aa r X i v : . [ phy s i c s . op ti c s ] F e b Analytical description of high-apertureSTED resolution with 0-2 π vortex phasemodulationSupplementary Information Hao Xie , , Yujia Liu , , , Dayong Jin , and Peng Xi ∗ Department of Biomedical Engineering, College of Engineering, PekingUniversity, Beijing 100871, China; School of Life Sciences and Biotechnology, Shanghai Jiao Tong University,No. 800 Dongchuan Road, Shanghai 200240, China; Advanced Cytometry Labs, MQphotonics Research Centre, MacquarieUniversity, NSW 2109, Sydney, Australia; Wallace H Coulter Department of Biomedical Engineering, Georgia Instituteof Technology and Emory University, Atlanta, USA *Email:[email protected]
To analytically derive the field of circular polarized incident laser, it is nec-essary to note that in the rotated coordinations, Eq. (1) still holds with phasemask function e ik Φ( θ ′ ,φ ′ ) = 1 for the excitation beam and e ik Φ( θ ′ ,φ ′ ) = e i ( φ ′ + π ) for the depletion beam. It yields e ′ x ′ ( P ) = − iA y ( I − I cos 2 φ p ) e ′ y ′ ( P ) = iA y I sin 2 φ p e ′ z ′ ( P ) = − A y I sin φ p for excitation and e ′ x ′ = A y [ I exp( iφ p ) + 12 I exp( − iφ p ) − I exp( i φ p )] e ′ y ′ = iA y I exp( − iφ p ) + I exp( i φ p )] e ′ z ′ = − A y [ I + I exp( i φ p )]for depletion. 1n Eq. (4) the parameters are defined as I = a + a ( k ρ p ) I = a ( k ρ p ) I = a ( k ρ p ) I = b ( k ρ p ) I = b ( k ρ p ) I = 0 I = b + b ( k ρ p ) I = b ( k ρ p ) (S1)where a = [ B ( s, , /
4) + B ( s, , / / a = [ B ( s, , /
4) + B ( s, , / / a = B ( s, , / / a = [ B ( s, , ) − B ( s, , )] / b = [ B ( s, / , /
4) + B ( s, / , / / b = [ B ( s, / , / − B ( s, / , / / b = B ( s, / , / b = B ( s, / , / / b = B ( s, / , / / I dep = D a . If we defined the depletion power as D s when peak fluoresceintensity into half, we also has I s = D s a . So depletion later power D /D s in Eq.(4) is equivalent to peak intensity ratio I dep /I s in Eq.(5). For continuouswave, I ex ( ρ ) η ( I dep ( ρ )) = I ex (0) / η ( I dep ) = 1 / (1 + I dep /I s ), I ex = C [ a +2( a a + a )( k ρ ) ]and I dep ( ρ ) = D b ( k ρ ) yield F W HM = λ ex πn √ − a a + a ) a q b a a + a ) ( λ ex λ de ) I STED I s (S2)And for continuous wave, I ex ( ρ ) η ( I dep ( ρ )) = I ex (0) / η ( I dep ) = exp ( − ln I dep /I s ), I ex = C a exp a a + a ) /a ( k ρ ) and I dep ( ρ ) = D b ( k ρ ) yield F W HM = λ ex πn q − a a + a )( ln a q ( ln b a a + a ) ( λ ex λ de ) I STED I s (S3)Then in Eq.(5) factor α and β can be obtained.To verify our code STED3D, first we simulated the excitation intensity onxy and xz plane in Fig. S1. The FWHM is 256.8 nm and 605.7 nm respectively,in good agreement with result from PSFlab [19], thus verifying our code. Thenwe plotted in Fig. S2 the intensity on the xy plane of the focus for the depletionbeam with left-circular polarized (a), linear polarized (b), and right-circular po-larized laser (c). It confirms our conclusion that only the left-circular polarizeddepletion beam results in zero intensity in the focus. Also we plotted in Fig.S3 intensity of left-circular polarized depletion beam with Eq. (S2) in (b). Thesimulated STED distribution in both linear and circular polarization agreed wellwith the results of [23], and again the matching results verified our derivation.In the discussion part we referred the saturation power P s correspondsto about twice the saturation intensity I s . To illustrate this, we plotted the2igure S1: Excitation intensity on (a) xy plane and (b) xz plane.Figure S2: Intensity on the xy plane of the focus for the depletion laser with(a)left-circular polarized, (b) linear polarized, and (c) right-circular polarizedlaser.nonlinear curves of integration depleted fluorescence efficiency and depletionintensity in Figure S4. And we also listed different factor ξ with various depletionwavelengths of STED dyes in Table S1.3igure S3: Intensity of the left-circular polarized depletion beam with Eq. (1)in (a) and Eq. (2) in (b)Figure S4: Integration depleted fluorescence efficiency and depletion intensity.Simulation results are plot in blue solid for CW-STED and red dash for pulsed-STED. Then fractal (blown dot) and exponential (green dash and dot) fits areperformed for two conditions respectively.4ye Wavelengths/nm Pulsed STED CW STEDExcitation Depletion χ χ ATTO 532 470 603 0.31 0.40Alexa Fluor 488, Chromeo 488 488 592 0.36 0.45FITC, DY-495, CitrineGFP 490 575 0.42 0.53Citrine, YFP 490 600 0.35 0.44Pyridine 2, RH414 554 745 0.26 0.33Alexa 594, Dylight 594, ATTO 594 570 690 0.36 0.45ATTO 633, ATTO 647N, 635 750 0.42 0.52Abberior STAR635Table S1: Typical values of β for different STED sys-tems. Dates of excitation and depletion wavelengths are from http://nanobiophotonics.mpibpc.mpg.de/old/dyes/ . Numerical aper-ture of the lens is set to be N A = 1 .
4, and α = 2 .
15 for pulsed STED, α = 1 . r X i v : . [ phy s i c s . op ti c s ] F e b Analytical description of high-apertureSTED resolution with 0-2 π vortex phasemodulation Hao Xie , , Yujia Liu , , , Dayong Jin , and Peng Xi ∗ Department of Biomedical Engineering, College of Engineering, PekingUniversity, Beijing 100871, China; School of Life Sciences and Biotechnology, Shanghai Jiao Tong University,No. 800 Dongchuan Road, Shanghai 200240, China; Advanced Cytometry Labs, MQphotonics Research Centre, MacquarieUniversity, NSW 2109, Sydney, Australia; Wallace H Coulter Department of Biomedical Engineering, Georgia Instituteof Technology and Emory University, Atlanta, USA *Email:[email protected]
Abstract
Stimulated emission depletion (STED) can achieve optical super-resolution,with the optical diffraction limit broken by the suppression on the peripheryof the fluorescent focal spot. Previously, it is generally experimentally acceptedthat there exists an inverse square root relationship with the STED power andthe resolution, yet without strict analytical description. In this paper, we haveanalytically verified the relationship between the STED power and the achiev-able resolution from vector optical theory for the widely used 0-2 π vortex phasemodulation. Electromagnetic fields of the focal region of a high numerical aper-ture objective are calculated and approximated into polynomials, and analyticalexpression of resolution as a function of the STED intensity has been derived.As a result, the resolution can be estimated directly from the measurement ofthe saturation power of the dye and the STED power applied.1 ntroduction In 1873, Ernst Abbe derived the diffraction limit of conventional microscopyas d Abbe = λ/ N A , which is also regarded as the resolution limit for opticalmicroscopy [1]. It was not until the past few decades with the development ofsuper-resolution microscopy techniques that this limit has been surpassed [2].One of these techniques, stimulated emission depletion (STED), belongs to theoptical nanoscopy family of reversible saturable optical fluorescence transitions(RESOLFT) [3], in which the point spread function (PSF) of the fluorescence isengineered to obtain a higher resolution which breaks the optical diffraction limit[4]. In particular, a spatially modulated depletion beam is spatially overlappedon the excitation focal spot to effectively remove the peripheral fluorescencedistribution with either STED [5], or by exciting the electron to a triplet state(ground state depletion, GSD) [6], or through excitation state absorption (ESA)to excite the electron to a higher energy level [7]. To achieve super-resolution inthe lateral plane, a widely used method is to place a 0-2 π vortex phase modulatorin the optical path of the circular polarized depletion beam [8, 9, 10, 11].To estimate the theoretical super-resolution power of STED, the most widelyused expression is the square root law in which the full-width at half-maximum F W HM = h / p I/I s which was first developed by Westphal and Hell [12]and later by Harke et al. [13]. Although partially explains the experimen-tal results, it is an empirical theory without solid theoretical foundation, asthe depletion intensity distributions discussed in previous work are not derivedfrom the diffraction theory. The scalar solution to the depletion intensity of a0-2 π phase plate has been derived by Watanbe et al. [14]. Yet without con-sidering the vectorial nature of light, such theory can only be applied with asmall aperture objective where the paraxial approximation is applicable. As aresult, this theory fails for objectives with large apertures. On the other hand,it is possible to numerically simulate the intensity distribution at the focal spot,without analytical expression [15, 23]. So far, despite the huge progress in theexperimental instrumentation and application of STED in optical nanoscopy,accurate estimation of the performance of a STED system under a known de-pletion power is still lacking. It is therefore of great importance to establisha new formula describing the maximum achievable resolution of STED opticalnanoscopy, based on the relationship between the power of the depletion beamand the measured power when the detected fluorescence intensity is depleted toits half.In this paper we use a vectorial theory to investigate the laser field in theneighborhood of focus. We present the diffraction analysis in Results, whereanalytical excitation and depletion light intensity distributions are given andthen approximated with polynomials. It is demonstrated that only the chiralityof the circular-polarization matching the direction of the 0-2 π vortex phase platecan generate a central zero doughnut intensity distribution. Furthermore, witha second-order approximation, expressions of FWHM with different saturationfunctions are derived from system parameters in both CW-STED and pulsed-STED. The previous resolution law F W HM = h / p I/I s is validated, and2he relationship between the depletion power and the half fluorescence inhibitionpower are developed. Then we numerically simulate the electromagnetic fields,and compare its results with our analytical results. Finally we discuss the errorinduced by our approximation, and the scope and limitation of the applicationto this theory. Results
Wolf et al. derived the vectorial theory of diffraction under asymptotic andthe optical rays approximation [17, 18]. For a linear polarized excitation laserbeam, the electric beam in objective space can be written as: e x = − iAπ Z α Z π cos θsinθ { cos θ + (1 + cos θ ) sin φ } e ikr p cos ǫ dθdφe y = iAπ Z α Z π cos θ sin θ (1 − cos θ ) sin φ cos φe ikr p cos ǫ dθdφe z = iAπ Z α Z π cos θ sin θ cos φe ikr p cos ǫ dθdφ (1)The coordinates are illustrated in Fig. ?? . In an idealistic Gaussian imagesystem, the above expression yields: e x ( P ) = − iA x ( I + I cos φ p ) e y ( P ) = − iA x I sin φ p e z ( P ) = − A x I cosφ p where A x = k f e x / k = 2 π/λ ex and e x is the linear excitation electricfield along the x-axis, and on focal plane integration I , I , I are given by I = Z α cos θ sin θ (1 + cos θ ) J ( k ρ p sin θ ) exp( ik z p cos θ ) dθI = Z α cos θ sin θJ ( k ρ p sin θ ) exp( ik z p cos θ ) dθI = Z α cos θ sin θ (1 − cos θ ) J ( k ρ p sin θ ) exp( ik z p cosθ ) dθ where ρ p = ( x p + y p ) / . In STED a 0-2 π phase mask is generally used, as itcan generate the most efficient inhibition pattern for fluorescence depletion [3].The electric field of a depletion beam with distinct polarization states can becalculated [15]. Notice R π exp { inφ + ρ cos φ } dφ = 2 πi n J n ( ρ ), and substituteexp( ikr p cos ǫ ) = e ikz p cos θ e ikρ p sin θ cos( φ − φ p ) e ik Φ( θ,φ ) into Eq. (1), where the3hase function of 0-2 π phase plate is e ik Φ( θ,φ ) = e iφ , we derive: e x = A x [ I exp( iφ p ) − I exp( − iφ p ) + 12 I exp( i φ p )] e y = − iA x I exp( − iφ p ) + I exp( i φ p )] e z = iA x [ I − I exp( i φ p )] (2)Here A x = k f e x / k =2 π/λ de . The angular integrations I − I are defined as I = Z α dθ cos θ sin θJ ( k ρ p sin θ )(1 + cos θ ) exp( ik z p cos θ ) I = Z α dθ cos θ sin θJ ( k ρ p sin θ )(1 − cos θ ) exp( ik z p cos θ ) I = Z α dθ cos θ sin θJ ( k ρ p sin θ )(1 − cos θ ) exp( ik z p cos θ ) I = Z α dθ cos θ sin θJ ( k ρ p sin θ ) sin θ exp( ik z p cos θ ) I = Z α dθ cos θ sin θJ ( k ρ p sin θ ) sin θ exp( ik z p cos θ )Comparison of the 3D intensity profile with result of PSF Lab [19] can beseen in Fig. S1.For circular polarization incidence, the electric field can be obtained by co-ordinate rotation [20]. The electromagnetic field now consists of contributionsof two linear polarized incident fields along the x and y axis: E x ( r p , θ p , φ p ) = e x ( r p , θ p , φ p ) − e ′ y ′ ( r ′ p , θ ′ p , φ ′ p ) E y ( r p , θ p , φ p ) = e y ( r p , θ p , φ p ) + e ′ x ′ ( r ′ p , θ ′ p , φ ′ p ) E z ( r p , θ p , φ p ) = e z ( r p , θ p , φ p ) + e ′ z ′ ( r ′ p , θ ′ p , φ ′ p ) (3)where r ′ p = r p , θ ′ p = θ p , φ ′ p = φ p − π .Then the intensity of laser field couldbe calculated by I = E x E ∗ x + E y E ∗ y + E z E ∗ z for both the excitation and thedepletion beam, and residual intensity is I residual = I ex ∗ η ( I de ), where η ( I de ) =1 / (1 + I de /I s ) for continuous beams and η ( I de ) = exp[ − (ln 2) I de /I s ] for pulsedSTED [21].Now consider the light intensity distribution on focal plane z p = 0. Since J ν ( x ) = Σ ∞ k =0 ( − k k !Γ( k + ν +1) ( x ) k + ν , the approximation to the second-order can beused: J ( x ) = 1 − x , J ( x ) = x , and J ( x ) = x . Define s = sin α and B ( s, p, q ) = R s t p − (1 − t ) q − dt = 2 R α sin p − θ cos q − θdθ , which is the in-complete Beta function, we can express I − I as polynomials of ρ p with theircoefficients given by a − b , and their coefficients are incomplete Beta functionsof s , as shown in Supplementary Information. For a linear or elliptically polar-ized laser, it is always possible to rotate coordinates to make the polarization4 standard ellipse with A x = C , A y = iC , A x = D , A y = iD , where C , C , D , D are real. Then the excitation and STED doughnut intensities canbe expressed as: I ex ( ρ p , φ p ) = ( C + C ) a + 2[( C + C )( a a + a )+ ( C − C )( a a + a ) cos 2 φ p ]( k ρ p ) I dep ( ρ p , φ p ) = ( D − D ) b + ( D + D ) b ( k ρ p ) + ( D − D ) ( 12 b + 2 b b )( k ρ p ) − ( b b + 2 b b )( D − D ) cos 2 φ p ( k ρ p ) (4)This expression gives an analytical description of the excitation and depletionintensities in the vicinity of the focus. A special case is that the depletion beam isleft-hand circularly polarized (the same as the direction of the vortex direction)with C = C = √ C and D = D = √ D , which indicates that the centerof donut has zero intensity. Then I ex = C [ a + 2( a a + a )( k ρ ) ] and I dep = D b ( k ρ ) . For an elliptically polarized laser or linearly polarized laser[22], D = D leads to I de = 0 at the focus. This is derived from second-orderapproximation, but still holds true for a strict analytical solution because allthe higher-orders annihilate at r = 0. The results obtained with our methodscan also be validated with [23], as shown in Fig. S2. The full-width at half-maximum (FWHM) of STED PSF could be estimated accordingly, with thedecay functions for CW or pulse cases described in SI. Discussion
The intensity I s could be measured as the following procedures: (1) measure thepeak fluorescence intensity of excitation beam (here the size of the fluorophoreis assumed to be very small, i. e. a delta function); (2) introduce the depletionbeam without phase modulation, whose PSF should be overlapped with thePSF of the excitation; (3) adjust the depletion power until the peak fluorescenceintensity is depleted to its half [13]. As a result, the peak depletion intensityis defined as I s . When the CW saturation functions η ( I de ) = 1 / (1 + I de /I s ) isconsidered, to its second-order [21]: F W HM = λ ex αN A p βI dep /I s (5)where I dep is the non-modulated peak intensity of the depletion beam focus,and α = π p − a a + a ) /a s , β = b a a + a ) ( λ ex λ de ) . For N A = 1 . n = 1 . λ ex = 635 nm , λ de = 760 nm , this expression yields F W HM = λ ex / (2 . N A p . I dep /I s ). For pulsed-STED with saturation function η ( I ) = exp( − ln I/I s ), it is more convenient to rewrite excitation into an ex-ponential function [21] and thus α = π p − a a + a ) / ( ln /a s , β =5 ln b a a + a ) ( λ ex λ de ) . With the same system parameters, this expression yields F W HM = λ ex (1 + 0 . I dep /I s ) − / / . N A . We found the inverse root lawstill holds in vectorial theory except for modification factors α and β , which areincomplete beta functions with aperture angle as their parameters. The rela-tionship between the depletion intensity ratio I/I s and the fluorescence intensityratio P d et/P is shown in Fig. S4.In experiment, the measurement of intensity is often replaced with measuringof the laser power, and then calculating the intensity distribution for modulatedand unmodulated STED PSFs [24]. Stimulated emission intensity I s is oftendefined as the intensity of fluorescence dropped into its half [13, 21], or 1/e[12, 25]. The resolution of STED is often described with the relationship of P/ P s , where P is the depletion power and P s is defined by the depletionpower when the detected fluorescence intensity is reduced by half, if we overlapthe non-phase modulated depletion PSF with the excitation PSF [24, 13].We used STED3D to simulate the electromagnetic field distributions aroundthe focus. In Fig. we plotted the excitation (a) and the depletion (b) intensitieson x-axis of the focal plane. In (a) the simulation intensity was plotted in red, thepolynomial approximation of Eq. (4) was plotted in blue, and the exponentialapproximation was plotted in green. In Fig. (b) the numerical solution to therigid EM field intensity was plotted in red, and polynomial approximation ofEq. (4) was plotted in blue. As expected, approximations agreed well with thenumerical solution of the EM field in the vicinity of focus, but diverged with theincrease of the radius. Then we plotted in Fig. the simulated FWHM (red)and the calculated results in Eq. (5) (blue). Figure (a) is the CW case withfraction saturation function and Figure (b) is the pulsed case with exponentialsaturation function [21]. In both figures the two curves overlap better with theincrease of depletion intensity.In above discussion we followed Wolf’s diffraction theory and Harke’s ap-proximation to derive an analytical resolution expression. Yet, it is difficult tomeasure the focal intensity of a high-NA objective. Practically, the intensity iscalculated with the measurement of the laser power and diffraction distribution[13, 24, 26]. First the total fluorescence power with excitation beam is record bya confocal point detector [13, 24], and then depletion beam is introduced with-out phase modulation. Reduced detected fluorescence power is recorded withrespect of the depletion power, and the exponential-fitted saturation power isdefined as P s [26]. Therefore, it is of great importance to develop relationshipbetween the STED resolution and the STED power [26]. Numerical evaluationof the factor reveals P = P s approximately corresponds to I = 2 I s (AppendixFig. 6). Then the resolution vs. STED power relationship becomes [24, 27] F W HM = λ ex αN A p χP/ P s (6)where P is the depletion power, P s is the saturation power when the fluo-rescence is reduced by half. Our representation of resolution then becomes F W HM = λ ex / (2 . N A p . P/ P s ) and F W HM = λ ex / (1 . N A p . P/ P s )6or CW-STED and pulsed-STED, respectively. As an example typical values ofparameter χ for different dyes are listed in Supplementary Information Tab. S1.In this paper we have applied a polynomial approximation of the Bessel in-tegrals, to yield an analytical expression of STED with vectorial waves. The re-sults were validated with previous reported numerical simulations of the electro-magnetic wave. The relationship of 1 / p I/I s has been validated in both CWand pulsed STED cases. We have given new sets of resolution estimation equa-tions, to predict the resolution achievable for a STED system: the expressionof resolution, incident depletion intensity and half-peak-fluorescence depletionintensity is analytically derived with incomplete Beta functions, and the rela-tionship of half-peak-fluorescence depletion intensity and half-total-fluorescencedepletion power was established by simulation. The result can be extended toother RESOLFT type super-resolution techniques where the fluorescence pointspread function is modulated with another doughnut shape point spread func-tion through 0-2 π phase modulation.In the derivation of Eq. (5), asymptotic expansion at r = 0 makes theapproximation only available in the vicinity of the focus. In Fig. we foundthat the curves of analytical and simulation laser intensity overlap well when r was small but diverged with the increase of r . In Fig. we found that when r < nm , the approximated excitation and depletion intensity will induceerrors of 5% and 8%, respectively. As a result, our theory applies to highaperture objective lenses with N A >
1, or those with high depletion intensity,roughly when the FWHM is less than λ/
3. For smaller aperture objective lens,Watanabe’s theory [14] with scalar diffraction theory [28, 29] can provide anacceptable estimation. For the same reason, in Fig. the analytical FWHMcurves better predicts results with error less than 5% when
I > I s . Withthe development of the high efficiency fluorescent dye and through applicationof larger STED power, STED resolution of 6 nm has been demonstrated [4],and typical STED system resolution is < nm . As usually the best STEDnanoscopy resolution at a certain depletion power is concerned, rather than theresolution at moderate power, our result has a wide scope of application in theaccurate estimation of the STED (or RESOLFT) system performance.There are several improvements that can be made in the future. One isintroducing the Gaussian spatial distribution of the incident lasers, which canbe done by inserting a Gaussian factor in the integration of I − I . Since thebeam is generally expanded for overfilling the back aperture of the objective, theplanar wave can serve as a very good approximation to the experimental situa-tion. Another possible improvement would result from retaining higher ordersin the Bessel summation. It may reduce the error when the radius increases, atthe cost of much complexity in expression. References [1] Abbe, E. Beitr¨age zur theorie des mikroskops und der mikroskopischenwahrnehmung.
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Appendix B Sec. B (1968).9 cknowledgments PX thanks Prof. Stefan W. Hell for mentoring and training on STED nanoscopyinstrumentation. We thank Dr. Thomas Lawson for critical proofreading ofthe manuscript. This research is supported by the “973” Major State Ba-sic Research Development Program of China (2011CB809101, 2010CB933901,2011CB707502), and the National Natural Science Foundation of China (61178076).
Methods
Home-made codes are accessible at http://bme.pku.edu.cn/~xipeng/tools/STED3D.html (replace with Nature document url once accepted). The parameters we used arelisted as follows: numerical aperture of the objective lens
N A = 1 .
4, refractiveindex of the objective space n = 1 .
5, the focus length of the lens f = 1 . λ ex = 635 nm, the depletion wavelength λ de = 760nm, and the incident excitation beam was set to be circularly polarized. Author contribution statement
P. X. conceived the project. H. X. accomplished the theory and program coding.H. X. and YJ. L. performed the data analysis. H. X., DY.J, and P. X. drafted themanuscript together. All authors commented and approved of the manuscript.
Competing financial interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to: Peng Xi. 10 igure captionsigure captions