If a molecule is

*F nr k*

*k k* (1)

phenanthrene (Lakowicz, 2006).

(Rendell, 1987; Lakowicz, 2006).

transitions; (2) lifetime of the excited state, the transition probability measured by the molar absorbitivity (ε), large ε implies short lifetime, thus largest fluorescence are observed from short lifetime and high ε state, as an example π\* → π > π\*→ n (10-9-10-7 s > 10-7-10-5 s); (3) structure of the molecule, emission of light is favored in aromatic molecules (conjugated systems) involving n → π\* and π → π\* transitions, and fluorescence increased by number of fused rings and substitution on or in the ring, such as pyridine, pyrrole, quinoline and indole; (4) rigidity of the structure, fluorescence quantum yield increases with increasing the rigidity of the molecules specially with chelation, such as fluorine and biphenyl; and (5) the fluorescence quantum yield is highly dependence on the temperature, pH and solvent (Rendell, 1987; Lakowicz, 2006).

The total fluorescence intensity (*I*) is given by Equation 2:

$$I\_F = \text{2.303I}\_o \Phi \text{εct} \tag{2}$$

Where *Io* is the incident radiant power, the term εcb is deduced from the well known Beers' law expression for the absorption (ε is the molar absorptivity, c is the molar concentration of the fluorescent substance, and b is the path length of the cell). The Beers' law is valid only for diluted solutions (εbc< 0.05) (Guilbault, 1990)

When an emission spectrum is obtained, data are typically collected for more than 0.1 sec. at each wavelength increment (typically 1nm), but since fluorescence lifetimes typically is measured in nanoseconds, it follows that the obtained spectrum is a time-average of a many events. The time averaging loses much information, and time-resolved experiments are often the more interesting when a system is investigated. The fluorescent lifetime of the excited state, τF, is the average time a molecule stays in the excited state before returning to ground state. Thus τF can be expressed as the inverse of the total depopulation rate as in Equation 3 (Rendell, 1987; Lakowicz, 2006).

$$\sigma\_F = \frac{1}{k\_F + k\_{nr}}\tag{3}$$

Where kF and knr are the rate constant of fluorescence and non-radiative processes, respectively.

Typically fluorescence lifetime values are in the 5-15 ns range. The expression in Eq. 3 is related to the expression for ΦF, in that way that they have a common denominator. Actually an approximation of τF can be obtained by measuring ΦF in aired and degassed solutions.

In the absence of non-radiative relaxation (*k*nr= 0), the lifetime becomes the inverse of *k*r and is often called the natural lifetime, denoted τN. For many compounds the natural lifetime can be calculated from the measured lifetime τ and the measured quantum yield ΦF, Equation (4) (Rendell, 1987; Lakowicz, 2006).

$$
\pi\_N = \frac{\pi\_F}{\Phi\_F} \tag{4}
$$

It is important to notice that the fluorescent lifetime is what is experimentally obtained, and the natural lifetime can be calculated.

Current Achievement and Future Potential of Fluorescence Spectroscopy 221

By combining anisotropy with time-resolved measurements it is possible to measure the mobility of a fluorophore. Immediately after excitation all excited molecules will be oriented along a common axis. In the solid phase the system will retain its anisotropy until emission. However, if the fluorophores are free to move, the anisotropy of the system will decrease

The principal sketch of a typical fluorescence spectrophotometer is shown in Figure 11. It consists of a light source, an excitation and emission monochromator (*grooves/mm*), polarizers (*prisms*), sample chamber and a detector (*such as photomultiplier tube*). For steady state measurements the light source usually consists of a 450W xenon arc lamp, and for time resolved measurements it is equipped with nanosecond flash lamp. Most simple spectrometers have a similar geometry, but often extra detectors and/or light sources are

Fig. 11. Schematic representation of a fluorescence spectrophotometer.

The light source produces light photons over a broad energy spectrum, typically ranging from 200 to 900 nm. Photons impinge on the excitation monochromator, which selectively transmits light in a narrow range centered about the specified excitation wavelength. The transmitted light passes through adjustable slits that control magnitude and resolution by further limiting the range of transmitted light. The filtered light passes into the sample cell causing fluorescent emission by fluorphors within the sample. Emitted light enters the

before emission.

**2.2 Instrumentation** 

fitted resulting in a T- or X-geometry.

The fluorescence lifetime, τF, is determined by observing the decay in fluorescence intensity (decay profile) of a fluorophore after excitation. Immediately after a molecule is excited the fluorescence intensity will be at a maximum and then decrease exponentially according to Equation 5 (Rendell, 1987; Lakowicz, 2006).

$$I(t) = I\_o e^{-t/\tau\_F} \tag{5}$$

Thus after a period of τF the intensity has dropped to 37% of *I0*, that is 63% of the molecules return to the ground state before τF. In many cases the above expression needs to be modified into more complex expressions. First of all it is assumed that the instrument yields an infinite (or very) short light pulse at time zero. In cases where τF is small *I0* must be replaced by a function, which describes the lamp profile of the instrument. Also, more than one lifetime parameter is often needed to describe the decay profile, which is I(t) must be expressed as a sum of exponentials. Finally the concept of anisotropy should be mentioned. Anisotropy is based on selectively exciting molecules with their absorption transition moments aligned parallel to the electric vector of polarized light. By looking at the polarization of the emission the orientation of the fluorophore can be measured. The anisotropy of the system is defined as (Equation 6) (Rendell, 1987; Lakowicz, 2006):

$$r = \frac{I\_{\parallel} - I\_{\perp}}{I\_{\parallel} + 2I\_{\perp}} = \frac{3\left\{\cos^{2}(\theta)\right\} - 1}{2} \tag{6}$$

The oval in Figure 10 symbolized the absorption transition moment. Vertical polarized excitation light enters along the x-axis and *I*<sup>⊥</sup> and *I*║ are measured along the y-axis, setting the emission polarizer perpendicular and parallel to the excitation polarizer respectively. Θ is the angle of the emission to the z-axis (see Figure 10), the squared brackets indicates that it is the average value. If all absorption transition moments are aligned along the z-axis then *I*<sup>⊥</sup> = 0 and θ = 0, leading to *r* = 1, the maximum anisotropy.

Fig. 10. The absorption dipole is aligned along the z-axis. The excitation light is vertically aligned and follows the x-axis. Emission is measured along the y-axis.

By combining anisotropy with time-resolved measurements it is possible to measure the mobility of a fluorophore. Immediately after excitation all excited molecules will be oriented along a common axis. In the solid phase the system will retain its anisotropy until emission. However, if the fluorophores are free to move, the anisotropy of the system will decrease before emission.
