Stimulated Emission Depletion (STED) microscopy is a fluorescence microscopy super-resolution technique that is able to circumvent the optical diffraction limit. STED microscopy was first described in theory by Stefan Hell [1]. Stefan Hell was one of the three persons to win the Nobel Prize in Chemistry 2014 for their separate important achievements in optical super-resolution microscopy.

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Easy STED performance

Deconvolution applied to STED images is especially powerful because it naturally compensates for the drawbacks of STED microscopy, by providing improved resolution and signal-to-noise and drift correction. You choose how to optimally use those benefits, whether you want longer (live) recordings via less photobleaching and less phototoxicity or just get the ultimate resolution, Huygens STED deconvolution gives you premium results while taking care of many problems for you. Click below to get an imporession what it can do for you, or see the STED deconvolution page to find the settings that work best on your data.

Huygens STED software

Deconvolution applied to STED images is especially powerful because it naturally compensates for the drawbacks of STED microscopy, by providing improved resolution and signal-to-noise and drift correction. You choose how to optimally use those benefits, whether you want longer (live) recordings via less photobleaching and less phototoxicity or just get the ultimate resolution, Huygens STED deconvolution gives you premium results while taking care of many problems for you. Click below to get an imporession what it can do for you, or see the STED deconvolution page to find the settings that work best on your data.

Huygens STED software

*Image description: Primary hippocampal neurons with cytoskeleton proteins labelled (magenta, alpha-Adducin, Abberior STAR 635P and green, ßII spectrin, Alexa 594). Imaged with Abberior Instruments’ STEDYCON and deconvolved with Huygens.*

## The STED Concept

The idea presented in Hell's first STED paper [1] is based on depleting fluorophores in the outer region of the diffraction limited spot with a second laser, the STED-laser, which has a red-shifted wavelength (λSTED ) and a small spatial offset with respect to the excitation laser. In the region where both lasers overlap, the fluorophores are excited by the excitation laser but are stimulated to emit at the wavelength of the second laser (λSTED ). If the stimulated emission rate is high enough, the fluorophores in the overlapping region will emit photons with wavelength λSTED due to stimulated emission rather than emitting fluorescent photons with wavelength λF (spontaneous emission). The process of stimulated emission effectively depletes the fluorophores in the overlapping region leaving only the fluorophores in the non-overlapping region able to fluoresce. The fluorophores in the overlapping region are thus not switched into an ’off-state’ by the STED beam, but are merely stimulated to emit photons with the same wavelength as the STED laser. The photons that arise from stimulated emission are suppressed in the emission path with an appropriate filter, while the fluorescent photons are able to reach the detector. This filtering ultimately leads to a size reduction of the effective fluorescent spot as is illustrated in Figure 1b.*Figure 1. (a) Jablonski diagram illustrating the process of excitation, stimulated emission and fluorescence. (b) When two diffraction limited STED spots are overlapped with a diffraction limited excitation spot with offset Δx and when the STED intensity is high enough, the fluorophores in the outer region of the excitation spot are effectively depleted to the ground state by stimulated emission. In the region where there is no STED field present, the fluorophores are still able to fluoresce. As a result, the effective fluorescence spot is reduced to below the diffraction limit, but only in the direction of the offset [1]. Figure adopted from [2] with permission.*

## STED deconvolution

STED microscopy allows super-resolution imaging in the 50nm range. Unfortunately, this increased optical resolution also leads to a drawback: because many fluorophores are depleted by the depletion laser, this also results in a lower signal (fewer photons) being captured by the detector. Because of the Poisson nature of photon statistics, the signal-to-noise ratio of the resulting image will decrease compared to normal confocal imaging. Therefore the SNR value in STED images will be much lower compared to their confocal counterpart. Luckily, Huygens deconvolution of STED images offers a great solution. Since 2012, the Huygens software is able to generate a theoretical PSF based on STED microscope parameters. This theoretical PSF can be used in the deconvolution process, which will significantly reduce the noise in the image, and will increase the contrast of up to 10 times. Additionally, the Huygens deconvolution algorithms are able to increase the resolution in the image with a factor of 2 in both lateral and axial direction. See also our Microscopy Today article on Huygens STED deconvolution.## STED Beam Shaping

The offset method as described in the previous section allows resolution increase only in the lateral direction of the offset. This limitation could be partially solved by using four offset beams. However, this requires either multiple STED lasers or splitting a single high intensity STED beam into four separate beams. Both options are experimentally challenging, since it requires very precise alignment of multiple beams. A more delicate solution would be to shape a single STED beam in such a way that it creates a symmetrical high intensity profile around a zero intensity center in the focus. This doughnut-like shape (Figure 2 (c)) allows efficient and symmetrical confinement of the depleted region. To create a perfect null intensity in the focus of a high NA lens, it is quite intuitive that the phase front of the incoming beam must be shaped such that the focused light interferes destructively in the focus center. Over the last decade, different options have been implemented to create such a focal spot, and the development is still ongoing.One way to create a perfect null intensity in the focus is by using an annular phase plate [3]. This shape modification can also be realized with a vortex phase plate (Figure 2 (c) top figure). The vortex phase plate is most widely used for super-resolution in the lateral direction, while the annular phase plate is used for depletion in the axial direction, leading to super-resolution in the Z dimension (see Figure 3). Also a combination of the two phase-plates can be used, as is implemented in for example the Leica SP8 STED 3X system.

There are also alternative methods to obtain a suitable STED focus, such as using a spatial light modulator (SLM) or by using other types of phase-plates or phase modulators [4, 5]. However, the annular and vortex phase plates are still most widely used in STED microscopy.

*Figure 2. Simulation results of the normalized intensity in XY and XZ planes in the focus of the objective with perfect circularly polarized light. As a result of the symmetry, the ZY plane is similar to the ZX plane and is therefore not shown here. Black represents a normalized intensity of zero, while white represents a normalized intensity of one. The simulations are performed in a 800 nm × 800 nm area for each plane with 10 nm step size. The top illustrations show the type of phase adjustment that was used. (a) When the incoming beam is not spatially phase-adjusted (like a confocal excitation spot), the focus is a diffraction limited spot in XY plane and has a cigar shape in ZX and ZY planes. (b) When an annular phase plate is used to spatially adjust the phase of the beam, the focus in XY plane will have a doughnut shape. ZX and ZY plane will contain two cigars aligned in Z with a null intensity in the center. (c) A vortex phase plate will also lead to a doughnut in XY plane, but with much higher absolute intensity compared to the doughnut formed by the annular phase plate. In ZX and ZY plane the focus shows two cigar shapes aligned in respectively X and Y direction.*

*Figure 3. Schematic of STED 3X excitation bundles. The lateral depletion (red donut) and the axial depletion (two purple lobs) deplete everywhere except for the green fluorescent bundle.*

*Figure 4. Cartoon illustrating the spatial resolution increase by STED. Two thin wire-like structures are labeled with fluorophores (gray dots). In conventional confocal microscopy most of the fluorophores in the focal region are excited (red) and are able to fluoresce, resulting in a large blob in the image. When STED is applied, a doughnut-shaped beam is overlapped with the excitation spot in the focal region. This leads to depletion of most of the excited fluorophores in the overlapping region (yellow dots), leaving only the fluorophores in the center of the doughnut to fluoresce (green). [2]*

## STED Resolution

Figure 4 illustrates how the overlap of a doughnut shaped STED beam leads to an increase in resolution. In this figure, two thin wire like structures are labeled with fluorophores and are in close vicinity of each other (< diffraction limit). In conventional fluorescent microscopy the two wires cannot be spatially separated due to the diffraction limit as a large area of fluorophores is excited and therefore able to fluoresce. When STED is applied, the overlapping STED doughnut results in depletion of the fluorophores in the outer region of the excitation spot. This depletion results in a confinement of the area in which the fluorophores are still allowed to fluoresce.When the sample is raster scanned, the reduced size of the effective fluorescent spot enables the spatial separation of the two closely spaced wires in the resulting image. The increase in spatial resolution is dependent on the STED excitation intensity and can be described mathematically. With a doughnut-shaped STED focus, the FWHM Δr of the effective fluorescent spot can be described by:

$$\Delta r \approx \frac{0.5 \lambda }{n \sin{\theta} \sqrt{I^{\text{max}}_{\text{STED}}/I_{\text{sat}} +1 }} = \frac{\lambda}{2\text{NA}} \frac{1}{\sqrt{I^{\text{max}}_{\text{STED}}/I_{\text{sat}} +1 }}$$

**(1)**

Equation (1) is the familiar inverse square root intensity law for STED: the FWHM resolution approximately scales with the inverse square root of the STED intensity. The definition of Δr is also illustrated in Figure 5. Also note that the first part of the right hand of Equation 1 is the conventional diffraction limit, i.e., when the STED intensity ISTEDmax = 0 Equation 1 reduces to Abbe’s diffraction limit. The amount by which this conventional limit is reduced is determined by the Saturation Factor (ISTEDmax / Isat). This is the ratio between the maximal intensiy in the STED depletion beam (ISTEDmax) and the intensity at which the probability of fluorescence emission is reduced by half (Isat). The full derivation of this equation can be found below. In theory, the maximum obtainable resolution as stated by Equation 1 would be infinitely small. In practice however, the extreme intensities required to reach extreme high resolutions would lead to unwanted effects such as photo-bleaching, optical trapping, multi-photon absorption, sample heating or even sample destruction. However the setup would most likely be practically limited due to stage drift or become dark noise limited due to the decreasing intensity that inevitably comes with increasing resolution. Moreover in the ultimate environment where STED is to be used, i.e. in biological samples, other processes such as diffusion play an important role.

The chosen fluorophore will also largely determine the maximal obtainable resolution. Some fluorophores are more sensitive to the STED depletion lasers than others. Regardless of the STED intensity: there will always be a fraction of fluorophores in the depletion area that are not efficiently being depleted. Additionally, some fluorophores in the depletion area may be re-excited (anti-Stokes) by the depletion light. If the emission filters are non-perfect, then also some stimulated light will still reach the detectors. These three factors will lead to an immunity fraction, which is the percentage of fluorophores that is not being depleted, and thus still able to fluoresce. This property is fluorophore dependent, and can be determined experimentally. Gated STED (g-STED) is a technique that makes use of the lifetime information of the emitted light, and can thereby help to minimize the immunity fraction and increase the resolution.

*Figure 5. Top: overlapping a doughnut-shaped STED spot with a diffraction limited excitation spot will result in a confinement of the area in which fluorophores are able to fluoresce. Bottom: (a) Graph illustrating the overlap of the intensity of the STED donut shaped spot ISTED (r) (orange) and excitation spot (blue) in one dimension. (b) Graph illustrating the decrease in the width of the effective fluorescence probability h(r) (green) as a result of STED. Δr is the FWHM of the effective fluorescence probability. [2]*

### Mathematical Derivation of the STED Equation

This derivation is based on the works in references [6] and [7]. The increase in spatial resolution in STED microscopy is dependent on the STED depletion intensity and can be described mathematically. After the passage of a STED pulse, the fluorescence probability \(\eta (r)\) of a molecule has decayed to:$$ \eta (r) = \exp{\left[-\sigma t_{\text{s}} I_{\text{STED}}(r) \right]} $$

**(2)**

with \(\sigma\), \(t_{\text{s}}\) and \(I_{\text{STED}}(r)\) denoting the cross section for stimulated emission, STED pulse duration and the focal intensity of the STED pulse, respectively. The \(I_{\text{STED}}(r)\) is given by number of photons per unit area per unit time. As a result of the overlapping STED beam, the probability to detect a photon from location \(r\) is proportional to:

$$ \label{eq:STED2} h(r)= h_{\text{exc}}(r) \eta(r) $$

**(3)**

with \(h_{\text{exc}}(r)\) the normalized excitation probability in the focal plane resulting from the excitation pulse. Given a lens with semi aperture \(\theta\), we can describe the spatial excitation probability \(h_{\text{exc}}(r)\) and the STED intensity \(I_{\text{STED}}(r)\) by:

$$
\label{eq:STED3}
h_{\text{exc}}(r) = C \cos^2{\left(\frac{\pi r n \sin{\theta}}{\lambda_{\text{exc}}} \right)}
$$

**(4)**and

$$
\label{eq:STED4}
I_{\text{STED}}(r) = I^{\text{max}}_{\text{STED}} \sin^2{\left(\frac{\pi r n \sin{\theta}}{\lambda_{\text{STED}}}\right)}
$$

**(5)**Normalized cross sections of \(h_{\text{exc}}(r)\) and \(I_{\text{STED}}(r)\) are also illustrated in Figure 5a as the blue and yellow curve respectively.

\(I^{\text{max}}_{\text{STED}}\) is the maximum STED intensity in the doughnut shape.

For further simplification we can define the constant \(\varsigma\) as:

$$
\label{eq:STED5}
\varsigma = \sigma t_{\text{s}} I^{\text{max}}_{\text{STED}}
$$

**(6)**If we combine equations 2, 4, 5 and 6, and use the approximation \(\lambda_{\text{exc}} \approx \lambda_{\text{STED}} \equiv \lambda\), Equation 3 can be written as:

$$
\begin{eqnarray}
\label{eq:STED6}
{h(r)} &=& C \cos^2 {\left( \frac{\pi r n \sin{\theta}}{\lambda_{\text{exc}}}\right)} \exp{\left[-\sigma \tau I^{\text{max}}_{\text{STED}} \sin^2{\left( \frac{\pi r n \sin{\theta}}{\lambda_{\text{STED}}}\right)}\right]}\\
{} &\approx& C \cos^2{\left( \frac{\pi r n \sin{\theta}}{\lambda}\right)}
\exp{\left[- \varsigma \sin^2{\left(\frac{\pi r n \sin{\theta}}{\lambda}\right)}\right]}
\end{eqnarray}
$$

**(7)**\(h(r)\) is the FWHM of the effective fluorescent spot, which cross section is also represented as the green curve in Figure 5b.

We can further approximate Equation 7 with a Taylor series \((r < \lambda/2n \approx 0)\) to the second order:

$$
\label{eq:STED7}
h(r) \approx 1 - \left(\frac{\pi n r \sin{\theta}}{\lambda}\right)^2 \left(\varsigma + 1 \right)
$$

**(8)**At the FWHM we have \(h(r) = 0.5\), so we can solve for \(r\):

$$
\label{eq:STED8}
r = \pm \frac{\lambda \sqrt{1-0.5}}{\pi n \sin{\theta} \sqrt{\varsigma +1}}
$$

**(9)**The FWHM \(\Delta r = 2r\) can then be described with:

$$
\label{eq:STED9}
\Delta r = 2\frac{\lambda \sqrt{0.5}}{\pi n \sin{\theta} \sqrt{\varsigma +1}} \approx \frac{0.45 \lambda }{n \sin{\theta} \sqrt{\varsigma +1}} = \frac{0.45 \lambda }{n \sin{\theta} \sqrt{\sigma t_{\text{s}} I^{\text{max}}_{\text{STED}} +1 }}
$$

**(10)**From Equation 2 we can see that for \(I_{\text{STED}} = 1/\sigma t_{\text{s}} = I_{\text{sat}}\) the fluorescence has dropped to \(1/e\) of the initial value. By introducing \(I_{\text{sat}}\) in Equation 10 we obtain:

$$
\label{eq:STED10}
\Delta r = \frac{0.45 \lambda }{n \sin{\theta} \sqrt{I^{\text{max}}_{\text{STED}}/I_{\text{sat}} +1 }}
$$

**(1)**Equation 1 is the familiar inverse square root intensity law for STED: the FWHM resolution approximately scales with the inverse square root of the STED intensity. In case of gated-STED there is an additional factor that takes the effect of gating into account, leading to an additional resolution increase.

## Gated-STED (g-STED)

Gated-STED is a type of STED microscopy that makes use of the fluorescent lifetime of fluorophores. In the STED depletion area there will always be some fluorophores that are not being depleted by the STED depletion laser, and as a result emit fluoresence light. This leads to a lower obtainable resolution compared to what would be possible with STED in case of an ideal fluorophore that is perfectly being depleted. Although some fluorophores are not fully being depleted, they are still effected by the STED depletion beam in their fluorescence lifetime. By filtering the photon detection based on the photon arrival time, a higher resolution can be obtained, especially when CW-depletion lasers are being used. This filtering based on photon arrival time is called 'gating' and is applied in gated-STED. Alternatively, the gating can be used as a tunable parameter to reduce the STED depletion power without compromising the resolution. This helps to avoid photo-bleaching and photo-toxicity.Gated-STED has the advantage that a higher super-resolution can be obtained compared to STED without gating. However, because also less photons will be detected compared to when no gating is used, the images will also have a lower signal-to-noise ratio. Therefore deconvolution is highly advised when working with gated-STED images in order to significantly improve the image contrast and resolution.

## Pulsed-STED

STED can be implemented either with pulsed lasers systems or with CW laser sources [9-11]. Pulsed STED requires tight synchronization of both the excitation and STED depletion pulses. Furthermore, the extremely short pulses emitted by most pulsed laser sources need to be stretched to make stimulated emission more efficient. Continuous wave STED does not suffer from these drawbacks, but it does require a higher average power laser source for efficient depletion compared with pulsed-STED.The need for tight pulse synchronization can be illustrated with the use of a Jablonski diagram (Figure 6a) and the rates at which different energy transitions occur. An excitation pulse excites fluorophores (L0 -> L1) after which the fluorophore quickly decays (~ ps) to L2 by vibrational relaxation.

The STED pulse directly following the excitation pulse will induce the transition L2 -> L3 and thereby depletes L2 by stimulated emission. In order to do this most efficiently, the time between the excitation and STED pulse

*Δt*must be well tuned. In fact, depletion is most efficient when the STED pulse arrives directly after the excitation pulse has left, as in this case the level L2 is not longer pumped and can be efficiently depleted by the STED pulse. Furthermore, it is advantageous that the laser pulse durations are significantly shorter than the fluorescent lifetime, which is in the order of 2 ns. The STED pulse should be longer than the vibrational lifetime,

*τvib ~ 1-5*ps, to avoid absorption back to L2 by the same pulse. Therefore, pulses in the 100 ps range are typically used.

*Figure 6. a) Jablonski diagram illustrating stimulated emission. The fluorophore is pumped from L_0 to L_1 by the excitation pulse, after which it quickly relaxes to L_2 by vibrational relaxation. The fluorophore is then depleted to L_3 by the STED pulse. L_3 is then quickly depleted back to the ground state L_0 before re-absorption of another STED photon can take place. b) Diagram illustrating the simplified temporal configuration of excitation and STED pulses. A rectangular excitation pulse with duration t_exc ends at t = 0 and is directly followed by a rectangular STED pulse with duration t_s. [6]*

### Mathematical Derivation of the Pulsed STED Equation

The effect of STED pulse duration ts on the resolution increase can be derived mathematically. The derivation presented here is analogue to the derivation presented by Moffitt*et al.*[11]. We start with the simple case of a single rectangular excitation pulse that ends at

*t=0*followed directly by a rectangular STED pulse starting at

*t=0*as is illustrated in Figure 6b.

We can consider a total amount of fluorophores n0 with stimulated emission cross-section

*σ*and decay rate \(k=1/\tau_{\text{fl}}\), where \(\tau_{\text{fl}}\) is the unstimulated fluorescent lifetime.

Since excitation is absent after the excitation pulse, the total number of excited fluorophores

*n(r,t)*after

*t=0*is decreasing due to fluorescence and stimulated emission, and can be described with the following differential equation:

$${dn(r,t)\over dt} = \left\{
\begin{array}{l l}
-kn(r,t) - \sigma I(r) n(r,t) & \quad 0 \leq t \leq t_{\text{s}}\\
-kn(r,t) & \quad t > t_{\text{s}}\\
\end{array} \right .
$$

**(11)**Where

*k*is the fluorescent decay rate,

*n(r,t)*is the number of excited molecules,

*σ*is the stimulated emission cross-section and

*I(r)*is the spatially dependent STED photon-flux (recall the doughnut shape). In Equation 11, alternative non-radiative decay processes are neglected, since these will only change the amount of excited fluorophores and will leave the temporal dynamics unchanged [11].

For simplification the term

*σI(r)*can be replaced with

*K(r)*which can be thought of as the stimulated emission rate coefficient which is dependent on the STED photon flux

*I(r)*at location

*r*.

This differential equation can be solved by integration, which gives us the number of excited molecules as a function of time:

$$
n(r,t) = n_{0}\left\{
\begin{array}{l l}
e^{-(k+K(r))t} & \quad 0 \leq t \leq t_{\text{s}}\\
e^{-K(r)t_{\text{s}}}e^{-kt} & \quad t > t_{\text{s}}\\
\end{array} \right.
$$

**(12)**The term \(e^{-K(r)t_{\text{s}}}\) in the

*t > ts*solution is a result of the fact that the number of excited molecules must be equal for both solutions at time

*t = ts*.

The number of emitted fluorescent and stimulated photons will grow as time increases and can be expressed by the following equations:

$$
{{d}N_{\text{s}}(r,t)\over{d}t} = \left\{
\begin{array}{l l}
\sigma I(r) n(r,t) & \quad 0 \leq t \leq t_{\text{s}}\\
0 & \quad t > t_{\text{s}}\\
\end{array} \right.
$$

**(13)**$$
{{d}N_{\text{f}}(r,t)\over{d}t} = kn(r,t)
$$

**(14)**With the solutions found for

*n(r,t)*(Equation 12) we can again solve these equations by integration and by using the fact that \(N_{\text{f}}=N_{\text{s}}=0\) at

*t = 0*.

The total number of STED photons will first be solved:

$$
\label{eq:pulse5}
N_{\text{s}}(r,t) = n_{0}\left\{
\begin{array}{l l}
\frac{K{(r)}}{k+K{(r)}}\left(1-e^{-(k+K(r))t}\right) & \quad 0 \leq t \leq t_{\text{s}}\\
\frac{K{(r)}}{k+K{(r)}}\left(1-e^{-(k+K(r))t_{\text{s}}}\right) & \quad t > t_{\text{s}}\\
\end{array} \right.
$$

**(15)**The total number of fluorescent photons can now also be determined:

$$
N_{\text{f}}(r,t) = n_{0}\left\{
\begin{array}{l l}
\frac{k}{k+K{(r)}}\left(1-e^{-(k+K(r))t}\right) & \quad 0 \leq t \leq t_{\text{s}}\\
\frac{k}{k+K{(r)}} + \frac{K{(r)}}{k+K{(r)}}e^{-(k+K(r))t_{\text{s}}} - e^{-K(r)t_{\text{s}}} e^{-kt} & \quad t > t_{\text{s}}\\
\end{array} \right.
$$

**(16)**To solve for the last equation, the relation between the total number of photons in time (fluorescent and stimulated emission photons) and the number of excited molecules (\(N_{\text{s}}+N_{\text{f}} = n_{0} - n(r,t)\)) was used.

When the STED intensity is zero, i.e. \(I(r) \propto K(r) =0\), Equation 16 (t > t

_{s}reduces to \(N_{\text{f}} = n_{0}\left(1-e^{-kt}\right)\), in which \(e^{-kt}\) is the fluorescent lifetime. However, when the STED intensity is non-zero, i.e. \(I(r) \propto K(r) \neq 0\), there will be a spatial dependence in the fluorescent lifetime (\(e^{-K(r)t_{\text{s}}} e^{-kt}\)) according to Equation 16. This means that the presence of the STED light will influence the lifetime of the fluorophore. Hence there is also spatial information encoded in the lifetime signal of the fluorophores. This fact is used to increase CW-STED resolution with time-gating (g-STED) [10,11].

In this case we are only interested in the total number of photons originating from position

*r*that arrive at the detector after a relative long time (t>>t

_{s}), since the detection time per pixel is much longer than the STED pulse duration. For this use we can take the infinite-time limit for equations 15 and 16):

$$
\lim_{t \to \infty} N_{\text{s}}(r,t)/n_{0} = \frac{K{(r)}}{k+K{(r)}}\left(1-e^{-(k+K(r))t_{s}}\right)
$$

**(17)**and

$$
\lim_{t \to \infty} N_{\text{f}}(r,t)/n_{0} = \eta_{\text{STED}}(r) = \frac{k}{k+K{(r)}} + \frac{K{(r)}}{k+K{(r)}}e^{-(k+K{(r)})t_{\text{s}}}
$$

**(18)**The effective fluorescent spot

*h*is described by the product of the spatial excitation probability

_{fl,eff}(r)*h*with the spatially varying STED probability

_{exc}(r)*η*:

_{STED}(r)$$
h_{\text{fl,eff}}(r) = \eta_{\text{STED}}(r) h_{\text{exc}}(r)
$$

**(19)**The excitation probability

*h*can be described by:

_{exc}(r)$$
h_{\text{exc}}(r) = \cos^2{\left(\frac{\pi r n \sin{\theta}}{\lambda_{\text{exc}}} \right)},
$$

**(20)**The stimulated emission rate function

*K(r)*can be described by:

$$
K(r) = k^{\text{max}}_{\text{STED}} \sin^2{\left(\frac{\pi r n \sin{\theta}}{\lambda_{\text{STED}}}\right)}
$$

**(11)**Here \(k^{\text{max}}_{\text{STED}} \equiv \sigma \Phi_{\text{max}}/t_{\text{s}}\), which is the peak STED depletion rate (number of photons per unit time) at the dougnhut maximum. Here

*Φ*stands for the number of STED photons per unit area at the doughnut maximum. With Equation 9 simulations were performed to investigate the influence of the STED pulse duration on

_{max}*h*. The results of these simulations are shown in Figure 7a. In this figure it can be clearly observed that an increasing STED pulse length will increase the effective spot size (FWHM). Consider a STED pulse repetition rate

_{fl,eff}(r)*f*of 80 MHz, so that the pulses are spaced

_{rep}= 1/t_{r}*t*= 12.5 ns apart. If we stretch the pulse length to 12.5 ns we effectively have a CW source. By keeping the number of photons within each pulse constant, the average power will also remain constant. However only the STED photons that fall within the fluorescent lifetime really attribute to stimulated emission [8]. If the fluorescent lifetime is 3 ns, only a fraction

_{r}*τ*~ 1/4 of the total amount of STED photons will attribute to depletion. To obtain a similar FWHM for the effective spot as for the pulsed case, the average power \(P_{\text{avg}} \propto \Phi_{\text{max}}/t_{\text{r}}\), needs to be increased with a factor of 4. This is exactly what was simulated in Figure 7b. When Φ was increased with a factor of four, the FWHM for the CW case was indeed almost similar to the 0.25 ns pulse case, which means a similar resolution. However, note that in the pulsed case the PSF intensity quickly reduces to zero at the base, whereas the CW PSF has a significant wider base. The FWHM of the cw-case can be further reduced by increasing the average power, however the PSF base will always remain relatively broad as compared to the pulsed case. Still, this can be further improved by implementing gated-STED. Both pulsed-STED and gated-STED will have a higher saturation factor compared to CW-STED with similar STED intensity, and will thus also lead to a better STED resolution.

_{fl}/f_{r}*Figure 7. a) Simulation results of the effect of STED pulse duration t_s on the effective fluorescent spot size h_{fl,eff}. When the STED pulse duration increases while the average power is kept constant, the effective spot size (FWHM) increases. Also notice that an increase in pulse duration leads to a broadening of the tails of the effective point spread function. b) A remedy for the increasing spot size with increasing pulse duration is to increase the average power. The average power is here considered as the number of photons per pulse Φ divided by the pulse repetition rate t_r. [6]*

## References

[1] Stefan W. Hell and Jan Wichmann.

*Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy.*, Opt. Lett., 19(11):780–782, Jun 1994.

[2] Remko R.M. Dijkstra,

*Design and realization of a CW-STED super-resolution microscope setup*, Master Thesis , University of Twente, 2012''

[3] Thomas A. Klar, Stefan Jakobs, Marcus Dyba, Alexander Egner, and Stefan W. Hell.

*Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission.*Proceedings of the National Academy of Sciences, 97(15):8206–8210, July 2000.

[4] Matthias Reuss.

*Simpler STED setups.*PhD thesis, Combined Faculties for the Natural Sciences and for Mathematics of the Ruperto-Carola University of Heidelberg, Germany, December 2010.

[5] Matthias Reuss, Johann Engelhardt, and Stefan W. Hell.

*Birefringent device converts a standard scanning microscope into a STED microscope that also maps molecular orientation.*Opt. Express, 18(2):1049–1058, Jan 2010.

[6] Remko R.M. Dijkstra,

*Design and realization of a CW-STED super-resolution microscope setup*, Master Thesis , University of Twente, 2012

[7] Volker Westphal and Stefan W. Hell, ''Nanoscale Resolution in the Focal Plane of an Optical Microscope, Phys. Rev. Lett. 94, 143903, April 2005

[8] Katrin I. Willig, Benjamin Harke, Rebecca Medda, and Stefan W. Hell.

*STED microscopy with continuous wave beams.*Nature Methods, 4(11):915–918, October 2007.

[9] Gael Moneron, Rebecca Medda, Birka Hein, Arnold Giske, Volker Westphal, and Stefan W. Hell.

*Fast STED microscopy with continuous wave fiber lasers.*Opt. Express, 18(2):1302–1309, Jan 2010.

[10] Giuseppe Vicidomini, Gael Moneron, Kyu Y. Han, Volker Westphal, Haisen Ta, Matthias Reuss, Johann Engelhardt, Christian Eggeling, and Stefan W. Hell.

*Sharper low-power STED nanoscopy by time gating.*Nat Meth, 8(7):571–573, July 2011.

[11] Jeffrey R. Moffitt, Christian Osseforth, and Jens Michaelis.

*Time-gating improves the spatial resolution of STED microscopy.*Opt. Express, 19(5):4242–4254, February 2011.