1 Background

Fig. 1 illustrates a three level energy level structure of commonly used fluorescent molecule, where the \(S_{0} \rightarrow S_{1}\) transition represents the singlet-singlet transition to a higher energy vibration state induced by incident light of energy \(h \nu\). After the molecule relaxes to the lowest electronic excited state (\(S_{1}\)), fluorescent molecules will be emitted if relaxation occurs through the pathway \(S_{1} \rightarrow S_{0}\). However, additional nonradiative relaxation pathways such as internal conversion, external conversion, inter-system crossing (\(k_{ICS}\) in Fig. 1) into a triplet state, etc. may be possible. In general, relaxation to the ground state (\(S_{0}\)) include energy loss steps which induce a redshift (Stokes shift) to lower energy, longer wavelength emission.

The probability a single dye will absorb an incident photon is given by Eq. \(\ref{eqn:pdf_abs}\):

\[\begin{equation} \label{eqn:pdf_abs}\tag{1} P_{\text{absorption}} = \frac{\sigma_{p}}{A} \end{equation}\]

where \(\sigma_{p}\) is the absorption cross section of the dye and \(A\) is the cross section area of the incident light. Incident photon flux (photons per unit time per square area) can be calculated using Eq. \(\ref{eqn:flux}\):

\[\begin{equation} \label{eqn:flux}\tag{2} H\left(\frac{W}{m^{2}}\right) = \eta \frac{hc}{\lambda} \end{equation}\]

where \(H\) is the power density, \(\eta\) is the photon flux, \(h\) is Plancks constant, \(c\) is the speed of light, and \(\lambda\) is the wavelength of incidence light.

The dye absorption cross section can be thought of as the area of the molecule capable of capturing incident photons. The peak cross sectional absorbance of a randomly oriented molecule is given by Eq. \(\ref{eqn:abs_cs}\):

\[\begin{equation} \label{eqn:abs_cs}\tag{3} \sigma_{p} = 2\pi \left(\lambda/2 \pi\right)^{2}\left(\frac{\gamma_{r}}{\Gamma_{tot}}\right) \end{equation}\]

where \(\lambda\) is the indecent light wavelength, \(\gamma_{r}\) is the spontaneous fluorescence rate, and \(\Gamma_{tot}\) is the total frequency width of the absorption. For most organic dyes used in fluorescence spectroscopy/microscopy, \(\gamma_{r} \sim 10^{-3} \text{cm}^{-1}\) at room temperature, in aqueous solution, using \(\lambda \sim 500 \text{nm}\). This corresponds to a fluorescent radiative lifetime of a few nanoseconds and \(\Gamma_{tot} \sim 1000 \text{cm}^{-1}\), yielding a \(\sigma_{p} \sim 4 \ \mathring{A}^{2}\), which is similar to the molecular size of the dye. At room temperature and in the same solvent used for extinction coefficient determination (\(\epsilon\)), one can also relate \(\sigma_{p}\) to \(\epsilon\) through the equation \(\sigma_{p} = \frac{2.303 \epsilon}{N_{A}}\) where \(N_{A}\) is Avogadros number.

The probability of photon emission per absorption event is another key metric controlling the quantity of emitted fluorescence. This probability is defined by the fluorescence quantum yield \(\left(\phi_{F}\right)\),

\[\begin{equation} \label{eqn:qy}\tag{4} \phi_{F} = \frac{k_{rad}}{\left(k_{rad}+k_{\text{nonrad}}\right)} = \frac{\tau_{F}}{\tau_{rad}} \end{equation}\]

where \(k_{rad}\) is the radiative fluorescence rate constant, \(k_{\text{nonrad}}\) is the sum of all rate constants for nonradiative relaxation pathways (inter-sytem crossing, etc.), \(\tau{F}\) is the fluorescence lifetime, and \(\tau_{rad}\) is the radiative lifetime. Typical dyes used in fluorescence microscopy have \(\phi_{F} \sim 0.45-0.6\) in water and \(\tau_{F} \sim \text{a few nanoseconds}\). For an example dye with \(\phi_{F} \sim 0.45, \tau \sim 4 \text{ ns}\), the maximum emission rate would be \(\sim 10^{8} s^{-1}\).

A number of factors limit the maximum emission rate of any dye. One major limiting factor is optical saturation of the ground-excited state transition. As laser power goes up, so does the photon emission rate (per second) until transition becomes saturated. At this point, the effective absorption cross section of the dye shrinks and any further incident light power only acts to increase background photons. The effective cross section of absorption depends on the incident light intensity according to Eq. \(\ref{eqn:lp}\):

\[\begin{equation} \label{eqn:lp}\tag{5} \sigma_{p} = \frac{\sigma_{p}^{0}}{\left(1+\frac{I}{I_{S}}\right)} \end{equation}\]

where \(I_{S}\) is the characteristic saturation intensity, which itself depends on the dye energy level structure and indicates the fact that if the absorption rate exceeds a certain threshold, the dye will not relax to ground state fast enough (limiting the ability to absorb photons). Assuming the dye approximates a three level system (Fig. 1),

\[\begin{equation} \label{eqn:Is}\tag{6} I_{S} = \frac{h \nu}{2 \sigma \tau_{21}}\frac{1+\left(\frac{k_{ISC}}{k_{21}}\right)}{1+\left(\frac{k_{ISC}}{2 k_{T}}\right)} \end{equation}\]

where \(k_{21}=\frac{1}{\tau_{21}}\) is the rate of direct relaxation for \(S_{1} \rightarrow S_{0}\), \(k_{T}\) is the total decay rate from \(T_{1} \rightarrow S_{0}\), and \(k_{ISC}\) is the inter-system crossing rate constant. This smaller saturation intensity indicates that the laser power that can be employed to effectively probe a dye is limited for a fixed focal point.