2.1.

The Photobleaching Process of Fluorescein

A detailed description of the photobleaching kinetics of fluorescein has been published by Song ²⁵ As the photobleaching process in the computer simulations of this study is based on the work of Song, we will repeat the main conclusions here for convenience of the reader. A fluorophore can make an electronic transition from the ground state S to the excited singlet state ${\mathrm{S}}^{*}$ (see Fig. 1 ) by absorbing a photon of wavelength $\lambda$ (m) and corresponding energy $hc∕{\lambda }_{\mathit{ex}}$ , where $h$ is Planck’s constant and $c$ is the speed of light. If the excitation light has an irradiance $H$ $(\mathrm{W}∕{\mathrm{m}}^2)$ and the fluorophore has an absorption cross-section ${\sigma }_a$ $({\mathrm{m}}^2∕\text{molecule})$ , the rate of photon absorption $k_a$ $\left({\mathrm{s}}^{-1}\right)$ is

Fig. 1

A simplified Jablonski energy diagram for a typical fluorophore showing possible transitions between the ground state S, the excited singlet state ${\mathrm{S}}^{*}$ , and the excited triplet state ${\mathrm{T}}^{*}$ , as explained in the main text.

For fluorescein it has been demonstrated that the short-lived excited singlet state does ordinarily not contribute to the photobleaching mechanism. The long-lived triplet excited state, at the other hand, becomes depopulated via two major pathways: (1) the reaction between a triplet and another triplet or a ground state fluorescein molecule, and (2) the reaction between a triplet fluorescein molecule and an oxygen molecule. These two mechanisms are termed the dye-to-dye (D-D) and dye-to-oxygen (D-O) mechanisms and can lead to either reduced (R) or semi-oxidized (X) radical forms. These radicals can again react with triplet state molecules and revert either to stable nonfluorescent photoproducts or back to the ground state. Song ²⁵ have incorporated all relevant photochemical reactions into a theoretical model comprising the following six coupled differential equations:

The individual photochemical reactions and the corresponding rate constants as reported in Song ²⁵ are shown in Table 1 . This set of differential equations has to be solved numerically for a given rate of photon absorption $k_a$ and initial values for the concentrations $N_{\mathrm{S}}$ , $N_{{\mathrm{S}}^{*}}$ , $N_{{\mathrm{T}}^{*}}$ , $N_{\mathrm{X}}$ , $N_{\mathrm{R}}$ , and $N_{{\mathrm{O}}_2}$ . The rate of photon absorption $k_a$ can be calculated from Eq. 6 for a given irradiance $H$ .

Table 1

Photochemical reactions of fluorescein.

2.2.

Photobleaching of Fluorescein by a Focused Laser Beam

When a laser beam is focused down through an objective lens, a complex 3-D intensity distribution results around the focal point. For clarity and because most FRAP models are limited to diffusion in two dimensions, the study presented here will be restricted to two dimensions as well (focal plane). The conclusions from this study equally apply to the third dimension as well. The radial irradiance distribution in the focal plane of a (confocal) microscope’s objective lens is in good approximation given by the Fraunhofer–Airy formula, which is valid for small diffraction angles:^{27, 28, 29}

Fig. 2

The illumination intensity distribution in the focal plane. (a) The diffraction pattern in the focal plane of the $10\times$ objective lens has been recorded by imaging a $190\text{-}\mathrm{nm}$ fluorescent nanosphere with the confocal microscope. The confocal diaphragm was opened completely to allow the light coming from higher-order diffraction rings to enter the detector. Besides the central maximum, the first- and second-order diffraction rings are visible as well after applying a logarithmic color map. The intensity distribution along the vertical direction in the image is also shown (solid line). The central peak can be very well approximated by a Gaussian intensity distribution (dashed line). (b) For comparison, the diffraction pattern according to the Fraunhofer–Airy formula is shown as well using the same color map. Again the central peak of the theoretical curve (solid line) can be very well approximated by a Gaussian intensity distribution (dashed line).

For most fluorescence applications it is sufficient to take the central part of the diffraction pattern into account. For example, the central peak of the Fraunhofer–Airy distribution contains 84% of the total amount of light in the diffraction pattern while the subsequent higher-order maxima have a relative intensity of only 1.7%, 0.4%, 0.1%, etc. Hence, also for FRAP models a more simple Gaussian distribution is ordinarily used to describe the BID:

FRAP models that explicitly take the BID into account never use the relatively complicated Eq. 8 but rather the simplified Gaussian intensity distribution Eq. 9 ^{2, 16, 17, 18, 21} or possibly a 3-D extension with an additional axial (modified) Gaussian distribution.^{11, 15, 23} However, as we will show further on, the higher-order diffraction maxima cannot be neglected any longer in case of fluorescein being bleached by a high-intensity laser beam. Under the extreme illumination conditions of the bleaching phase, even the higher-order diffraction maxima will be able to populate the excited triplet state ${\mathrm{T}}^{*}$ significantly, resulting in a complex expanded bleaching profile that can no longer be explained by simple first-order photobleaching with a Gaussian intensity distribution.

3.1.

The Bleaching and Imaging System

The FRAP experiments are performed on a CSLM (model MRC1024 UV, Bio-Rad, Hemel Hempstead, UK) equipped with the SCAMPER module for photobleaching experiments.^{31, 32} The module consists of a $4\mathrm{W}$ Ar-ion laser (model Stabilite 2017; Spectra-Physics, Darmstadt, Germany) and an acousto-optical modulator (AOM) (Brimrose, Campbell, Baltimore, MD, USA), controlled by a computer and dedicated software. The AOM controls the intensity of the laser beam that is sent to the CSLM. A high-power laser beam (“bleaching beam”) is used for photobleaching, while a low-power beam (“detection beam”) is used for the recording of the confocal images. Bleaching masks can be designed in the software that controls the AOM. While recording an image with the CSLM, low- and high-power laser light is sent to the sample according to the selected bleaching mask.

All bleaching experiments have been performed with the $488\text{-}\mathrm{nm}$ line from the Ar-ion laser. The intensity of the laser beam at the sample can be controlled over a range of $0\text{to}10\mathrm{mW}$ . A $10\times$ objective lens (CFI Plan Apochromat; Nikon, Badhoevedorp, The Netherlands) with a numerical aperture (NA) of 0.45 was used for bleaching. On the Bio-Rad confocal microscope, the back aperture of this lens is only partially filled, however, resulting in a lower effective NA of about 0.2 for the illumination beam and a corresponding resolution $w=1.5\mu \mathrm{m}$ (and $r_0=1.0\mu \mathrm{m}$ ). The resolution was determined by directly imaging $190\text{-}\mathrm{nm}$ fluorescent nanospheres (Molecular Probes, Leiden, The Netherlands) with a completely opened confocal diaphragm. Imaging of the bleached profiles was done with a high-resolution $60\times$ water immersion lens of 1.2 NA (CFI Plan Apochromat; Nikon, Badhoevedorp, The Netherlands) having a resolution $w$ of approximately $0.25\mu \mathrm{m}$ .

3.2.

Photobleaching Experiments

Fluorescein isothiocyanate dextrans (Sigma-Aldrich, Bornem, Belgium) of $2000\mathrm{kDa}$ molecular weight (FD2000) were incorporated in an acrylamide gel. The acrylamide gel was prepared by radical polymerization of an aqueous acrylamide solution (30% w/w). The solution was prepared by dissolving acrylamide in deoxygenated phosphate buffer ( $10\mathrm{mM}$ ${\mathrm{Na}}_2\mathrm{H}\mathrm{P}{\mathrm{O}}_4$ , 0.02% sodium-azide, adjusted with $1\mathrm{N}$ hydrochloric acid to pH 7.0). Prior to addition of the gelation reagents, $3.915\mathrm{mg}∕\mathrm{g}$ gel of FD2000 was added. The polymerization reagents were $\mathrm{N},\mathrm{N},{\mathrm{N}}^{\prime },{\mathrm{N}}^{\prime }$ -tetramethylene-ethylenediamine (TEMED; Fluka, Buchs, Switzerland) and ammonium persulphate (10% w/v) (APS; VWR International, Leuven, Belgium). The procedure involves adding $0.5\mu \mathrm{L}$ TEMED solution to $1\mathrm{g}$ hydrogel. After mixing, $2.5\mu \mathrm{L}$ APS solution was added to initiate gelation. The gelating mixture was transferred into a cuvette with a glass bottom for use with an inverted microscope (Nalge Nunc International, Naperville, IL, USA). The gelation required $1\text{hour}$ at room temperature. FRAP measurements using the CSLM¹¹ revealed a very low diffusion coefficient of $0.020\pm 0.005\mu {\mathrm{m}}^2∕\mathrm{s}$ and a large fraction of immobile fluorescent FITC-dextrans of $90\pm 5\%$ .

Bleaching experiments were performed with a stationary laser beam (“beam parked”) or a line scanning beam. The laser power at the sample was between $1\mu \mathrm{W}$ and $5\mu \mathrm{W}$ for imaging and up to $10\mathrm{mW}$ for bleaching. The time between the bleaching phase and imaging of the bleached profiles was approximately $30\min$ to allow diffusion of the (small) mobile fraction to complete. After that time, the remaining bleached profile is due to the immobile fraction and reflects directly the bleaching profile immediately after bleaching. Confocal images of the bleaching profiles were recorded with the $60\times 1.2$ NA objective lens to obtain the best possible resolution. Before normalizing the confocal images to the background fluorescence (i.e., unbleached fluorophores), they were convolved with a median smoothing mask to reduce the noise.

3.3.

Photobleaching simulations

To study the bleaching of fluorescein at the focal plane, a computer program was written to solve the six coupled differential Eqs. 7 numerically. For the simulation of the spot photobleaching experiments, the absorption rate $k_a\left(r\right)$ is calculated from Eqs. 6, 8. At first a set of radial coordinates $r$ is defined (the spatial mesh) together with the initial concentrations of $N_{\mathrm{S}}$ , $N_{{\mathrm{S}}^{*}}$ , $N_{{\mathrm{T}}^{*}}$ , $N_{\mathrm{X}}$ , $N_{\mathrm{R}}$ , and $N_{{\mathrm{O}}_2}$ . In the simulations reported here, it is always assumed that all fluorescein molecules are initially in the ground state $(N_{{\mathrm{S}}^{*}}=N_{{\mathrm{T}}^{*}}=N_{\mathrm{X}}=N_{\mathrm{R}}=0)$ . Next, the differential equations are solved for each of the radial coordinates and the corresponding value of $k_a\left(r\right)$ . When the illumination time $t$ has passed, in general there still will be a number of fluorescein molecules in the excited singlet and triplet state: $N_{{\mathrm{S}}^{*}}$ & $N_{{\mathrm{T}}^{*}}>0$ . Therefore, even when the illumination is switched off, the photobleaching reactions still can go on for some time until all the molecules are either bleached or back in the ground state. This has been taken into account in the simulation program by setting $k_a$ to zero after the bleaching time $t$ and continue to solve the differential equations until the number of bleached molecules $(N_X+N_R)$ together with the number of ground state molecules $N_{\mathrm{S}}$ are equal to the initial number of molecules.

In FRAP experiments on the CSLM, a scanning beam is often used for bleaching rather than a stationary beam. ^{11, 17, 23, 32, 33, 34, 35, 36} Therefore, the program for the simulation of the photobleaching of fluorescein was extended for a line-scanning beam as well. When bleaching a line segment, the bleaching will be uniform along the scanning direction (except at the beginning and the end of the line). Only in the direction perpendicular to the scanning direction a bleaching gradient will exist. It is therefore sufficient to calculate the bleaching profile perpendicular to the scanning direction for one position on the bleached line segment only. In the simulation algorithm, the line scanning is taken into account by first defining a start and end position of the laser beam ( $x_s$ and $x_e$ ) relative to the position on the line at which the bleaching profile is calculated $\left(x_0\right)$ : the laser beam starts at position $x_s$ and is moved at a scanning speed $\nu$ in discrete steps $\Delta x$ toward the position $x_e$ . At each step the laser beam is stationary for a time $\Delta t=\Delta x∕\nu$ and the effect is calculated at position $x_0$ . The smaller the step size $\Delta x$ , the better the continuous movement of the laser scanning beam will be approximated (at the cost of an increased calculation time). All line scanning simulations reported here were calculated with a step size that is one fifth of the resolution of the bleaching beam $(\Delta x=w∕5)$ since no appreciable differences in the calculated bleaching profiles were observed for smaller step sizes. The start and end positions of the bleaching beam were always taken symmetrically at 10 times the resolution of the bleaching beam: $x_s=x_0-10w$ and $x_e=x_0+10w$ .

4.1.

Photobleaching of Immobilized FD2000

To visualize the bleaching profiles at different bleaching conditions, FITC-dextrans of molecular weight $2000\mathrm{kDa}$ were incorporated into an acrylamide gel. Using a stationary laser beam, spots were photobleached in the gel with a low NA objective lens for different bleaching times. Confocal images were subsequently recorded using a high NA objective lens to obtain the best possible resolution. Some examples are shown in Fig. 3 .

Fig. 3

Using a stationary laser beam, spots are photobleached in an acrylamide gel loaded with FITC-dextrans. All images are shown to the same scale. (a) At a low intensity of $2\mu \mathrm{W}$ and a bleaching time of $125\mathrm{ms}$ , only the central peak of the BID contributes to the bleaching process. (b) At a high bleaching intensity of $10\mathrm{mW}$ and a bleaching time of $100\mu \mathrm{s}$ , also the first-order diffraction ring is able to cause photobleaching. (c) For a bleaching time of $500\mu \mathrm{s}$ and (d) $1000\mu \mathrm{s}$ at $10\mathrm{mW}$ , the contribution of the first and second diffraction ring is becoming even more pronounced. (e) When increasing the illumination time to $100\mathrm{ms}$ , the fourth diffraction ring is clearly showing up as well, together with various diffraction spots. The square root of the original image is displayed here for better visibility of all features.

To be able to relate the results of this study directly to the existing FRAP models (see Discussion), we will now define the “apparent BID” as the intensity distribution one would have to take into account to explain the experimentally observed bleaching profile if the bleaching process would follow first-order kinetics. In other words, the “apparent BID” is the intensity distribution $I_{\mathrm{app}}$ that has to be used in Eq. 2 to correctly calculate the observed real bleaching profile. In case first-order photobleaching is valid, $I_{\mathrm{app}}$ should be identical to the intensity distribution as shown in Fig. 2a. In the experiments below we will examine if this is true or not under a variety of photobleaching conditions. If $C_b$ denotes the fluorescence concentration distribution after photobleaching, the normalized “apparent BID” can be calculated from

The first spot (Fig. 3a) was bleached at a low intensity of $2\mu \mathrm{W}$ , which is typical for confocal imaging conditions, and a bleaching time of $125\mathrm{ms}$ . The apparent BID calculated from Fig. 3a using Eq. 10 is shown in Fig. 4a . A best fit of Eq. 9 shows there is a very good correspondence of the apparent BID with a Gaussian intensity profile, in accordance with what is assumed theoretically in the FRAP models. We note that the bleached spot was not perfectly circular but was stretched along the $y$ -direction resulting in a slightly increased Gaussian radius. This is probably caused by a small difference in resolution in $x$ and $y$ because of the polarized laser light.³⁷ To check the validity of a Gaussian BID for high-intensity laser beams, spots were bleached at $10\mathrm{mW}$ and different bleaching times: $25\mu \mathrm{s}$ (image not shown), $50\mu \mathrm{s}$ (image not shown), $100\mu \mathrm{s}$ (Fig. 3b), $500\mu \mathrm{s}$ (Fig. 3c), $1000\mu \mathrm{s}$ (Fig. 3d), and $100\mathrm{ms}$ (Fig. 3e). A Gaussian apparent BID is still found for very short bleaching pulses of $25\mu \mathrm{s}$ (Fig. 4b) and $50\mu \mathrm{s}$ (Fig. 4c), although a slight broadening of the profile is already observed compared to Fig. 4a. When bleaching for $100\mu \mathrm{s}$ (Fig. 4d), the higher-order diffraction rings are already starting to show up as indicated by the arrows and the corresponding apparent BID starts to deviate from a pure Gaussian distribution. For bleaching times of $500\mu \mathrm{s}$ (Fig. 4e) and $1000\mu \mathrm{s}$ (Fig. 4f) the higher-order diffraction rings become much more pronounced, expanding the apparent BID even more. Although the apparent BIDs do no longer correspond to a simple Gaussian distribution, it is worth noting that it is still possible to roughly approximate the apparent BIDs by an (expanded) Gaussian intensity distribution. Figure 3e, finally, shows the situation if the bleaching time is prolonged to $100\mathrm{ms}$ . Besides the fourth-order diffraction ring, all kinds of diffraction spots take part in the bleaching process. Fitting of a Gaussian intensity distribution to the apparent BID in this case (data not shown) yielded a highly increased Gaussian resolution of $r_0=7.6\mu \mathrm{m}$ . These experiments clearly show that the bleaching profiles obtained at high laser powers cannot be explained by first-order photobleaching alone. Assuming first-order kinetics leads to an apparently variable BID, which physically is nonsense. In reality, a higher laser power results in a higher amplitude of the BID but cannot change the shape of the distribution.

Fig. 4

The apparent BID in case of fluorescein depends on the specific bleaching conditions. (a) At a low bleaching intensity of $2\mu \mathrm{W}$ , only the central peak of the diffraction pattern effectively takes part in the bleaching process. Hence, the apparent BID corresponds to a Gaussian distribution. (b) At $10\mathrm{mW}$ and $25\mu \mathrm{s}$ bleach time, the apparent BID can still be approximated by a Gaussian distribution, although an expansion is already observed. (c) For a bleaching time of $50\mu \mathrm{s}$ , the apparent BID expands even more and (d) for $100\mu \mathrm{s}$ , the individual diffraction rings are becoming visible as indicated by the arrows. (e) Bleaching for $500\mu \mathrm{s}$ and (f) $1000\mu \mathrm{s}$ increases the relative contribution of the higher-order diffraction rings even more, causing a substantial expansion of the original BID.

Besides being dependent on the bleaching time, the apparent BID also depends on the bleaching intensity. Since many FRAP techniques make use of a scanning beam for bleaching, a line photobleaching experiment instead of spot photobleaching was chosen here to explicitly show that the results from this study equally apply to a scanning beam as well. Three line segments of $80\mu \mathrm{m}$ are bleached at different bleaching powers, but with an equal total amount of light received. A first line is bleached by scanning two times at $10\mathrm{mW}$ . A second one is bleached at $1\mathrm{mW}$ by scanning 20 times. A third one is bleached at $75\mu \mathrm{W}$ by scanning 267 times. A confocal image of the right part of each of the line segments is shown in Fig. 5 . The fluorescence intensity profile (along the $y$ -direction) and the corresponding apparent BID are shown for each line in Figs. 5a, 5b, 5c. Assuming first-order bleaching kinetics, the bleaching profile of a single line segment [a, b] can be calculated from Eq. 3 with $\Delta y=1$ and $K(x,y,z)={\int }_a^b{I}_b(x-x^{\prime },y,z)d{x}^{\prime }$ .¹¹ For a Gaussian BID according to Eq. 9 and a line segment that is much longer than the resolution $r_0$ , the resulting bleaching intensity distribution finally becomes

Fig. 5

Three line segments are photobleached at different bleaching intensities but with an equal total amount of light received. (a) Scanning twice at $10\mathrm{mW}$ , (b) 20 times at $1\mathrm{mW}$ , and (c) 267 times at $0.075\mathrm{mW}$ results each time in a different bleaching profile. Although all lines have received the same amount of light, bleaching is most efficient when scanning multiple times at a lower laser intensity. Bleaching at low intensities has the additional advantage of preserving a Gaussian distribution for the apparent BID, while an expansion is observed at high intensity levels.

4.2.

Simulation of Photobleaching Immobilized Fluorescein

The photochemical reactions according to Table 1 and the corresponding mathematical model Eq. 7 can be used to explain the observations of the previous section. We will show that the expansion of the bleaching profiles—which is unexpected from classical FRAP models, which use first-order photobleaching to calculate the bleaching profile—can indeed be explained by the complex photobleaching reactions of fluorescein in combination with the very high laser powers that are used for the bleaching phase of an FRAP experiment. We note that, as most of the FITC-dextrans are immobilized in the acrylamide gel, the contribution of the D-D interactions to the bleaching of fluorescein is expected to be very small and virtually all bleaching will be due to the D-O reactions. This can be taken into account in the simulations by setting the rate constants $k_2$ , $k_3$ , $k_4$ , $k_5$ , $k_6$ , and $k_7$ to zero. The initial concentration of fluorescein in the gel was approximately $250\mu \mathrm{M}$ (which can be calculated from the concentration of FD2000 in the gel and the loading degree of fluorescein on the dextran chains). The initial oxygen concentration is taken to be $250\mu \mathrm{M}$ as well.²⁵ Assuming that all fluorescein molecules are initially in the ground state, the simulations therefore always start from the initial condition: $N_{\mathrm{S}}=250\mu \mathrm{M}$ , $N_{{\mathrm{O}}_2}=250\mu \mathrm{M}$ , and $N_{{\mathrm{S}}^{*}}=N_{{\mathrm{T}}^{*}}=N_{\mathrm{X}}=N_{\mathrm{R}}=0$ .

We simulated the photobleaching of immobilized fluorescein as a function of time first to examine the evolution of the photobleaching process. The solid lines in Fig. 6 are calculated for a bleaching intensity of $10\mathrm{mW}$ and a bleaching time of $1\mathrm{ms}$ . From ${10}^{-11}\mathrm{s}$ on, the ground state S is being depleted to the advantage of the excited singlet state ${\mathrm{S}}^{*}$ that peaks at around ${10}^{-8}\mathrm{s}$ . After ${10}^{-8}\mathrm{s}$ , the excited singlet state ${\mathrm{S}}^{*}$ starts to decay to both the ground state S and the long-lived excited triplet state ${\mathrm{T}}^{*}$ . Between ${10}^{-8}$ and ${10}^{-6}\mathrm{s}$ , almost all fluorescein molecules are gradually getting trapped in the long-lived excited triplet state ${\mathrm{T}}^{*}$ . Once the triplet state population starts to rise, the D-O reactions can start producing the photobleaching products X and R, which are formed from ${10}^{-7}\mathrm{s}$ on. The formation of the photobleaching products X and R cause the triplet state to be depleted, together with the number of oxygen molecules. At ${10}^{-4}\mathrm{s}$ , almost no molecules are left in the ground state because all are either photobleached or in the triplet state. At ${10}^{-3}\mathrm{s}$ , the illumination is switched off $(k_a=0)$ and the reactions are allowed to continue until all molecules are either photobleached or back in the ground state.

Fig. 6

The evolution over time of fluorescein molecules in the various energy levels is simulated for an illumination intensity of $10\mathrm{mW}$ (solid lines) and $0.17\mathrm{mW}$ (dashed lines). The evolution of the number of oxygen molecules and the photobleaching products is shown as well. All populations are normalized to the initial fluorophore population at the ground state. The bleaching time is $1\mathrm{ms}$ , after which the illumination is switched off $(k_a=0)$ and the reactions are allowed to continue until all fluorescein molecules are photobleached or back in the ground state.

The first-order maximum of the BID has a relative intensity of 1.7% (see Fig. 2b). To see what happens at this intensity level, the same calculations were repeated at a bleaching power of $0.17\mathrm{mW}$ (Fig. 6, dashed lines). Essentially the same processes are taking place, albeit somewhat delayed because of the reduced illumination intensity. It is important to note that the triplet state still becomes significantly populated to approximately 30%, despite the very much reduced illumination intensity. As a result, about the same level of bleaching is reached after ${10}^{-3}\mathrm{s}$ , which explains why the higher-order diffraction rings are showing up in the bleaching experiments.

To see this more clearly, calculations have been performed for a BID according to Eq. 8 (also see Fig. 2b) at a bleaching power of $10\mathrm{mW}$ and for different bleaching times. Figure 7a shows the result for $t=25\mu \mathrm{s}$ . The apparent BID is again calculated from the bleaching profile using Eq. 10. For comparison with the experimental data (Fig. 4b), it should be realized that the simulated bleaching profile is not what is observed directly through a microscope. When observing an object through a microscope, it is modulated by the microscope’s total point spread function (PSF), which depends directly on the objective lens being used.^{29, 38} The detection PSF of an objective lens in the focal plane of a confocal microscope can be approximated by a Gaussian intensity distribution corresponding to the central diffraction peak. This is allowed because the higher-order diffraction rings only have a small contribution to the collected fluorescence compared to the central peak and because the higher-order diffraction rings are largely blocked by the small confocal apperture (see, e.g., Fig. 2 of Jonkman and Keller²⁹). The $60\times$ objective lens used to image the bleaching profiles has a Gaussian resolution $r_0$ of approximately $0.2\mu \mathrm{m}$ . To obtain a more direct comparison between the simulated and experimental data, the simulated bleaching profile is therefore convolved with a corresponding Gaussian mask. The result is shown in Fig. 7b. The individual maxima seem less pronounced because of the limited resolving power of the objective lens. Together with the presence of noise, this explains why the individual maxima are not always easy to discern in the experimental apparent BIDs (compare with Fig. 4). The same calculations have been performed for $t=50\mu \mathrm{s}$ (Fig. 7c), $100\mu \mathrm{s}$ (Fig. 7d), $500\mu \mathrm{s}$ (Fig. 7e), and $1000\mu \mathrm{s}$ (Fig. 7f). Only the convolved apparent BIDs together with a Gaussian fit are shown for clarity. Comparing Figs. 4b, 4c, 4d, 4e, 4f with Figs. 7b, 7c, 7d, 7e, 7f shows that the theory corresponds well with what is observed experimentally: an apparent BID that expands with increasing bleaching time due to the increasing contribution to the bleaching process of the higher-order diffraction rings. Again we find that each of the expanded apparent BIDs can roughly be approximated by a Gaussian distribution.

Fig. 7

The bleaching of immobilized fluorescein by a stationary laser beam is simulated for a laser power of $10\mathrm{mW}$ and different bleaching times. The dotted line shows the intensity distribution of the illumination BID according to the Fraunhofer–Airy formula [Eq. 8] for a resolution of $1.5\mu \mathrm{m}$ . (a) The solid line shows the bleaching profile for $t=25\mu \mathrm{s}$ . The bold line is the corresponding apparent BID. (b) When an object is observed by a microscope, it is modulated by the total PSF, which depends primarily on the objective lens that is used. The results of (a) are therefore convolved with a Gaussian mask having a resolution radius of $0.2\mu \mathrm{m}$ in accordance with the properties of the $60\times$ objective lens used in the experimental section. As a result, the individual maxima appear less pronounced. The dashed line is a best fit of a Gaussian intensity distribution to the apparent BID. The fitted Gaussian resolution is specified in the legend. (c–f) The same calculations are repeated for bleaching times of $50\mu \mathrm{s}$ , $100\mu \mathrm{s}$ , $500\mu \mathrm{s}$ , and $1000\mu \mathrm{s}$ . The longer the bleaching time, the more the higher-order diffraction maxima contribute to the apparent BID. As a result, the apparent BID expands with increasing bleaching time.

It is worth noting that the simulation results not only correspond qualitatively to the experimental observations, but that there is also a reasonable quantitative correspondence between both. The simulation parameters used to obtain the results from Fig. 7 were chosen thus to directly reflect the experimental conditions leading to the results presented in Fig. 4. Figure 7b is in fact the simulation of the experiment in Figs. 4b and 7c is the simulation of 4c and so on (the experiment of Fig. 4a was not simulated). In all cases a Gaussian distribution was fitted to the apparent BID to quantify the expansion through the Gaussian resolution parameter $r_0$ . The values for $r_0$ are reported in each of the figures, from which it can be seen that there is a fairly good correspondence between the experimental and simulated values. The simulated results deviate from the experimental ones only by 19%, 9%, 7%, 12%, and 7% for the Figs. 7b, 7c, 7d, 7e, 7f, respectively. This further supports the validity of the model being used and the accuracy of the computations presented here.

The line photobleaching experiments have been simulated as well for bleaching powers of $10\mathrm{mW}$ , $1\mathrm{mW}$ , and $0.075\mathrm{mW}$ . The simulation program does not allow us to calculate repeated line scans because of the extensive computation times involved. Alternatively we have simulated line profiles using those three intensities but with different scanning speeds. The scanning speed at $1\mathrm{mW}$ was $1∕10$ of the scanning speed at $10\mathrm{mW}$ , and $1∕133$ at $0.075\mathrm{mW}$ . Hence, the same amount of light will be received in all three cases. The resulting bleaching profiles are shown in Fig. 8a . In correspondence with the experimental observations, a better bleaching efficiency is obtained when using a low bleaching intensity with an increased bleaching time. This effect can be explained as follows. The high-intensity beam quickly depletes the ground state in favor of the triplet state, which will soon become saturated. The low-intensity beam, on the other hand, will only partly populate the triplet state at any one time, but integrated over the entire time span a larger number of triplet states will have existed, effectively leading to a larger number of photobleached molecules. Although in both cases an identical total amount of light dose was received, the lower light intensity applied for an extended period of time effectively causes more bleaching than the short saturating high-intensity bleach pulse. The apparent BIDs are shown in Fig. 8b. Again we find an expansion of the apparent BID with increasing bleaching intensities. Fitting of a Gaussian distribution to the simulated profiles (not shown in Fig. 8b) gives $r_0$ -values of $1.32\mu \mathrm{m}$ , $1.72\mu \mathrm{m}$ , and $2.95\mu \mathrm{m}$ for the bleaching powers of $75\mu \mathrm{W}$ , $1\mathrm{mW}$ , and $10\mathrm{mW}$ , respectively. As indicated in Fig. 5, the experimental $r_0$ -values are (in the same order) $1.51\mu \mathrm{m}$ , $1.82\mu \mathrm{m}$ , and $2.94\mu \mathrm{m}$ . Thus we obtain a deviation of the simulated from the experimental results of 13%, 6%, and 0%. Again we find a good quantitative correspondence between the simulated and experimental results.

Fig. 8

The bleaching of three lines is simulated for three different intensities: $75\mu \mathrm{W}$ , $1\mathrm{mW}$ , and $10\mathrm{mW}$ . To obtain the same total amount of light for each of the lines, the scanning speed is changed accordingly. (a) The bleaching profiles show that a better bleaching efficiency is obtained when using a low power beam in combination with a reduced scanning speed. (b) From the corresponding apparent BIDs it is clear that at high intensities the higher-order diffraction maxima are able to contribute significantly to the bleaching process. As a result, the apparent BID expands with increasing bleaching intensity.