Fluorescence correlation spectroscopy with a doughnut-shaped excitation profile as a characterization tool in STED microscopy Charmaine Tressler,1 Michael Stolle1 and C´ecile Fradin1,2,∗ 1 Department

of Physics and Astronomy, McMaster University, Hamilton, ON L8S 4M1, Canada 2 Department of Biochemistry and Biomedical Sciences, McMaster University, Hamilton, ON L8N 3Z5, Canada ∗ [email protected]

Abstract: The resolution of stimulated emission depletion (STED) microscopes is ultimately limited by the quality of the doughnut-shaped illumination profile of the STED erase beam. We show here that in the focal plane this illumination profile is well approximated by an analytical expression - a difference of Gaussian functions, which tends towards a first order Laguerre-Gaussian profile in the case of a well aligned beam with a true zero-intensity central minimum. We further show that along the optical axis the maximum intensity profile is reasonably approximated by a Gaussian decay away from the focal plane. The result is a fully Gaussian analytical approximation of the three-dimensional point-spread function of STED erase beams. This allows the derivation of an analytical form for the autocorrelation function of the fluorescence generated by fluorophore diffusion through the STED depletion volume. We verified this form to be correct by performing fluorescence correlation spectroscopy (FCS) experiments in solutions of the dye Alexa Fluor 532. Since the quality of the illumination profile is reflected in the shape of the autocorrelation function, we propose that fluctuation analysis can be used as a tool to assess the quality of STED erase beams. © 2014 Optical Society of America OCIS codes: (140.3300) Laser beam shaping; (140.7300) Visible lasers; (180.2520) Fluorescence microscopy; (260.2510) Fluorescence; (300.2530) Fluorescence, laser-induced.

References and links 1. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulatedemission-depletion fluorescence microscopy,” Opt. Lett. 19(11), 780–782 (1994). 2. V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94(14), 143903 (2005). 3. A. Rittweger, K. Y. Han, S. E. Irvine, C. Eggeling, and S. W. Hell, “STED microscopy reveals crystal colour centres with nanometric resolution,” Nat. Photonics 3, 144–147 (2009). 4. L. Kastrup, H. Blom, C. Eggeling, and S. W. Hell, “Fluorescence fluctuation spectroscopy in subdiffraction focal volumes,” Phys. Rev. Lett. 94(17), 178104 (2005). 5. C. Eggeling, C. Ringemann, R. Medda, G. Schwarzmann, K. Sandhoff, S. Polyakova, V. N. Belov, B. Hein, C. von Middendorff, A. Schonle, and S. W. Hell, “Direct observation of the nanoscale dynamics of membrane lipids in a living cell,” Nature 457(7233), 1159–1162 (2009).

#225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31154

6. D. Wildanger, E. Rittweger, L. Kastrup, and S. W. Hell, “STED microscopy with a supercontinuum laser source,” Opt. Express 16(13), 9614–9621 (2008). 7. G. Moneron, R. Medda, B. Hein, A. Giske, V. Westphal, and S. W. Hell, “Fast STED microscopy with continuous wave fiber lasers,” Opt. Express 18(2), 1302–1309 (2010). 8. D. Ganic, X. Gan, and M. Gu, “Focusing of doughnut laser beams by a high numerical-aperture objective in free space,” Opt. Express 11(21), 2747–2752 (2003). 9. N. Bokor, Y. Iketaki, T. Watanabe, and M. Fujii, “Investigation of polarization effects for high-numerical-aperture first-order laguerre-gaussian beams by 2d scanning with a single fluorescent microbead,” Opt. Express 13(26), 10440–10447 (2005). 10. X. A. Hao, C. F. Kuang, T. T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010). 11. G. Donnert, J. Keller, C. A. Wurm, S. O. Rizzoli, V. Westphal, A. Schonle, R. Jahn, S. Jakobs, C. Eggeling, and S. W. Hell, “Two-color far-field fluorescence nanoscopy,” Biophys. J. 92(8), L67–69 (2007). 12. B. Harke, J. Keller, C. K. Ullal, V. Westphal and S. W. Hell, “Resolution scaling in STED microscopy,” Opt. Express 16(6), 4154–4162 (2008). 13. D. McBride, C. Su, J. Kameoka, and S. Vitha, “A low cost and versatile STED superresolution fluorescent microscope,” Mod. Inst. 2, 41–48 (2013). 14. E. B. Kromann, T. J. Gould, M. F. Juette, J. E. Wilhjelm, and J. Bewersdorf, “Quantitative pupil analysis in stimulated emission depletion microscopy using phase retrieval,” Opt. Lett. 37(11), 1805–1807 (2012). 15. D. Koppel, “Statistical accuracy in fluorescence correlation spectroscopy,” Phys. Rev. A 10, 1938–1945 (1974). 16. Z. Petr´asˇek and P. Schwille, “Precise measurement of diffusion coefficients using scanning fluorescence correlation spectroscopy,” Biophys. J. 94(4), 1437–1448 (2008). 17. P T¨or¨ok and P Munro, “The use of gauss-laguerre vector beams in STED microscopy,” Opt. Express 12(15), 3605–3617 (2004). 18. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London. Series A, Mathematical and Physical Sciences 253, 358–379 (1959). 19. L. Kastrup, D. Wildanger, B. Rankin, and S. W. Hell, “STED microscopy with compact light sources,” in Nanoscopy and Multidimensional Optical Fluorescence Microscopy, (CRC Press, 2010), 1.1–1.13. 20. B. Zhang, J. Zerubia, and J. C. Olivo-Marin, “Gaussian approximations of fluorescence microscope point-spread function models,” Appl. Opt. 46(10), 1819–1829 (2007). 21. R. W. Cole, T. Jinadasa, and C. M. Brown, “Measuring and interpreting point spread functions to determine confocal microscope resolution and ensure quality control,” Nat. Protoc. 6(12),1929–1941 (2011). 22. H. Xie, Y. Liu, D. Jin, P. J. Santangelo, and P. Xi, “Analytical description of high-aperture STED resolution with 0–2π vortex phase modulation,” J. Opt. Soc. Am. A 30(8), 1640–1645 (2013). 23. M. J. Nasse and J. C. Woehl, “Realistic modeling of the illumination point spread function in confocal scanning optical microscopy,” J. Opt. Soc. Am. A 27(2), 295–302 (2010). 24. S. W. Hell, G. Reiner, C. Cremer, and E. H. K. Stelzer, “Aberrations in confocal fuorescence microscopy induced by mismatches in refractive index,” J. Microsc.-Oxford 169, 391–405 (1993). 25. S. T. Hess and W. W. Webb, “Focal volume optics and experimental artifacts in confocal fluorescence correlation spectroscopy,” Biophys. J. 83(4), 2300–2317 (2002). 26. N. L. Thompson, “Fluorescence correlation spectroscopy,” in Topics in Fluorescence Spectroscopy, (Springer, 1999), 337–378.

1.

Introduction

Amongst optical super-resolution techniques, stimulated emission depletion (STED) stands out as the most adapted to study dynamical processes, since it does not necessitate image reconstruction. It relies instead on the spatial overlay of a red-shifted ring-shaped depletion volume on the diffraction-limited excitation volume, reducing the effective size of the latter by stimulated emission [1, 2]. In ideal cases it can result in spatial resolutions as good as 10 nm [3]. The potential of STED for dynamic studies can be tapped into via combination with fluorescence correlation spectroscopy (FCS) [4]. STED-FCS was notably applied to the study of the small-scale dynamics of single lipid molecules [5]. The implementation of STED microscopy has become more cost-effective, notably thanks to the introduction of new high-power laser sources [6, 7]. Yet technical hurdles still exist on the road to building a STED microscope. In particular, the all-important production of a symmetric depletion volume with a true zero-intensity central minimum requires flawless alignment of the depletion beam with a number of optical elements and perfect control of its polarization [8–10].

#225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31155

Typically, assessment of the quality of the STED depletion volume is done by simple visual inspection of the image formed when imaging fluorescent beads [9,11,12], gold nanoparticles [6] or small apertures [13]. More elaborate methods, e.g. quantitative pupil analysis, have started to emerge [14]. However, they still rely on retrieving and analyzing the full illumination profile of the depletion beam at the focus. For FCS instruments, the alignment of the confocal excitation beam is usually assessed by inspecting the auto-correlation function (ACF) of the fluorescent signal collected for a wellcharacterized fluorophore diffusing through the excitation volume. A diagnostic can be made based on the specific brightness and transit time of the fluorophore. Both these quantities are immediately accessible by fitting the ACF, and they are respectively maximal and minimal for a symmetric diffraction-limited excitation volume [15]. Here, we consider which quantities can be extracted from the ACF resulting from the diffusion of fluorophore through a ring-shaped excitation volume, and which can be used as benchmarks for assessing the quality of STED depletion volumes. To this end, we first look for a Gaussian analytical approximation of the full STED depletion profile, allowing the derivation of a general expression for the ACFs resulting from fluorophore diffusion in ring-shaped excitation volumes. We then provide a comparison of these expressions to experimental data, demonstrating that the proposed analytical Gaussian form indeed provides a correct description of actual STED depletion profiles, and that it can be used to analyze the result of FCS experiments in doughnut-shaped excitation profiles. 2. 2.1.

Methods Instrument

All experiments were carried out on a home-built confocal instrument based on an inverted microscope (Eclipse Ti, Nikon). Excitation at a wavelength λ = 532 nm was provided by a continuous wave laser (Verdi-V10, Coherent) operating in the TEM00 mode. When required, its phase was modulated with a 2π vortex phase plate (VPP-1, RPC Photonics), adding a +1 topological charge to the beam. The initially linear polarization of the beam was rendered lefthanded circular by a quarter-wave plate (WPQ05M-532, Thorlabs) placed right before the high numerical aperture (NA) water immersion objective lens (Plan Fluor 60×, NA 1.27, Nikon). The collected epi-fluorescence signal was focused by the microscope tube lens (which, together with the objective lens resulted in a magnification M = 60×) and, unless otherwise specified, passed at that point through a 50 μ m-diameter confocal pinhole to reduce out-of-focus background fluorescence. The signal was then split between two photo-multiplier tubes (H7421-40, Hamamatsu) using a polarizing beam-splitter. A digital correlator (Flex02-01D, correlator.com) provided both time-resolved fluorescence intensity and ACFs. Images and image stacks were obtained by scanning the sample with a piezoelectric stage (P 733.2CL, Physik Instrumente) and piezoelectric objective holder (P 721.17, Physik Instrumente). 2.2.

Visualization of illumination profiles

To directly visualize PSFs, we imaged R = 40 nm-radius orange carboxylate modified microspheres (FluoSpheres), with peak excitation 540 nm and peak emission 560 nm (F8792, Invitrogen). The beads were attached to a microscope coverslip before imaging. Typically, stacks of 10 images (50 × 50, with a 40 nm distance between two adjacent pixels in each image, and with two consecutive images spaced by 200 nm along the optical axis) were acquired with an excitation power around 10 nW and a ∼ 5 ms dwell time. Each image was analyzed using a dedicated home-written ImageJ plugin to extract an average radial intensity profile. Briefly, the centre of the PSF was first determined visually, after which the intensities of all pixels within a certain distance of this centre were averaged.

#225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31156

2.3.

Fluorescence correlation spectroscopy

All FCS experiments were carried out on solutions of the dye Alexa Fluor 532 C5 maleimide (A-10255, Life Technologies). For storage, the dye was diluted at high concentration in DMSO. For experiments, a small amount of the stock solution was then diluted in deionized water to reach a nanomolar concentration (∼ 5 nM) of dye. To the best of our knowledge a reliable value for the diffusion coefficient of Alexa Fluor 532 had not been published. We therefore measured it by comparing the correlation data obtained for Alexa Fluor 532 to that obtained in the same conditions for Alexa Fluor 546 succinimidyl ester (A-20002, Life Technologies), which has a known diffusion coefficient D = 341 μ m2 /s at 22.5◦C [16]. We found (data not shown) that the diffusion coefficient of Alexa Fluor 532 was 16 ± 2% larger than that of Alexa Fluor 546, i.e. D = 396 ± 10 μ m2 /s at 22.5◦C (the temperature used in our experiments). This is as expected, given the respective molecular weight of these two dyes, 821.88 g/mol for Alexa 532 C5 maleimide and 1159.6 g/mol for Alexa Fluor 546 succinimidyl ester. In all FCS experiments, fluorescence was excited using a laser power around 25 μ W. Usually, 10 ACFs were collected in short succession for each sample and each condition, each for a duration of 30 s. The data was analyzed using the program Kaleidagraph (Synergy Software). 3. 3.1.

Analytical approximation of ring-shaped illumination profiles Calculated STED depletion profile

The ring-shaped illumination profile used for depletion in STED experiments is usually produced by a 2π phase modulated left-handed circularly polarized laser beam focused through a high NA objective. The illumination profile in the focal region can be calculated from the expression of the electric field predicted by vectorial diffraction theory. Assuming that the optical system is aberration-free and that the electric field has no significant component along the optical axis (a condition realized for a left-handed circularly polarized incident beam given a +1 topological charge [9, 17]) the light intensity in the focal plane is given by [18, 19]:  θ 2  2π  max √  ikr sin θ cos φ   d θ cos θ sin θ d φ P ( ρ (θ ) , φ ) e I(r) = I0   . 0 0

(1)

In the above expression, r is the radial distance from the focal point, θ and φ are the inclination and azimuth angles of the light rays emerging from the objective lens, θmax = sin−1 (NA/n) the light wavenumber, and n is the index of is the semi-aperture of the lens, k = 2π n/λ is √ refraction of the focusing medium. The factor cos θ is the geometrical apodization factor associated with an aplanatic lens. P (ρ , φ ) is the normalized complex amplitude of the beam incident on the objective lens, where ρ (θ ) = ρmax tan θ / tan θmax is the radial distance from back-aperture. the optical axis at the objective lens and ρmax is the radiusof the objective  For a TEM00 Gaussian beam with radius w, P (ρ , φ ) = exp −ρ 2 /w2 . The numerical integration of Eq. (1) then produces a profile closely resembling the well-known Airy pattern (Fig. 1(a)). In the presence of a 2π vortex phase plate, the azimuthal phase modulation of the beam results in a modification of the intensity profile to accommodate the introduced phase singularity, generating a Laguerre-Gaussian beam. For a first-order Laguerre-Gaussian beam, √   P (ρ , φ ) = ( 2ρ /w) exp −ρ 2 /w2 exp (+iφ ) [17]. The illumination profile then assumes a doughnut shape (Fig. 1(b)). To facilitate comparison with experimentally obtained profiles, we also calculated the profiles expected when imaging a small spherical object (simulating the imaging of a fluorescence bead) with radius R up to 50 nm. We both considered the case where the fluorophores are distributed on the surface of the bead (”sphere”) and the case where they are distributed homogeneously throughout the volume of the bead (”ball”). In the first case, the intensity at a given radial #225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31157

Fig. 1. Calculated illumination profiles in the focal plane, in the absence (a) or in the presence (b) of 2π phase modulation. All profiles were obtained for λ = 532 nm, n = 1.3, NA = 1.27, θmax = 78◦ and w/ρmax = 1. The 2D illumination profiles are shown above the radial profiles (scale bar: 0.5λ ). Ideal profiles (expected when imaging infinitely small objects with R = 0 nm) calculated directly according to Eq. (1) (dark symbols) are compared to profiles convoluted with the shape of a ball with radius R = 50 nm (light symbols). Note that the radial profiles have all been normalized to the height of the ideal profile in the absence of phase modulation, and that the vertical scale is different in (a) and (b). The 2D illumination profiles, on the other hand, have all been normalized by the highest intensity point in the image. Solid lines are Gaussian fits, using Eq. (2) in (a) and either Eq. (4) (for R = 0 nm) or Eq. (3) with σ = 0.99 (for R = 50 nm) in (b). Residuals are shown in the lower panels. (c) Height of the central minimum relative to maximum height as a function of bead size, for profiles obtained in the presence of phase modulation. (d) Characteristic profile size (obtained from the Gaussian fits) as a function of bead size.

position, r, was calculated as an average of the actual illumination intensity over the sphere of radius R centered at r in the focal plane (where the illumination profile, who varies slowly along the optical axis, was taken to be constant along z). In the second case, the integration was performed over the volume of the sphere. The results are presented in Fig. 1, where convoluted profiles are shown for the case of the ball and R = 50 nm. The most striking difference between the ideal and convoluted profiles is that the central minimum of the doughnut shape profile is no longer 0 in the non-ideal case (Figs. 1(b) and 1(c)). 3.2.

Approximation of ring-shaped profiles by a difference of Gaussian functions

In the absence of phase modulation, e.g. for confocal microscopy, the point spread function (PSF) is often approximated by a three-dimensional Gaussian profile [20]: 2

IG (r, z) =

2

2z 2P − 2r 2 − 2 2 e ωG e SG ωG , 2 πωG

(2)

where P is the total power carried by the beam through the focal plane at z = 0, and where ωG and SG ωG are the transverse and longitudinal 1/e2 radii of the PSF, respectively. Using Eq. (2) to fit the calculated illumination profile in the absence of phase modulation gave ωG = 0.33λ (Fig. 1(a)), very close to the often cited value ωG = 0.51 (2 ln 2)−1/2 λ /NA, i.e. 0.34λ for NA = 1.27 [21]. It should be noted that the exact value of ωG depends on how much the incident beam fills the objective back aperture, i.e. on the value of w/ρmax , which was chosen here to be 1, to roughly reproduce our experimental conditions. Although it fails to reproduce the fringes of the Airy pattern (which in confocal imaging are eliminated anyway by the presence of a #225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31158

Table 1. Characteristic dimensions of the PSFs Parameter Fit of calculated PSFs Fit of experimental PSFs no pinhole (n = 1) P = 200μ m (n = 1) P = 50μ m (mean, n = 6) Fit of experimental ACFs P = 50μ m (mean, n = 45)

Gaussian PSF ωG zG 0.33λ 0.54λ 0.53λ (0.49λ ±0.02)λ

0.8λ 0.6λ 0.8λ

(0.52 ±0.02)λ

Doughnut PSF rD,max zD 0.26λ

SG

ωD 0.37λ

3.0 1.7 2.1 ±0.7

0.65λ 0.61λ (0.54λ ±0.03)λ

0.46λ 0.43λ (0.38λ ±0.02)λ

5.7 ±2.5

(0.54 ±0.05)λ

(0.38 ±0.04)λ

1.5λ 1.0λ 1.1λ

SD

3.2 1.8 3.2 ±0.7 6.0 ±1.9

pinhole in the detection path), the Gaussian approximation is always within 0.03IG,max of the calculated profile (see residuals in Fig. 1(a)). Using Eq. (2) to fit the convoluted profiles showed that the apparent size of the PSF should only be very slightly affected when imaging a finite size fluorescent bead (Fig. 1(d)). By analogy, we looked to approximate ring-shaped PSFs by a difference of two Gaussian functions with a slight difference in amplitude ( f ) and in width (width ratio σ ):    2r2  2 2 − 2 − 2r − 22z 2 2P/ πωD2 2 σ, f ωD σ 2 ωD SD ωD e ID (r, z) = e − (1 − f ) e . (3) (1/σ 2 − 1 + f ) This expression can capture the shape of PSFs with or without a central minimum (i.e. either ring-shaped or peak-shaped), depending on the values of f and σ , as illustrated below in section 4.2. The best fit of the ideal calculated profile in the presence of 2π phase modulation was obtained for f  0, reflecting the presence of a zero-intensity central minimum, and for σ → 1. In this limit, Eq. (3) becomes: 2

ID (r, z) =

2

2z 2P 2r2 − 2r 2 − S2 ω 2 ωD D D, e e πωD2 ωD2

(4)

an expression calling to mind the intensity profile of a first-order Laguerre-Gaussian beam. Using Eq. (4) to fit the calculated illumination profile in the focal plane in the presence of 2π phase modulation returned ωD = (0.358 ± 0.003)λ (Fig. 1(b)). Convoluted profiles were better fit with the more general expression, Eq. (3), which can account for a non-zero central minimum. Doing so confirmed that the profile characteristic length, ωD , is only very slightly affected by convoluting the profile with the shape of a bead (Fig. 1(d)). Just as for the Airy disc, this simple analytical approximation fails to reproduce the presence of secondary maxima, yet it provides an approximation of the calculated profile that is always within 0.13ID,max of the calculated profile. √ The profile captured in Eq. (4) is maximal at a distance rD,max = ωD / 2 from the optical axis, reaching a value ID,max = 2P/(πωD2 e). For comparison, if the 2π phase modulated beam carries the same power, P, as the non phase modulated beam, then ID,max = IG,max (ωG /ωD )2 e−1 . Using the values of ωD and ωG given by the fits of the calculated profiles yields ID,max = 0.22IG,max , close to the actual amplitudes of the calculated profile. The proposed approximation (Eq. (4)) captures two important features of the PSFs of STED erase beams. The first is that, just like the excitation beam, it remains diffraction-limited with a single characteristic length scale ωD ∼ λ /2NA. Thus, the increase in resolution achieved in STED is solely due to a combination of #225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31159

Fig. 2. Representative PSFs measured by imaging single fluorescent beads in the focal plane, without (a) or with (b) phase modulation. The left panels show images obtained with a 50 μ m-diameter pinhole (scale bars: 400 nm). The right panels show the normalized average radial intensity computed from images acquired with or without a confocal pinhole. Solid lines are fits of the profiles with simple Gaussian approximations (Eq. (2) in (a) and Eq. (3) with the value of σ fixed to 0.99 in (b) - in both cases a constant background term was added).

diffraction-limited patterns. The second is that the illumination profile is parabolic close to the optical axis, a well-known attribute that can be used to derive the theoretical resolution power of STED microscopy [12, 22]. 3.3.

Experimental illumination profiles

We next verified that the proposed analytical form of the illumination profile with 2π phase modulation (Eq. (4)) is indeed a good approximation for experimental STED beam illumination profiles. To visualize PSFs, we imaged fluorescent beads (r = 20 nm) attached to a microscope coverslip, both with and without phase modulation. The imaged profiles differ from the ideal calculated illumination profiles given by Eq. (1) for at least two reasons. First, the imaging of finite-size beads will modify the shape of the profile, as discussed in the previous section. Second, the presence of a confocal pinhole in the detection path modifies the detected profile by cutting out light, especially above and below the focal plane. For this reason, the experimental profiles presented below are referred to as ”detection” profiles rather than ”illumination” profiles. To assess the effect of the confocal pinhole on the detection profiles, images were obtained without pinholes and with pinholes of different sizes (50 μ m = 1.6 Airy unit and 200 μ m = 6.5 Airy unit, where the Airy unit for the used confocal set-up is d = (1.22λ /NA)M = 30 μ m). As expected, in the absence of phase modulation the detection PSF in the focal plane resembled an Airy disc, while in the presence of a 2π vortex phase plate it assumed a doughnut shape (Fig. 2). The measured ring-shaped profiles had a low but non-zero central minimum (with I(0, 0)/Imax  5%). This can at least in part be attributed to the finite size of the imaged beads, since for 20 nm-radius spheres our simulations show that at best I(0, 0)/Imax = 2% (see Fig. 1). When fitting the experimental PSFs with Eqs. (2) and (3) with σ = 0.99, respectively, we found there was a very good agreement between the shapes of the experimental and Gaussian profiles, as long as a constant background term was added. The most conspicuous consequence of including a confocal pinhole was a significant reduction in background noise. For both types of profiles, the widths of the experimental PSFs decreased slightly as the confocal pinhole size was reduced (see Table 1). This effect was stronger for the doughnut-shape profile, as expected since it extends further than the peak-shaped profile. However, even with a 50 μ m pinhole, these widths remained  40% larger than that of the calculated ones. Although this discrepancy can seem large, it is within the range of what is usually observed when measuring PSFs experimentally [21]. It can be attributed to a combination of optical aberrations [23], medium

#225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31160

Fig. 3. Representative three-dimensional PSFs obtained (a-c) in the absence and (d-f) in the presence of phase modulation. (a,d) Image stack showing the enlarging and dimming of the PSF away from the focal plane (at z ≈ 0nm). (b,e) Examples of radial intensity profiles at different distances from he focal plane (solid circles). Solid lines are fits of the data with Eq. (2) (b) or Eq. (4) (e). In (a,d) and (b,e), the confocal pinhole diameter was 50μ m. (c,f) Maximum intensity profile (left panel) and characteristic size of the PSF (right panel) as a function of distance from the focal plane. Data is shown for three different pinhole sizes. The continuous lines are fits of the data with the expected dependence for a propagating Gaussian beam (Eq. (5) in the left panel and Eq. (6) in the right panel). The dashed lines in the left panel show an alternate fit with a Gaussian function.

stratification [24], imperfect beam shape and/or underfilling of the objective back-aperture [25], and the finite size of the imaged particles (although this last effect is small, as shown in section 3.1). To assess the shape of the PSFs along the optical axis we acquired image stacks (Fig. 3). In the absence of phase modulation, the radial profile of the PSF was very well approximated by a Gaussian function at all distances from the focal plane (Fig. 3(b)). Similarly, in the presence of phase modulation (Fig. 3(e)), the radial profile of the PSF remained very well approximated by a difference of Gaussian functions (Eq. (3), with σ = 0.99). The amplitude of the central minimum (relative to the profile maximum) remained less than 10%. In the fits, this was reflected in the fact that the obtained values of f remained less than 0.001. This confirmed that in the case of a well-aligned STED erase beam, both Eqs. (3) and (4) provide excellent approximations of the detection PSF. The peak intensity of a propagating Gaussian or Laguerre-Gaussian beam is expected to quickly decrease away from the focal plane according to: Imax,G/D (z) =

2 πωG/D

2P 2 ,  1 + z/zG/D



(5)

where zG/D is the Rayleigh range. In parallel, the characteristic size of the PSF should slowly #225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31161

increase according to:

2  ωG/D (z) = ωG/D 1 + z/zG/D .

(6)

Using Eqs. (5) and (6) to fit the experimentally obtained values of Imax (z) and ω , for both the Gaussian beam (Fig. 3(c)) and the Laguerre-Gaussian beam (Fig. 3(f)), confirmed that these expressions gave a good description of the data and that the value of the Rayleigh range, zG/D , 2 /λ . approached the theoretical value, πωG/D Because ωG (z) varies slower than IG (0, z) close to the focal plane, ωG is often considered to be constant. Further, for practical purpose, the Lorentzian form of Eq. (5) is often replaced by a Gaussian. In the case of the Gaussian beam, these two simplifications lead to the often-used purely Gaussian form of the three-dimensional PSF captured in Eq. (2). The data we acquired confirmed that these simplifications could also be used in the case of the phase modulated beam. The intensity profile along the optical axis was equally well approximated by a Gaussian or a Lorentzian decay (Fig. 3(f)), while the variation of the PSF size was reasonably approximated by Eq. (6) (the asymmetry in the profile is likely due to refractive index mismatch [24]) and slow enough compared to the variation in intensity that the simplified forms of the vertical intensity variations captured in Eqs. (2) and (3) are justified (Fig. 3(f)). The obtained value of the aspect ratio (SD  3) was comparable to that obtained for the Gaussian PSF (SG  2). 4. 4.1.

Fluorescence correlation spectroscopy in a doughnut Derivation of the autocorrelation function for diffusion in ring-shaped excitation volumes

Since Eq. (4) represents a good analytical approximation of the doughnut-shaped PSF obtained by 2π phase modulation, we next used its more general form (Eq. (3)) to calculate the ACF of the fluorescence signal collected for particles diffusing through ring-shaped excitation volumes, and to compare it to that obtained for the excitation volume generated by a Gaussian beam. For a three-dimensional Gaussian confocal detection volume with 1/e2 radii ωG in the focal plane and SG ωG along the optical axis (Eq. (2)), the ACF takes the form: GG (τ ) =

γG 1

. NG (1 + τ /τ ) 1 + τ / S2 τ  G G G

(7) 

NG is the average number of fluorescent particles in the detection volume, V = (I (r) /Imax ) dr.  The geometrical factor γ is defined as γ = (I (r) /Imax )2 dr/V . It is often omitted in the FCS literature, through absorption in the definition of the excitation volume. It is, however, an interesting quantity when considering changes in detection volume geometry, because it indicates how sharply the intensity profile of that volume decays to zero, and, as explained below, because it allows calculating the true specific brightness of the fluorophores [26]. For a Gaussian profile we have VG = (π /2)3/2 SG ωG3 and γG = 2−3/2 ≈ 0.35. The characteristic time, τG = ωG2 / (4D), is related to the transit time of the diffusing particles through the excitation volume, where D is the particles diffusion coefficient. To calculate the ACF in the case of doughnut-shaped PSFs, we used Eq. (3) instead of Eq. (2) to describe the illumination profile. This leads to the modified expression: σ ,f = GD

γDσ ,f NDσ ,f

   2   1 + Aσ ,f /Dσ ,f ττD + Bσ ,f /Dσ ,f ττD 

,



 τ σ 2τ 2τ 1 + τD 1 + τD 1 + 1+1/σ 2 τ 1 + S2ττ ( )D D

(8)

#225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31162

where:

Aσ ,f Bσ ,f Dσ ,f

  = σ 2 1 − σ12 − f 1 − σ14 − f 1 + σ32 ,  2 = 2σ 2 1 − σ12 − f ,  = σ12 + σ14 + (1 − f ) 1 − σ32 − f 1 + σ12 .

ND is the average number of particles in the ring-shaped excitation volume, VD . The factor γD is defined as above. Both the values of VD and γD depend on σ and f . In the limit where f → 0 and σ → 1, that is for the detection profile best representing a STED doughnut (Eq. (4)), the previous ACF simplifies into:  2 τ τ 1 + 2 + 2 τD τD γD . GD (τ ) =

3

ND 1 + ττD 1 + S2ττ

(9)

D

In this case VD = (π /2)3/2 eSωD3 and γD = γG e/2 ≈ 0.48 (with γD > γC indicating a sharper decay of the profile than in the simple Gaussian case). As expected the ACF captured in Eq. (9) has a single characteristic time, τD = ωD2 / (4D), since the STED doughnut is defined by a single characteristic length-scale. An important quality factor in confocal FCS experiments is the specific brightness of the fluorophore, B, defined as the fluorescence detected for a particle placed at the position of maximum excitation intensity. It is obtained by dividing the average detected fluorescence signal, F , by the number of particles in the detection volume, N, such that B = F /N = F G(0)/γ . For a Gaussian volume BG = GG (0) F /0.35, while for a STED depletion volume BD = GD (0) F /0.48. An apparent specific brightness, Bapp = G(0) F = γ B can be obtained straightforwardly from the FCS data, without any knowledge of the detection volume geometry (necessary to calculate γ ). Bapp acts as a quality factor in FCS experiments, because its increase is linked directly to an increase in statistical accuracy [15]. 4.2.

Expected signature of ring-shaped volumes in fluorescence correlation spectroscopy

One of the challenges encountered when setting up a STED experiment is the production of a PSF with true zero-intensity central minimum. We therefore asked whether such a PSF had a distinctive signature in the corresponding ACF. For this, we generated and examined a series of profiles with a single length-scale (such as the one used for STED depletion beams) by using the general expression proposed for the PSF (Eq. (3)) in the limit where σ → 1 (in practice we used σ = 0.99). As f varies from 0 to 1 the PSF changes continuously √ from a perfect STED doughnut (with zero-intensity central minimum and maximum at r = ωD / 2) to a Gaussian profile with 1/e2 radius ωD (Fig. 4(a)). This roughly represents the type of shape changes expected when the polarization of the incident beam goes from left-circularly polarized to linear, or in other words when the quarter waveplate controlling the beam polarization becomes misaligned. The switch between ring-shaped profiles (with central minimum) and peak-shaped profiles occur for f = 1 − σ 2 (Fig. 4(b)). A number of changes are observed in parallel in the corresponding ACFs (shown in Fig. 4(c)). We specifically concentrate here on quantities that can be calculated directly from the FCS data, without making any assumptions on the actual shape of the detection PSF, and therefore not depending on a particular analysis model. One such quantity is the apparent specific brightness, Bapp . As the profile switches from Gaussian to doughnut-shaped the amplitude of the ACF, G(0), is reduced by about half (Fig. 4(c)), and therefore so is the apparent specific brightness of fluorophores (Fig. 4(d)). A second easily extracted quantity is the ACF half-time, τ1/2 , defined #225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31163

Fig. 4. (a) Normalized illumination profiles generated using the difference of Gaussian model (Eq. (3)) with σ = 0.99 and f varying from 0 (ring-shaped profile) to 1 (Gaussian profile). (b) Depth of the central minimum as a function of f . The transition between ringshaped and peak-shaped PSFs occurs at f = 1 − σ 2 = 0.02. (c) ACFs calculated for the profiles shown in a). (d) Amplitude and (e) half-time of the ACFs as a function of f . (f) ACFs (same as in c)), normalized with respect to both G(0) and τ1/2 . (f) Shape factor of the ACFs as a function of f . Inset shows the shape factor as a function of the depth of the central minimum.

as the lag time at which the amplitude of the ACF falls to G(0)/2. τ1/2 increases almost 2fold when a shallow central minimum appears in the PSF, but it does not change much as this central minimum deepens (Figs. 4(c) and 4(e)). Unfortunately, as variations in Bapp and τ1/2 mostly occur as soon as a shallow central minimum appears, a decreased value of Bapp or an increased value of τ1/2 can signal the presence of a central minimum, but they do not guarantee a central zero-intensity. In addition, a decrease in Bapp may be caused by many other different factors (e.g. increase in background noise, incorrect positioning of the objective collar, etc...). A more specific marker of the presence of a zero-intensity minimum appears to be instead the conspicuous shape change observed for the doughnut ACF in the two decades preceding the average decay time (Fig. 4(f)). This change can be quantified by comparing the value of the normalized ACF at τ1/2 /5 with that same value for the ACF obtained with a Gaussian volume,       defining a shape factor: G(τ1/2 /5)/G(0) / GG (τ1/2 /5)/GG (0) = G(τ1/2 /5)/G(0) /0.835. #225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31164

Fig. 5. Experimentally determined ACFs without (orange symbols) and with (blue symbols) phase modulation. The measurement time in this case was 5 min. The ACFs halftimes, τ1/2 = 42μ s and τ1/2 = 82μ s, are indicated by arrows. Fits with Eq. (7) (orange line) and Eq. (9) (blue line), respectively, are shown. The inset shows the same ACFs after normalization by τ1/2 . The two arrows in the inset highlight the sameness of the curves at τ1/2 , and their dissimilarity around τ1/2 /5.

As Bapp and τ1/2 , this quantity can be calculated directly from the ACF, without making any assumption on the shape of the detection volume. It decreases regularly as the PSF central minimum deepens, reaching a minimum value of 0.95 for a zero-intensity minimum (Fig. 4(g)). It is therefore in principle a good indicator of the presence of a central minimum, although the relatively small change in shape factor (5%) means that high quality FCS data need to be acquired in order to detect it reliably, and might make its use as a quality factor difficult in practice. 4.3.

Comparison with experiments

To check whether a change in the shape of the ACF upon creation of a central minimum in the excitation volume was indeed discernible experimentally, we compared the results of FCS experiments performed with and without phase-modulation (Fig. 5). In both cases, a nanomolar solution of the dye Alexa Fluor 532 was used for the experiments, and its diffusion through different excitation volumes was characterized by examination of the ACF. The ACF obtained for a ring-shaped PSF had a ∼ 2.5-fold reduced amplitude and a ∼ 2-fold increased half-time compared to the ACF obtained for the traditional confocal volume, in agreement with the predictions made using the PSF analytical forms (and taking into account that the characteristic size of the experimental doughnut profile, ωD , was about 10% larger than that of the Gaussian profile, ωG , as shown in Table 1). Most importantly, as anticipated, there was a difference in shape between the two ACFs. The ACF obtained in the absence of phase modulation is perfectly fit by Eq. (7), with the addition of a photophysics term to explain the small decay present around ∼ 1 μ s. Using D = 396 μ m2 /s for the diffusion coefficient of the dye leads to the estimate ωG = 277 nm = 0.52 ± 0.02λ (mean ± stdev, n = 45), in good agreement with the image of the PSF shown in Fig. 2(a). The ACF obtained for the ring-shaped PSF is also very well fitted by the expression derived from the difference of Gaussian approximation with σ → 1 (Eq. (9)). On average, this fit yields ωD = 287 nm = 0.54 ± 0.04λ (mean ± stdev, n = 45), predicting that the maximum of the doughnut-shaped PSF is at 0.38λ , i.e. the same value as that obtained by directly imaging the profile (Table 1). From the normalized ACFs (Fig. 5, inset), the shape factor can be estimated:

#225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31165

In the absence of phase modulation it is 0.94, the value expected for a PSF with zero or near zero intensity central minimum. 5.

Conclusions

The results presented here show that the PSF used for depletion in STED experiments has a lateral intensity profile that closely resembles that of a first-order Laguerre-Gaussian beam, and an axial profile that is well approximated by a simple Gaussian function. Unlike previously proposed analytical approximations of this profile, this proposed three-dimensional approximations fully captures the shape of the STED PSF, including far from the optical axis. This approximate Gaussian form of the STED PSF allowed us to calculate an expression for the autocorrelation function associated with fluorophore diffusion in such a volume. This calculation, and the accompanying FCS experiments presented here, confirm that there is a detectable difference in the ACFs measured for a true doughnut-shaped detection volume and a traditional confocal volume. The increase in the width of the PSF results in a doubling of the ACF half-time, while the decrease in the PSF maximum intensity results in a significant decrease of the ACF amplitude and of the dye specific brightness. Most importantly, the appearance of a central intensity minimum in the PSF is accompanied by an ACF shape change in a specific region of the curve that is directly correlated to the presence and the depth of this minimum. Therefore, recording of ACFs corresponding to dye diffusion through a STED depletion beam allows a quick assessment of the quality of the STED beam through analysis of the shape of the ACF. This method presents the advantage of requiring only very simply prepared samples (nanomolar dye solution), and no imaging or image analysis (visual or automatic). It could thus be used as an online method, not as a replacement to imaging but as a complementary method, to confirm the formation of a central intensity minimum during certain steps of alignment such as the adjustment of the quarter waveplate controlling beam polarization. It can be implemented by adding a photon counting detector (if not already present) and a correlator to any existing STED set-up. In addition, a simple script to extract the value of the ACF around τ1/2 /5 will be required for automation. The successful analytical approximation of the STED doughnut illumination profile presented here can also be used in the future to calculate the cross-correlation function between excitation and depletion beams in a STED set-up, which is expected to help guide the alignment between the two beams. Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council of Canada. C.F. thanks Dr. D. O’Dell for useful discussions.

#225199 - $15.00 USD Received 23 Oct 2014; revised 20 Nov 2014; accepted 21 Nov 2014; published 8 Dec 2014 (C) 2014 OSA 15 December 2014 | Vol. 22, No. 25 | DOI:10.1364/OE.22.031154 | OPTICS EXPRESS 31166