# 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 fluorphores 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.
The information below explains with some physics what the difference is between CW-STED, pulsed-STED and where gated-STED comes in. If you would like to deconvolve either CW-STED, pulsed-STED or gated-STED images, then there is no direct need to understand the physics behind this. It is recommended to read the Quick Guide to STED deconvolution as a start.

## CW-STED vs. pulsed-STED

STED can be implemented either with pulsed lasers systems or with CW laser sources [2,3,4,5]. 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 1a) 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, since 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 1. (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. [1]

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. [5]. 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 1 (b).
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:

$${dn(r,t)\over dt} = \left\{ \begin{array}{l l} -kn(r,t) - \sigma I(r) n(r,t) & \quad 0 \leq t \leq t_{\text{s}}\\ -kn(r,t) & \quad t > t_{\text{s}}\\ \end{array} \right .$$(1)

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 (1), alternative non-radiative decay processes are neglected, since these will only change the amount of excited fluorophores and will leave the temporal dynamics unchanged [5].

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:

$$n(r,t) = n_{0}\left\{ \begin{array}{l l} e^{-(k+K(r))t} & \quad 0 \leq t \leq t_{\text{s}}\\ e^{-K(r)t_{\text{s}}}e^{-kt} & \quad t > t_{\text{s}}\\ \end{array} \right.$$(2)

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:

$${{d}N_{\text{s}}(r,t)\over{d}t} = \left\{ \begin{array}{l l} \sigma I(r) n(r,t) & \quad 0 \leq t \leq t_{\text{s}}\\ 0 & \quad t > t_{\text{s}}\\ \end{array} \right.$$(3)

$${{d}N_{\text{f}}(r,t)\over{d}t} = kn(r,t)$$(4)

With the solutions found for n(r,t) (equation (2)) 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:

$$\label{eq:pulse5} N_{\text{s}}(r,t) = n_{0}\left\{ \begin{array}{l l} \frac{K{(r)}}{k+K{(r)}}\left(1-e^{-(k+K(r))t}\right) & \quad 0 \leq t \leq t_{\text{s}}\\ \frac{K{(r)}}{k+K{(r)}}\left(1-e^{-(k+K(r))t_{\text{s}}}\right) & \quad t > t_{\text{s}}\\ \end{array} \right.$$(5)

The total number of fluorescent photons can now also be determined:

$$N_{\text{f}}(r,t) = n_{0}\left\{ \begin{array}{l l} \frac{k}{k+K{(r)}}\left(1-e^{-(k+K(r))t}\right) & \quad 0 \leq t \leq t_{\text{s}}\\ \frac{k}{k+K{(r)}} + \frac{K{(r)}}{k+K{(r)}}e^{-(k+K(r))t_{\text{s}}} - e^{-K(r)t_{\text{s}}} e^{-kt} & \quad t > t_{\text{s}}\\ \end{array} \right.$$(6)

To solve for the last equation, use was made of the fact that the total number of photons in time (fluorescent and stimulated emission photons) is related to the number of excited molecules by: $$N_{\text{s}}+N_{\text{f}} = n_{0} - n(r,t)$$.
When the STED intensity is zero, i.e. $$I(r) \propto K(r) =0$$, equation (6) (t > ts 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 (6). 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) [4,5].

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>>ts), 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 (5) and (6):

$$\lim_{t \to \infty} N_{\text{s}}(r,t)/n_{0} = \frac{K{(r)}}{k+K{(r)}}\left(1-e^{-(k+K(r))t_{s}}\right)$$(7)

and

$$\lim_{t \to \infty} N_{\text{f}}(r,t)/n_{0} = \eta_{\text{STED}}(r) = \frac{k}{k+K{(r)}} + \frac{K{(r)}}{k+K{(r)}}e^{-(k+K{(r)})t_{\text{s}}}$$(8)

The effective fluorescent spot hfl,eff(r) is described by the product of the spatial excitation probability hexc(r) with the spatially varying STED probability ηSTED(r):

$$h_{\text{fl,eff}}(r) = \eta_{\text{STED}}(r) h_{\text{exc}}(r)$$(9)

The excitation probability hexc(r) can be described by:

$$h_{\text{exc}}(r) = \cos^2{\left(\frac{\pi r n \sin{\theta}}{\lambda_{\text{exc}}} \right)},$$(10)

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 Φmax 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 hfl,eff(r). The results of these simulations are shown in figure 2 (a). 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 frep = 1/tr of 80 MHz, so that the pulses are spaced tr = 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 [2]. If the fluorescent lifetime is 3 ns, only a fraction τfl/fr ~ 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 2 (b). 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.

Figure 2. (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. [1]

[1] Remko R.M. Dijkstra, Design and realization of a CW-STED super-resolution microscope setup, Master Thesis , University of Twente, 2012 - All contents on this page have been used with permission of the author.
[2] 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.
[3] 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.
[4] 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.
[5] 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.