Cite this Article
Qin Haiyun, Zhao Wei, Zhang Chen, Liu Yong, Wang Guiren, Wang Kaige. Influence of fluorescence time characteristics on the spatial resolution of CW-stimulated emission depletion microscopy. Chinese Physics B, 2018, 27(3): 037803  
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Influence of fluorescence time characteristics on the spatial resolution of CW-stimulated emission depletion microscopy
Qin Haiyun1, †, Zhao Wei1, †, Zhang Chen1, Liu Yong2, Wang Guiren3, Wang Kaige1, ‡
Institute of Photonics and Photon-Technology, Northwest University, Xi’an 710069, China
School of Electronics and Information Engineering, Shanghai University of Electric Power, Shanghai 200090, China
Mechanical Engineering Department & Biomedical Engineering Program, University of South Carolina, Columbia, SC 29208, USA

 

† Corresponding author. E-mail: wangkg@nwu.edu.cn

Abstract

As one of the most important realizations of stimulated emission depletion (STED) microscopy, the continuous-wave (CW) STED system, constructed by using CW lasers as the excitation and STED beams, has been investigated and developed for nearly a decade. However, a theoretical model of the suppression factors in CW STED has not been well established. In this investigation, the factors that affect the spatial resolution of a CW STED system are theoretically and numerically studied. The full-width-at-half-maximum (FWHM) of a CW STED with a doughnut-shaped STED beam is also reanalyzed. It is found that the suppression function is dominated by the ratio of the local STED and excitation beam intensities. In addition, the FWHM is highly sensitive to both the fluorescence rate (inverse of fluoresce lifetime) and the quenching rate, but insensitive to the rate of vibrational relaxation. For comparison, the suppression function in picosecond STED is only determined by the distribution of the STED beam intensity scaled with the saturation intensity. Our model is highly consistent with published experimental data for evaluating the spatial resolution. This investigation is important in guiding the development of new CW STED systems.

PACS: 78.45.+h;78.55.-m
Keyword:stimulated emission depletion;continuous-wave laser;suppression function;numerical simulation
1. Introduction

Stimulated emission depletion (STED) microscopy is a novel super-resolution optical method that has been developed over the last two decades.[1,2] To achieve high spatial and temporal resolutions, various technical methods have been advanced.[3,4] In the beginning, the STED technique was realized by using picosecond pulsed lasers for both excitation and STED beams. By accurately synchronizing the excitation and depletion laser pulses, a highly depleted fluorescent spot with a significantly reduced size can be achieved. By using a doughnut-shaped STED beam, the spatial resolution can be enhanced from (confocal microscopy) to (STED),[5] where λ and NA denote the wavelength of the excitation beam and the numerical aperture of the objective, respectively. and are the maximum intensity of the STED beam and the saturation intensity of the fluorescence at which the probability of fluorescence emission is reduced by half, respectively. This theoretical model well describes the experimental studies of time-resolved fluorescent dynamics for dyes under exposure to picosecond pulses.

Meanwhile, a STED system based on continuous-wave (CW) lasers has also been developed.[6–9] Recently, the spatial resolution of CW-laser-based STED (CW STED) microscopes has been reported to typically be over 50 nm, which is less than that obtained for picosecond STED. However, CW STED is much easier to use than picosecond STED because it avoids the requirement of temporal synchronization of the excitation and STED beams, unlike picosecond STED. This is a great advantage for developing low-cost STED systems. Furthermore, many CW-laser-based techniques, such as the recently developed laser-induced fluorescence photobleaching anemometer, can be directly realized on the basis of a CW STED system.[6]

One of the primary obstacles in the development of a CW STED microscopy is the incomplete understanding of the suppression function. Currently, CW STED studies evaluate the suppression function and the corresponding spatial resolution based on theories advanced for picosecond STED, where the suppression function is normally expressed as . Here, Isted is the intensity of the STED beam. However, the instant and fast fluorescent dynamic process in a picosecond STED system is different from that in a CW STED system, where the fluorescent dynamic process is statistically steady. Therefore, the aforementioned theoretical resolution prediction for picosecond STED may not be accurate for CW STED.

In this study, we reanalyze the fluorescent dynamics of the CW STED system and theoretically derive the suppression function. The full-width-at-half-maximum (FWHM) of the point-spreading function (PSF) for the effective fluorescent spot in CW STED with a Gaussian excitation beam and a doughnut-shaped STED beam is studied in detail. The theoretical results are validated by published experimental data from other studies. Finally, the influence of the control time parameters, such as the rates of quenching, fluorescence, and vibrational relaxation, is investigated.

2. Theory of a CW STED system

The general excitation and emission processes of a fluorophore are plotted in Fig. 1(a). As suggested by Hell and Wichmann,[2] the corresponding probability ( ) of electrons on each level Li can be expressed as where and are the normalized intensities of excitation and STED beams in which νexc and νsted are the frequencies of excitation and depletion photons, respectively. is the reduced Planck constant. sexc and ssted are the cross sections for the absorptions. kvib, kfl, and kQ are the average rates of the vibrational relaxation, fluorescence, and quenching, respectively. In the CW STED system, the excitation and STED beam power intensities, hexc and hsted, are temporally constant. The fluorescence process is steady, and the probability ni in each state Li is also statistically constant in time. This can be seen by averaging Eq. (1) temporally as where the bar indicates temporal averaging. represents the mean probability for each excitation level. Let , , and then the solutions of Eq. (2) can be described as

Fig. 1. (color online) (a) Schematic of the excitation, emission, and depletion processes of a fluorophore. (b) Schematic profiles of Isted and γ for the lateral direction in the focus area of experimental case 2. Here, the wavelengths of the excitation and STED beams are 635 nm and 750 nm, respectively. To obtain these two profiles, both Iexc and Isted distributions on the focal plane are numerically computed by Debye integral method first, with the experimental conditions of case 2. Then, γ is calculated on the basis of Eq. (7). The profiles have been normalized for comparison.

Generally, the effective PSF for the STED microscopy can be described by the product of the PSF for the fluorescence on focus ( and a suppression function as .[10] Since ,[2] the effective PSF of STED microscopy becomes or equivalently, . In a typical CW STED system, , and kvib, kfl, and kQ are determined by the fluorescent dyes. Iexc, Isted, and the corresponding γ can be calculated by numerical methods.

It can be schematically seen from Fig. 1(b) that for picosecond STED, the suppression function is determined by the local intensity of the STED beam, i.e., , while in CW STED, the suppression function is dominated by γ, which is the ratio of the STED and excitation intensities. The peaks of γ are located farther away from the center with narrower lobes. The peaks of are relatively closer to the center and have wider lobes. The different control parameters and their distribution may lead to different suppression results. Intuitively, picosecond STED should show better suppression for the tails of Gaussian excitation beams compared to CW STED and result in a smaller FWHM for .

The FWHM of CW STED systems with a Gaussian excitation beam and a doughnut-shaped STED beam are theoretically studied below and compared with the FWHM of picosecond STED systems. Here, to simplify the analysis, we assume that the depletion due to the excitation beams has a high power ratio (e.g., ).

In general, the PSF of the excitation and STED beams can be described as where and are the peak values for and , respectively. dexc is the diameter of the Gaussian beam and dsted is the diameter of the doughnut-like spot at peak. It should be noted that due to the far-red tail of the emission spectrum of the STED beam, sexc is normally larger than ssted. To simplify the model for theoretical analysis, the approximation of equal absorption cross-sections, i.e., is applied. Then, we have

Normally, . By substituting Eq. (7) into Eq. (4), we obtain γ normally covers a large range of magnitudes. When is not sufficiently large, we have approximately . Then, equation (8) becomes

Since the FWHM of STED must be smaller than dexc and dsted in the range , using the Taylor expansion, we have

Thus, from Eq. (9), it can be seen that

Since , the denominator in Eq. (11) can be approximated by where and M=B/A. This approximation has also been validated by numerical methods in the range , as shown in Fig. 2. Then, from Eq. (11) can be approximated to Substituting Eqs. (6a) and (13) into Eq. (5), one has When x = 0, . Therefore, the FWHM can be determined at x where . By taking the logarithm of , we have

Fig. 2. (color online) Evaluation of the simplification of Eq. (11) to Eq. (13). It can be seen that in the range , a good agreement is achieved before and after the simplification.

Since and the second term on the left-hand side of Eq. (15) can be simplified by a Taylor expansion to the order of . This leads to Then, equation (15) becomes Finally, the FWHM represented by can be described as where is the factor of resolution improvement of the STED system compared to the confocal system. In the case of a high power ratio ( ) of depletion to excitation beams, normally, and . Therefore, , , and Here, is the maximum population of electrons in L2. This is proportional to the maximum fluorescent intensity.

Compared to the picosecond STED system whose resolution relies on the maximum and saturation intensities of the STED beam only, the spatial resolution of the CW STED is more complicated and is determined by γ, kQ, and kfl.

The FWHM of the CW STED beam at a large B is inversely proportional to . This is comparable to that for the picosecond STED beam, which implies that the CW STED beam has a potency equivalent to a picosecond STED beam for realizing a small FWHM. However, as shown in Fig. 3, the suppression function of the CW STED system is slightly different from that in the picosecond STED system. Around x = 0, n2 for the former is slightly narrower than for the latter. This implies that from a simple comparison of the suppression functions for CW and picosecond STED systems under the same conditions of case 2, one can also realize a sharp peak for in the CW STED system. Moreover, also exhibits a slower decreasing rate along x relative to . Therefore, the CW STED system with a doughnut-shaped STED beam should show a weaker suppression effect for the tails of Gaussian- type excitation beams. Consequently, the CW STED system should show relatively poorer suppression effect, spatial resolution, and signal-to-noise ratio compared to those of the picosecond STED system.

Fig. 3. (color online) Comparison of the suppression functions in the CW STED model ( and in the picosecond STED system (i.e., .[10] The simulations of the two suppression functions are both based on the conditions of case 2 in Table 1.

In addition, for the picosecond STED system, one can increase the resolution by increasing the intensity of the STED beam or decreasing the saturation intensity, which is an experimental parameter of the fluorescent dyes. However, the method for effectively decreasing the saturation intensity is not yet fully understood. For the CW STED system, we find which straight-forwardly elucidates the relationship between the fluorescent dye and spatial resolution and implies that one can enhance the spatial resolution by inhibiting kQ and kfl. This has the potential to be a significant advance for developing new CW STED systems.

3. Validation of the theoretical model
3.1. Numerical simulation of the excitation and depletion beams of CW STED

To give a straightforward explanation for the depletion process of CW STED and the resulting , a numerical method implemented by Matlab based on the vector diffraction theory and the generalized Debye integral[11] is adopted. The electric-field vector can be described as where rp, , and zp are the radial, azimuthal, and axial components in the cylindrical coordinates, respectively, as plotted in Fig. 4(a). is the electric-field vector at the point . represents the apodization factor resulting from energy conservation and geometric considerations.[12] NA is the numerical aperture of the objective. n =1.52 is the medium refractive index. θ denotes the angle between the ray direction and optical axis, where , with . and are the amplitude function and polarization vector of the incident beam, respectively. is the phase delay function. is the conversion matrix of the polarization from the object field to the image field, which is given as

Fig. 4. (color online) (a) Coordinate system of the simulation. rp, φp, and zp are the radial, azimuthal, and axial components in the cylindrical coordinates, respectively. θ denotes the angle between the incident light and the optical axis. AL represents an achromatic lens. z is the axial direction of the incident beam. x and y are the axes perpendicular to the optical axis. (b) Illustration of the excitation beam (green), STED beam (red), and the resulting (yellow). Here, experimental case 2 is plotted for reference.

The laser beam intensity evaluated by the magnitude of can be expressed as . Therefore, the corresponding distribution of I can be computed based on the optical system. In this investigation, several reported experimental studies[9,13,14] are revisited numerically. All of these reports have used a right-handed circularly polarized (RCP) beam, with , and a vortex phase plate (VPP) to generate a doughnut-shaped depletion beam. The phase delay function can be described by ( ).

In this simulation, both and are first numerically computed. Then, is computed accordingly. Depending on , , , and , the suppression function of CW STED is further investigated based on in Eq. (4). Subsequently, is obtained from Eq. (5) and the FWHM for is evaluated accordingly. In addition, can also be calculated from Eq. (18). Finally, the experimental resolution and the numerical and theoretical FWHM are compared and summarized in Table 1.

Table 1.

Comparison between the experimental and computational results for the resolution (or FWHM) of a STED system based on four experimental cases.

.
3.2. Results

The key parameters for the four revisited CW STED systems including the wavelength and power of the excitation (λexc) and STED (λsted) beams and fluorescent dyes with their relevant time constants are all summarized in Table 1. These STED systems all use Gaussian excitation beams and doughnut-shaped STED beams, as illustrated in Fig. 4(b). For instance, in the setup of Willig et al. (2007)[9] (case 2), the numerical simulations of the excitation and depletion beams at the focus are plotted in Fig. 4(b). Based on the numerically simulated depletion beam, the suppression function can be directly obtained, with its profile plotted in Fig. 3. The resulting based on the depletion theory of CW STED introduced in this manuscript is illustrated in Fig. 4(b). The corresponding FWHM along the dashed line, e.g., , is thus calculated. It can be seen that the spatial resolution ( measured in the experiment is 52 nm. The corresponding , which is equivalent to the spatial resolution of the STED system, is calculated to be 52 nm by numerical simulation. In comparison, by using Eq. (18), we obtain . Both determined by numerical simulation and calculated by Eq. (18) are sufficiently close to the experimental results. In all four cases, the theoretical model matches well with the experimental results, as shown in Table 1 and plotted in Fig. 5(a). The relative error evaluated by indicates that the difference between experiment and theory is within 10% in all four cases. Therefore, the four independent experimental datasets well support our theoretical model. The error could be due to several reasons, such as, inaccurate values for the power of the excitation and STED beams given in the references. Normally, laser powers are measured before the pupil of the objective, which can lead to an overestimation for the incident laser power. Additionally, the estimation of the spatial resolution of STED in the experiments can also be inaccurate. The FWHM of is sensitive to kQ and kfl, which can be influenced by many factors, such as temperature, dye concentration, laser power density, and so on.[18] This will also lead to errors between theory and experiments. In addition, the inevitable photobleaching can also result in error.

Fig. 5. (color online) (a) Comparison of the experimental and numerical results summarized in Table 1. (b) Comparison of the numerical results with the experimental results from Luo et al.,[17] where 405 nm ( ) and 532 nm lasers were used as excitation and STED beams, respectively. The fluorescent dye used is Atto 390. and represent the fluorescent signals before and after depletion, respectively. Here, (official data from ATTO-TEC), (nonlinearly fitted by the experimental data), and .

We also compare our theory with the investigation of Luo et al.,[17] who studied the remnant fluorescent intensity ( under various powers of the STED beam, as plotted in Fig. 5(b). It is interesting to note that except for the STED beam with low power (12 mW), all other experimental points are well matched by the theoretical predictions. Currently, for a sufficiently high STED beam power, our theory can be used to accurately predict the depletion process of a STED system. The discrepancy observed between theory and experiment at low powers is still not yet clear. One of the possible reasons for this difference is that the statistically steady state predicted in Eq. (2) does not apply in the low γ region. When the power of the STED beam is sufficiently high, the low γ region can be inhibited and its influence on the model is significantly limited. However, if the power of the STED beam is low, a large region with a low γ can exist. Consequently, the steady-state based model becomes invalid. This can result in a large discrepancy between the model and experiment.

4. Discussion

When the optical parameters, such as wavelength, power, and beam shape, are fixed, the spatial resolution (or FWHM) of the CW STED system is determined by the time constants of the fluorescent dye, i.e., kQ, kfl, and kvib.

We have also analyzed the influence of the time constants on the FWHM of , as shown in Fig. 6. It can be seen from Fig. 6(a) that for the kfl range considered, increases rapidly with kfl increasing from 46 nm to 117 nm. To reduce , one can use fluorescent dyes with a smaller kfl. This theoretical result is supported by the experimental observations of Neupane et al.,[7] who found that the FWHM of a CW STED system qualitatively increased with increasing kfl in the range of to .

Fig. 6. (color online) Influence of the time constants on the FWHM of , i.e., . The computation is based on case 2. (a) Influence of the rate of fluorescence, where and . (b) Influence of the rate of quenching, where and . (c) Influence of the rate of relaxation, where and .

Similar results are also shown in Fig. 6(b), where the relation between kQ and is plotted. For kQ in the range of to , increases from 26 nm to 114 nm. In the most commonly used range, i.e., to , varies from 30 nm to 52 nm. A variation in of more than 50% can be found. The results above show that both the fluorescent lifetime and quenching time of the dye can significantly influence the FWHM of a CW STED system. Therefore, these system parameters should be accurately determined by independent measurements. However, recent studies have ignored the influence of kQ when building CW STED systems. Furthermore, due to the complicated mechanism of quenching, it is difficult for researchers to measure the quenching rate of fluorophores.[18] We consider this to be the major obstacle in the application of our model.

The influence of vibrational relaxation has also been evaluated via kvib. For kvib in the range of to , the variation in for is less than 0.6 nm. Compared to kQ and kfl, kvib has negligible effect on the FWHM of CW STED, as plotted in Fig. 6(c).

In picosecond STED systems, where the fluorescence process in fluorophores is related to the time and pulse-width, the dependency of the spatial resolution (or FWHM of on and kfl is implicit, as discussed by Westphal and Hell.[19] However, is observed to be strongly dependent on kQ and kfl in the CW STED system for which the fluorescent process of the fluorophores is statistically steady. This is very interesting. From Eq. (1), when and kfl are sufficiently small (equivalent to a large fluorescent lifetime and quenching time), the population of the electrons in L2 is primarily dominated by n1 (which is determined by the intensity of the excitation beam) and the intensity of the STED beam. Hence, depletion at locations with high γ can be more easily achieved. Nevertheless, when and kfl are large, the decay process for n2 is proportional to the term . The population at L2 decreases much faster through the spontaneous fluorescence process instead of the depletion process. Consequently, depletion is weakened, which results in a larger FWHM for . It should be noted that the quenching mechanism and its time constant rely on many factors, such as the concentration of the fluorescent reagent, temperature, and additives.[20] As the concentration of the dye is increased, the molecular interaction is strengthened. This leads to fluorescent extinction through the process of fluorophore composition or non-radiative transition.[20] This may explain why it is normally difficult to reach high depletion efficiency at high concentrations of fluorescent dye.

5. Conclusion

In this manuscript, the depletion process for a CW STED system is reanalyzed. The results indicate that from the viewpoint of the suppression function, the depletion process in a CW STED system is not exactly the same as that in a picosecond STED system. The picosecond STED system is more dependent on the STED beam, while the CW STED system is more significantly affected by the intensity ratio of the STED to excitation beams, i.e., the relative intensity between the two beams. We also find the quenching and fluorescence rates, i.e., kQ and kfl, strongly influence the spatial resolution in a CW STED system. The FWHM of the effective fluorescent spot decreases rapidly with decreasing kQ and kfl. This observation is supported by the experimental results of Neupane et al.[7] Therefore, a proper selection (or development) of new fluorescent dyes with smaller kQ and kfl can be used as an effective method to increase the spatial resolution of a CW STED system. Based on numerical simulations, the theoretical model is determined to be highly consistent with reported experimental results. Despite the smaller power ratio of the STED and excitation beams, CW STED systems utilizing YFP citrine and pTDI dyes show better performance for realizing higher spatial resolutions compared with Atto 565 and 647N dyes.

As a fantastic super-resolution optical method, the STED technique has undergone rapid development. This investigation provides us with an important theoretical basis for the future development of CW STED microscopy. The results serve as a reminder that intrinsic dynamic processes with short lifetimes are still important even in a superficially steady fluorescent system.

In addition, since all of the relevant time constants are of the order of 1 ns or less, one can predict that the statistically steady state of a CW STED can also be realized by utilizing pulsed excitation and depletion lasers with pulse widths of the order of 10 ns. Therefore, we can construct a nanosecond STED system with a high spatial resolution without the need for precise temporal synchronization, as is required in a picosecond STED system, or the occurrence of large thermal effects, as can be found in a CW STED system.

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