Optical funnel to guide and focus virus particles for X-ray laser imaging
Salah Awel, Sebastian Lavin-Varela, Nils Roth, Daniel A. Horke, Andrei V. Rode, Richard A. Kirian, Jochen Küpper, Henry N. Chapman
OOptical funnel to guide and focus virus particles for X-ray laser imaging
Salah Awel,
1, 2
Sebastian Lavin-Varela, Nils Roth,
1, 4
Daniel A. Horke,
1, 5
Andrei V.Rode, Richard A. Kirian, Jochen K¨upper,
1, 2, 4 and Henry N. Chapman
1, 2, 4 Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron DESY,Notkestrasse 85, 22607 Hamburg, Germany Center for Ultrafast Imaging, Universit¨at Hamburg, Luruper Chaussee 149, 22761Hamburg, Germany Laser Physics Centre, Research School of Physics, Australian National University,Canberra, ACT 2601, Australia Department of Physics, Universit¨at Hamburg, Luruper Chaussee 149, 22761Hamburg, Germany Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135 ,6525 AJ Nijmegen, Netherlands Department of Physics, Arizona State University, Tempe, AZ 85287, USA (Dated: 2021-02-09) a r X i v : . [ phy s i c s . op ti c s ] F e b he need for precise manipulation of nanoparticles in gaseous or near-vacuumenvironments is encountered in many studies that include aerosol morphology,nanodroplet physics, nanoscale optomechanics, and biomolecular physics. Pho-tophoretic forces, whereby momentum exchange between a particle and sur-rounding gas is induced with optical light, were recently shown to be a robustmeans of trapping and manipulating nanoparticles in air. We previously pro-posed a photophoretic “optical funnel” concept for the delivery of bioparticlesto the focus of an x-ray free-electron laser (XFEL) beam for femtosecond x-raydiffractive imaging. Here, we describe the formation of a high-aspect-ratio op-tical funnel and provide a first experimental demonstration of this concept bytransversely compressing and concentrating a high-speed beam of aerosolizedviruses by a factor of three in a low-pressure environment. These results pavethe way toward improved sample delivery efficiency for XFEL imaging experi-ments as well as other forms of imaging and spectroscopy. XFEL facilities have the capability to enable atomic-resolution images of biomolecules atphysiological temperature and with time resolution down to the femtosecond regime.
Since2009 the serial femtosecond crystallography (SFX) method has yielded nearly 500 protein-structure entries in the protein data bank, many of which are from dynamic systems andare time-sequenced with time steps down to 100 fs.
Single-Particle Imaging (SPI) aims toenable similar capabilities, but with isolated biomolecules rather than microcrystals. Imagingsingle molecules at physiological temperatures would allow the observation of functionalmolecular motions that may otherwise be hindered in the crystal environment. Additionally,with the high data collection rates possible at XFEL facilities combined with the rapidshock-freeze method, this technique might enable the detection of rare intermediate states.The most significant present-day challenge in SPI is the production of high-densitynanoparticle beams that can be directed to an x-ray beam of 100–1000 nm diameter in alow-pressure environment. Nearly all SPI experiments have utilized aerodynamic focusinginjectors for particle delivery.
Such injectors are well developed and can generate particlebeams with diameters on the order of 10 µ m, but the fraction of x-ray pulses thatintercept a particle still remains at less than 0.1 %, for a 100 nm x-ray focus. At this rate,roughly one day of continuous data collection at a 10 kHz detector frame rate would beneeded for a full atomic-resolution dataset consisting of ≈ diffraction patterns. For these2easons, we consider optical forces as a means to increase the target precision and density ofSPI injection systems. Previously, we proposed and investigated an “optical funnel” that uses a focused hollow-core optical vortex beam to guide particles into a tight focus.
Our design utilized a laserthat counter-propagates against the particle beam to increase the particle density both byslowing the particles as well as by forcing them closer to the beam axis. The inclusion of asurrounding gas activates photophoretic forces, caused by light absorption and subsequentmomentum exchange with gas molecules, and which may be larger than optical scattering orgradient forces by orders of magnitude. The effectiveness of this optical funnel scheme iscomplicated by a number of factors that include the optical and thermal properties of theparticles, radiation damage, gas pressure as limited by background X-ray scattering, and the3D profile of the optical beam. While many questions remain, the basic feasibility of theoptical funnel concept was supported by our preliminary simulation and experimental studybased on photophoretic forces that we directly measured by counterbalancing microparticlesagainst the gravitational force.
Our previous efforts to demonstrate an optical funnel onhigh-speed particle beams yielded clear evidence of photophoretic forces, but no evidence ofparticle-beam compression, which may have been due to the rapid divergence of the opticalbeam and limited laser-particle interaction length.
FIG. 1. The basic experimental setup for aerosol particle beam imaging and focusing.
Here we demonstrate particle beam compression with an optical funnel constructed from alow-divergence hollow-core first-order Bessel beam that extends the particle–laser interactionlength by a factor of up to 1000 as compared with a Gaussian beam. The basic experimental3etup consists of assemblies for optical funnel beam shaping, particle injection and high-speedoptical imaging is shown in figure 1. The optical funnel profile was achieved by forming afirst-order quasi Bessel beam with a spiral phase plate and an axicon lens, and then re-imagingthe beam inside the chamber with a de-magnifying collimator. The resulting beam changesits size due to the continuously changing magnification along the propagation direction(ˆ z axis), as shown in figure 2. The diameter and propagation length of the optical funnel arecontrolled by the initial Gaussian beam diameter, the axicon geometry and refractive index,and the optical characteristics of the re-imaging collimator, namely the distance between theaxicon and the collimator, the de-magnification rate, and the distance between the lenses inthe collimator. Since optimizing the optical funnel geometry for guiding a particular streamof particles is a multi-parameter task, we carried out simulations based on Fourier optics, as discussed in detail in the methods section.Figure 2 shows a comparison of experimental laser beam profiles along with numericalsimulations. For this comparison, we formed an optical funnel with a 532 nm cw Gaussianbeam with an output waist w = 1 . l = 1 using a 16-step phase plate. The Besselbeam was formed using an axicon with a wedge angle α = 0 . ◦ . The beam was re-imagedby a de-magnifying collimator with f = 200 mm and f = 20 mm. The formation of thisbeam is illustrated in the supplementary material figure 1. The resulting optical funnel hada minimum peak-to-peak diameter of 7 . µ m and an angle of divergence in the first brightring of 1 . × − rad.We directed this optical funnel into a small chamber where we had prepared a counter-propagating particle beam in a low-pressure (0.4–0.9 mbar) helium gas environment as shownin figure 1. Samples of 265 × ×
445 nm Cydia pomonella granulovirus particles or2 µ m fluorescent polystyrene spheres were aerosolized with a gas-dynamic virtual nozzle atapproximately atmospheric pressure, and subsequently drawn through a gas nozzle/skimmerstage in order to control the gas pressure. A collimated beam of particles was ejected from a2 mm inner diameter capillary in the direction opposite that of the optical beam propagation.The particles had speeds in the range of 2–20 m/s, depending on the gas differential pressures.The particle-laser interactions were observed by pulsed-laser Rayleigh-scattering imaging,which localizes the coordinates of individual particles. Figure 3 shows Rayleigh-scattering images of granulovirus particles at 0.99 mbar pressure4 ) a) b) FIG. 2. Slow-diverging optical funnel formed by de-magnifying a quasi-Bessel beam with a Kepleriancollimator. (a) Simulated optical funnel; the distances are from the maximum intensity position.(b) Comparison between experimental intensity profile and a calculated profile in the first ringof the funnel at the total laser power of 1 W. The solid line describes a perfect axicon whereasthe shaded area between the dashed lines indicate the intensity profiles for imperfect axicons withradius of curvature within the range from 1 m to 2 m of the front-face on the axicon. (c) Beamprofiles at various cross sections of the optical funnel; the top row is the results of simulations, whilethe bottom row are the measured profiles. illuminated by 637 nm, 100 ns laser flashes repeating at 25 kHz. 532 nm light from the opticalfunnel beam was blocked by a band-pass optical filter (Thorlabs, FL635-10) in order to isolatethe Rayleigh-scattering in these images. Multiple exposures of the same particle appear in onecamera frame, which allowed us to calculate the velocities and accelerations from the particlecentroid positions, as shown in figure 3 (d,e). The granulovirus particles in figure 3 (b) wereexposed to a 0.5 W optical funnel, with the optical beam propagating in the +ˆ z direction.Based on the observed accelerations of 2-10 × m/s and estimated granulovirus mass of2.2 × − g, we infer forces of 0.044–0.22 pN. As detailed in the supplemental material, these5orces correspond to a temperature difference of ≈ ± FIG. 3. Granulovirus trajectories recorded with 25 kHz illumination and 0.99 mbar chamber gaspressure. (a) Background corrected raw images showing particle trajectories in a 2.0 W opticalfunnel. (b) Centroid positions of a single Granulovirus particle trajectory in a 0.5 W optical funnel.The optical axis of the optical funnel is indicated by the dashed red line. (c) Calculated x (blue)and z (red) positions, velocities and accelerations based on particle centroids in (b). Figure 4 (a) shows particle density maps that reveal the effect of a 0.5 watt optical funnelon the granulovirus particle beam at 0.4 mbar pressure. Panels (a) top and bottom show the2D particle densities with the optical funnel on and off, respectively, and panel (b) shows theradial profiles of the particle beams averaged over the z = 0 . ± . µ m diameter polystyrene particle beams focused by the optical funnel at 0.5 mbar chamber6 IG. 4. Focusing Granulovirus particles at 0.4 mbar chamber pressure and 2 µ m diameter polystyreneparticles at 0.5 mbar chamber pressure. (a) Granulovirus particle areal density normalized to thepeak density in the laser-off. Laser-off (top) and in the presence of a 0.5 W optical funnel (bottom).(b) Lorentzian fit to the areal particle densities in (a), averaged over the z = [0 . , .
6] mm region. (c)2 µ m diameter polystyrene particles areal densities normalized to the peak density in the laser-off.Laser-off (top) and the particle beams were illuminated by 2.5 W (middle) and 5.0 W (bottom)optical funnel, respectively. (d) Lorentzian fit to the areal particle density in (c), averaged over the z = [2 . , .
6] mm region. (e) and (f) are the velocity histograms of the laser-off particle densities in(a) and (c), respectively. pressure, at laser powers of 2.5 W and 5.0 W. Panel (c) top shows the areal particle densitywithout the laser beam, while middle and bottom show the particle density upon opticalfunnel illumination at 2.5 W and 5.0 W, respectively. The focus of the optical funnel islocated at the position x = z = 0, i.e, outside the FOV of the images. The radial profiles ofthe particle beams, averaged over z = [2 . , .
6] mm are plotted in panel (d). The Lorentzfits show that the peak particle densities improved roughly by a factor of 3 and 5 for 2.5 Wand 5 W, respectively, compared with the laser-off case. Relevant measurement conditions7re listed in the supplementary material table s1.In our observations, laser powers of the order of 1 W and beyond resulted in clearlyobservable changes to the particle trajectories. Notably, the particle beam compression effectshown in figure 4 did not confine the particles to the dark 7.5 µ m core of the optical funnel;the particle beam density increased within a broader ≈ µ m diameter region, partly as aresult of the broad (2 mm) particle beam incident on the laser. We also observed increasesin the particle beam density at progressively larger z values, i.e. the region the particlestraverse before reaching the laser focus, as the laser power was increased. Figure 5 shows theprofiles of polystyrene particle beam densities at z = 5 . z values between 5.2 mm and 7.2 mm.The observed z -dependence of the particle density suggests that significant particle-laserinteractions begin well before particles reach the laser focus. As particles decelerate, theyinteract with the laser beam for longer duration, and with sufficient laser power the particlesbegin to stop completely and reverse direction, as shown in supplemental material Fig.s4.The overall effects of the optical funnel are the result of complex, non-linear dynamicsand are sensitive to several factors that include particle speed, gas pressure, laser power, andmost importantly the precision of the alignment of the optical axis to the particle beam axis.While it is clear that more detailed calculations and measurements are needed in order togain a complete understanding of the particle dynamics, our observation of overall increasesin particle beam density is a significant milestone toward the development of an effectiveoptical funnel for x-ray imaging of biological particles. Furthermore, our high-throughputmanipulation of high-speed particles may stimulate developments and applications such as inattosecond dynamics, nanoscience, aerosol research, atmospheric physics, materials processingand x-ray imaging of cryo-cooled particle. Using cold particles has many fold benefits to ourapproach, besides enhancing the magnitude of photophoretic force (see Eq. s13) and reducingthe possible radiation damage on the optical funnel exposed particles, cooling the particlebeam enables better aerodynamic focusing, especially when smaller particles are used. IG. 5. Increase of 2 µ m diameter polystyrene particle number density at distances between 5.2 mmand 7.2 mm before the optical funnel focus, for laser power between 0 W and 5 W. (a) The radialcross-section of the particle beam density at z = 5 . I. METHODSA. Formation of the optical funnel
The optical funnel was generated using a cw 532 nm Gaussian beam from a Coherent VerdiV5 diode-pumped laser. A 16-step spiral phase plate (Holo/Or, VL-204-Q-Y-A) of topologicalcharge l = 1 produced a Laguerre Gaussian vortex beam with zero-intensity on the axis and apeak-to-peak diameter of the ring, w pp = 2 .
42 mm. This beam was passed through an axiconwith a wedge angle α = 0 . ◦ (Thorlabs AX2505-A) to produce a first-order Bessel beamof inner ring diameter w pp = 80 µ m. The optical funnel beam was produced by re-imagingthe quasi-Bessel beam inside the vacuum chamber using a 1:10 demagnifying collimator,composed of a plano-convex lens ( f = 200 mm) and a 10 × long working distance microscopeobjective (Mitutoyo, f = 20 mm). The optical funnel focus was formed 38 mm beyond the90 × objective, where w pp = 7 . µ m. The funnel extended ≈
55 mm after the focus with adivergence of the central peak of ≈ σ x = 84 nm and σ y = 72 nm, corresponding to a beam pointing stability below 4 µ rad. B. Sample preparation
Fluorescent polystyrene spheres of 2 ± . µ m diameter (density ρ PS = 1 .
05 g/cm ,FluoSpheres, Carboxylate-Modified Microspheres, yellow-green fluorescent) were used inthese measurements. The samples were supplied as a 2 % solid fraction suspended in waterplus 2 mM sodium azide, which were diluted to a concentration of roughly 1 × particle/mlbefore the injection.Granulovirus samples, which consist of individual viruses engulfed in crystallized proteinocclusion bodies, were prepared as described previously. Since the majority of thegranulovirus particles consist of protein crystals, we assumed its density to be similar tothat of a protein crystal, ρ GV = 1 . , so the mass of a single virus is estimatedas 2.2 × − g. The sample was suspended in water to a concentration of roughly 5 × particles/ml. An SEM image of Granulovirus particles is shown in the supplementary materialFig. 6. C. Aerosol injection
The particles first suspended in liquid were introduced into the gas phase by injectingthem into a small nebulization chamber upstream of the injector using a gas dynamic virtualnozzle (GDVN) (see figure 1). The GDVN produced liquid droplets with diameters ofapproximately 1–2 µ m with sample flow rates in the range of 1–2 µ L / min. The flow rateof the GDVN helium sheath gas was in the range 10–40 mg / min. The process of aerosolformation and transporting them into the injector is described in our previous work. Locatedupstream of the injector was a nozzle/skimmer stage that was used to control the pressure in10he injector. It consisted of an electropolished 300 µ m ID nozzle and a 500 µ m ID skimmer(Beam Dynamics, Inc.) with a 2–5 mm gap between them. The injector capillary (Swagelok × ×
110 mm and 45 × ×
110 mm in volume, respectively(see figure 1). D. Particle imaging
Particles were imaged with two distinct imaging systems as shown in figure 1. The firstwas a high-speed camera (Photron SA4) combined with a long working distance objective(Mitutoyo MY5X-802 – 5 × ) to provide a magnified field of view (FOV) of 2 × . Thesecond was a high quantum efficiency camera (photometrics prime 95B) combined witha variable-zoom objective (Thorlabs MVL6X12Z–6.5 × ) and was used to produce a lowermagnification with larger FOV in the range of 4 × – 10 ×
10 mm .The illumination for both cameras was provided by either a side illuminating Nd:YLFlaser (Spectra Physics Empower ICSHG-30, 527 nm, repetition rate 1 kHz, pulse duration150 ns, pulse energy 20 mJ, average power 20 W) or an oblique-illuminating fiber-coupleddiode laser (DILAS IS21.16-LC, 637 nm, 10–100 ns pulses, repetition rates up to 1 MHz,average power 10 W). The output of the Nd:YLF laser was focused by a cylindrical lens offocal length 75 mm to form a light-sheet of size 5 mm × µ m parallel to and intersectingthe axis of the particle beam as shown in figure 1. The intense and short pulse duration ofthe Nd:YLF laser produced single snapshots of many particles without significant motionblur. However, due to the relatively slow repetition rate of this illumination, images ofindividual particles were recorded at most once in a single frame. Therefore, such particleimages were used to count particles passing through the FOV in a given time and reconstructthe two-dimensional particle density map, as discussed in section I E below.The output from the fiber-coupled diode laser was collimated and focused into the chamberusing a fiber output collimator (Thorlabs F810SMA-635) and a plano convex lens, f = 50 mm.Typically, the diode laser was operated at a repetition rate between 25 kHz and 100 kHz,11hereas the camera recorded frames at 1 kHz and 1 ms exposure. Therefore, using thisillumination, a single particle could be imaged multiple times in a single frame, as depictedin figure 3 (a). Since the centroids of the images of particles could be accurately determinedfrom the frames, and the time between between each snapshot was accurately known, suchimages could be used to extract the particle trajectories and study the particle dynamics, asdiscussed in section I E below. Importantly, due to the broad range of particle speeds andresulting overlaps in the slower-moving particle images, it was not possible to quantitativelytrack all particle trajectories at a laser pulses repetition rate of 25 kHz when the opticalfunnel was turned on. E. Data analysis
Background noise in the particle images hampers their analysis and thus must be keptto a minimum. We found that the main source of noise in the recorded data was lightscattered from the stainless steel injector tip and the chamber walls. Most of the backgroundarising from the optical funnel was blocked by the optical filters shown in Fig. 1, whereasthe relatively constant background produced by the illumination lasers was reduced bysubtracting a time-integrated median background image from every raw frame. This wasfollowed by a spatial band-pass filtering of the images to smooth them and eliminate theremaining high frequency background.After the pre-processing of images, particles were identified by searching for connectedpixels with intensities and size above an empirically determined threshold. Intensity centroidswere then calculated for each group of connected pixels for every frame, and stored as list ofcoordinates. For particle number density determination, centroid positions were extractedfrom images collected over a period of time. These positions were accumulated into thetwo-dimensional particle density profile as shown in Fig 4 (a) and (c). To better represent theparticle density improvement by the introduction of the optical funnel we typically normalizethe laser on and the laser-off 2D particle beam profiles by the maximum particle beam densitymeasured in the laser-off. Note that in these measurements the particles were illuminated bythe Nd:YLF laser.For velocity and acceleration analysis, first the particle trajectories must be extracted fromeach frame of the analysed centroid data set. This was done by searching for clusters of particle12entroids in a frame which belong to the same particle trajectory, using a density-based spatialclustering of applications with noise (DBSCAN) clustering algorithm. Particle velocitiesand accelerations were then determined from each found trajectory, using finite differencecalculations based on the known illumination laser frequency, as shown in Fig 3 (c-e). Thischaracterisation of the particle dynamics can be used to calculate the forces and light-inducedtemperature changes on the particles, as discussed in the supplementary material. Theanalysis was done using a custom Matlab script.13
I. SUPPLEMENTAL CONTENTA. Modeling of the optical beam
Our numerical optical funnel simulation is based on a free-space propagation method usingFourier optics with phase shifts caused by the optical elements. The numerical simulationsstart with the description of the Gaussian beam. The electric field distribution is given, incylindrical coordinates, by: E g ( r, z = 0) = (cid:32) P tot πw (cid:33) / exp (cid:32) − r w (cid:33) , (1)where w and P tot are the waist and the total power of the initial Gaussian beam, respectively,and r = (cid:112) x + y is the radial coordinate. The complex-valued field at a particular plane z can be propagated by convolution with the Fresnel propagator or multiplied by the complexrepresentation of the phase modulation of a particular optical element. A full description ofthe phase modulation induced by each optical element is given below. Phase shift produced by a phase plate:
A vortex phase plate is a diffractive elementwith helical thickness variation that induces a linear phase shifts with respect to the azimuthalangle φ . This structure controls the phase of the transmitted beam azimuthally, transforminga Gaussian beam into a Laguerre-Gaussian vortex beam. The resulting field at a distance z v from the Gaussian beam source will be E l ( r, φ, z v ) = E g ( r, z v ) (cid:115) | l | ! rw | l | (2) × exp (cid:32) − r w + ilϕ (cid:33) where φ is the azimuthal coordinate and l is the azimuthal index, respectively.The index l , also called topological charge, refers to the number of 2 π cycles in the helicalstructure of the phase plate, making an increasing numbers of spiral staircases proportionalto the index. The topological charge is responsible for the amount of angular momentumof photons that compound the beam and the size of the hollow core of the vortex beam,both directly proportional to l . We note here that the diameter of the vortex beam ring atmaximum intensity, 2 w l , is related to the waist of the initial Gaussian beam by w l = w (cid:112) | l | /
2. 14
IG. 6. (a) Schematic illustration of the optical funnel formation. The initial Gaussian beam istransformed into a hollow-core Laguerre-Gaussian beam by the spiral phase plate. The first-orderBessel beam is formed by the axicon lens. The combined lenses f and f form the de-magnifyingcollimator that results in the slow-diverging optical funnel due to the varying magnifications ofthe Bessel beam image. (b) Propagation of a high-order Laguerre-Gaussian beam through anaxicon of thickness ∆ . The plane wavefront is refracted by the axicon, forming a new beam, whoseplane waves propagate over a surface of a cone with an angle β . These wavefronts interfere, creatingthe quasi-Bessel beam within a limited length z max . The round-shaped tip of the axicon (inset)is included into the simulations to achieve an accurate modeling of the experimental interferencepattern forming the optical funnel.(c) Constructing the optical funnel by de-magnifying a quasi-Bessel beam with a Keplerian collimator. The distances d , d , and d are the distance from theobject to first lens, the distance between lenses and the distance from the second lens to the image,respectively. Phase shift produced by an axicon:
To imprint the phase shift required to transforman incoming beam into a quasi-Bessel beam, an optical element of a conical shape is used,called an axicon. When the axicon is evenly illuminated, it refracts the incoming planewaves into waves that cover a conical surface with an angle α . After passing the axicon theinterfering wavefronts create an intensity profile described by several concentric rings, whose15istribution depends on the topological charge of the incident beam as well as the axicongeometry and refractive index.The angle between the optical axis and the normal of the refracted wavefront, β , is givenby axicon parameters as β = n − n n α = n − n n π − τ . (3)for a refractive index n of the axicon and the surrounding medium n , and τ is the apexangle of the axicon (Fig 6(b)).In order to compare the experimental results with the simulations we need to take intoaccount the imperfection of the conical surface of the axicon. This was done by introducinga radius of curvature on the front-face of the axicon, which affects the intensity distributionin the propagation of the optical field. To derive an expression for the transmitted fieldwe treat the axicon as a thin optical element. Thereby, we introduced a variable radius ofcurvature R on the front (ideally plane) face of the axicon to find the optimal match betweenthe simulation and experiment. The expression for the field modulation by the axicon is amodification of Brzobohaty et al. who represented its conical surface as a hyperboloid ofrevolution of two sheets. The field is written as follows, E ax ( r, z axi ) = E l ( r, z axi ) exp( ikn ∆ ) exp (cid:40) ik ( n − n ) (cid:34) R (cid:32) − (cid:114) − r R (cid:33) + (cid:32) a + r tan ( τ / (cid:33) / (cid:35)(cid:41) (4)where z axi is the distance from phase plate to the axicon and ∆ is the axicon maximumthickness. The field distribution E l ( r, z axi ) is the result of the free-space propagation over adistance z axi of the field produced by the phase plate E l ( r, z v ) using Eq. 3. The parameter a is of least importance in our case because the vortex beam has a singularity on its axis,which leads to negligible effect due to the minimal interaction with the central region ofthe axicon. This can be seen when we compare the incident beam, with a waist of severalmillimetres, to the parameter a , typically in the range of tens of micrometres.In order to give a fully quantitative description of the Bessel beam axial propagation, it isrequired to provide an expression for the length and the position of focus of the beam. Theseexpressions can be found in the work of Jarutis at al., where they described the focusingproperties of the high-order Bessel beams using an axicon,16 max = w tan α (5) z f ( l ) = w √ l + 12 sin α . (6) z max in Eq. 5 shows the maximum distance where the interference effect is still active, i.e. thelimited volume where the beam is created, while Eq. 6 describes the position of maximumintensity, also called the focus of the Bessel beam. Comparing Eqs.(5, 6), we note that thefocus position of the beam depends on the vortex order, whereas the length does not. Thisdifference is because with the increase of topological charge, the singularity of the vortexis increased, and consequently, the interference starts at a longer distance from the axicon,whereas the region of the interference will not be modified, because the ring width does notchange when the index l is modified. Based on this work, we can derive an expression forthe core radius of the Bessel beam when z (cid:29) r : r = j ( l,m ) λ π sin( α ) (7)where j l,m is the m -th maximum of the Bessel function of first kind and l -th order. Phase shift produced by a lens : After the modulation provided by the phase plateand the axicon, the beam enters the re-imaging system. A thin lens with focal length f i shapes an incoming beam by adding a phase equal to − ikr / (2 f i ). The resulting field justbehind the lens is obtained from: E Li ( r, z Li ) = E ( r, z Li ) exp (cid:18) − i k f i r (cid:19) (8)The distances z Li denote the position of the i -th lens relative to the previous opticalelement. Re-imaging of the Bessel beam:
Re-imaging is important part of the optical setup asit adds the divergence to the optical funnel and controls the minimum size of the beam andits position on the axis. The de-magnifying collimator is comprised by two thin lenses L L
2, with focal lengths f and f . The optical system can be optimised using the ABCDapproach for finding the optimal distances for the desired magnification and the focus of theoptical funnel. Schematically, the re-imaging setup is shown in Fig. 6(c).17he parameters that describe the geometry of the re-imaged beam are expressed as follows[20]: d = ( d f − f f ) d − d f f ( d − f − f ) d − d f + f f (9) M ( d ) = f ( d − f ) − d ( d − f − f ) f f (10)Θ = arctan (cid:34)(cid:32) d − f − f f f (cid:33) r (cid:35) (11)Eq. (10) presents the magnification of the object’s image depending on distance d , thusdetermining the radius of the optical funnel r (cid:48) ( d ) = M ( d ) r . It should be noted that theimaging system must satisfy the condition d > f + f in order to form a divergence alongthe propagation. The main purpose of the optimisation stage via the de-magnifying systemis to re-image the Bessel beam focus into the desired spot with chosen magnification, calledfrom now on as the focus of the optical funnel.The optimization procedure for the construction of optical funnels with parametersmatching the experimental conditions begins taking a fixed d . Next, with the help of Eqs. (5,10) we determine the desired magnification and, consequently, the focus size of the funnel.Finally, we calculate the distance where the object to be re-imaged should be located usingthe Eq. 9. There is a last variable that will help us in the characterization of the funnels,which is the divergence of the optical funnel determined by Eq. 11. B. Estimated temperature gradient on the particles
We estimate the temperature gradient across the granulovirus particles via our measuredforces and the expression relating the photophoretic force to the temperature gradient inthe high-Knudsen-number regime. The granulovirus particles have a size of approximately × ×
445 nm which is equivalent to the volume of a sphere with radius r = 162 nmand a mass of m = 2 . × − g. The Knudsen number Kn = 1320 is defined here as theratio of the mean-free-path of the gas to the particle radius. The mean-free-path λ = 429 µ mfor helium may be calculated by the formula λ = kT √ πP d m (12)18here k is the Boltzmann constant, T = 298 K is the gas temperature, P = 40 Pa is the gaspressure, and d m = 260 pm is the kinetic diameter of helium. The photophoretic force isderived from the gas kinetic theory and described by the following equation: F pp = π αP r ∆ TT (13)where α ≈ T is thetemperature difference across the particle. The equation of motion of a particle relatesacceleration to the combined photophoretic forces ( (cid:126)F pp ), optical scattering and absorptionforces ( (cid:126)F opt ), and gas drag drag force ( (cid:126)F d ). Assuming a symmetric optical beam, we expecttwo relevant force components for a particle in the x – z plane: (cid:40) ma x = F ppx + F optx + F dx ma z = F ppz + F optz + F dz (14)where the subscripts x and z refer to the transverse and axial components, respectively. Theslip-corrected drag force (cid:126)F d for a particle in the high-Kn regime is (cid:126)F d = 6 πµr ( (cid:126)v g − (cid:126)v p ) CC = 1 + Kn( c + c · e − c ) , (15)where C is the Cunningham slip-correction factor, introduced by Knudsen and Weber, withempirical coefficient of, c = 1 . c = 0 . c = 1 . v p and v g are thevelocity of the particle and the gas, respectively.In order to understand the importance of gas drag forces, we numerically simulated the gasvelocity using the injector geometry and the measured pressures upstream and downstream ofthe injector as an input. The Navier–Stokes equations were solved with a finite-element solverusing the COMSOL Multiphysics software suite. As shown in figure 7, close to the exit ofthe injector the gas expands with relatively high velocity, which also accelerates the particles.However, below 15 mm from the the injector, where our measurements were performed,the gas decelerates to negligible velocity. Under the assumption of negligible gas velocity,equation 15 predicts a deceleration of approximately 2 . × m/s for granulovirus particlesin absence of laser illumination at the typical measured speed 17.5 m/s. This predicted valueis not far from our measured acceleration of 1 . × m/s , which we obtained from theacceleration/velocity histograms shown in Figure 8. These histograms moreover reveal a19 IG. 7. The simulated He gas velocity between the exit of the aerosol injector and optical funnelfocus at 0.99 mbar chamber pressure.(a) The velocity along the transverse direction and (b) thevelocity along the axial direction. The injector exit is at z = 0. This simulations were performed asdescribed in our previous publication [10]. relatively small acceleration of a z = 43 m/s at zero velocity. Based on these calculationsand observations, we assume the gas velocity to be zero in subsequent calculations.We now consider the particle trajectory shown in main text figure 3, in which the peakacceleration at the time t = 0 .
15 ms is 9 . × m/s at a particle speed of 2.2 m/s. At0.99 mbar, the mean-free-path of helium gas is 172 µ m, and Knudsen number, Kn = 528,the relevant measurement parameters are listed in table II. After subtracting the slip-corrected drag force, the combined photophoretic and optical force is 6 . × − N, whichsuggests that the photophoretic force is dominant since the maximum possible radiationpressure is F optz ≈ πr I/c ≈ . × − N at the peak intensity in the optical funnel( I = 3 . × W/m ). Thus the peak acceleration suggests that ∆ T /T = 0 . T ≤ . T ≥
298 K. If we assume the thermal conductivity of granulovirusto be g = 0 . I heat ≈ g ∆ T /r ≈ . × W/m .20 . − . . v z [m/s] − a z [ k m / s ] . . . . . . N o r m . C o un t s × − . . . v x [m/s] − − a x [ k m / s ] . . . . . . N o r m . C o un t s × − . − . . v z [m/s] − a z [ k m / s ] . . . . . . N o r m . C o un t s × − . . . v x [m/s] − − a x [ k m / s ] . . . . . . N o r m . C o un t s × − . − . . v z [m/s] − a z [ k m / s ] . . . . . . N o r m . C o un t s × − . . . v x [m/s] − − a x [ k m / s ] . . . . . . N o r m . C o un t s × FIG. 8. Granulovirus velocities and accelerations under different optical funnel illuminationconditions, each at 0.99 mbar pressure (see table II for experimental parameters). Histogramsof particle acceleration and velocity for the z and x axes are shown in the top and bottom rows,respectively, with columns corresponding to 0, 1 and 3 Watt optical funnel powers, respectively.The velocities and accelerations were estimated from finite differences in centroid positions usingtriplets of sequential images determined from 25 kHz stroboscopic images. The green dashedline in the middle row corresponds to a linear fit to the function a z = av z + b . The resultingvalues for the { , , } -Watt optical funnel illumination were a = {− , − , − } s − and b = { . , . , − . } km/s . C. Mapping the position of the optical funnel in the camera FOV
To map the location of the optical funnel in the camera FOV we rely on Rayleigh scatteringimaging of high density particles illuminated by the optical funnel. In these measurements,the optical funnel was propagating opposite to the particles and the scattered intensity wasrecorded on the camera which was set to a relatively long exposure time of 20 ms. Thisproduce long streak-images of the particles, similar to streak-imaging described in. Theseimages were pre-processed to remove the background and the scattered image intensitieswere integrated through the measured frames. Then the beam profile is reconstructed bypixel by pixel weighted averaging of this integrated image intensity.Figure 10 (a) shows the two dimensional reconstructed optical funnel profile generatedfrom the 5000 frames each containing streaked scattering intensity of granulovirus particlesilluminated by a 0.2 watt optical funnel. The axial cross-section of the beam is shown in21
IG. 9. Selected background corrected stroboscopic images of 2 micrometer diameter polystyreneparticles trajectories in 5 W optical funnel. This measurement was performed at 1.5 mbar chamberpressure and the particles were illuminated by the DILAS laser operating at 100 kHz. In this images,the optical funnel was propagating in the vertical direction and the horizontal dashed-line indicatesthe position of focal plane in the images, how we determined this position is shown in Fig. 10. figure 10 (b). The maximum intensity position, indicated by the green dashed line, showsthe focus of the optical funnel. This position is used to define the center of our coordinatesystem. Similarly, the beam position can also be inferred by time integrating air scatteringintensity in the optical funnel path as seen in Fig 10 (c). Using these techniques, we candetermine the focal position of the optical funnel with a resolution better than a micrometer.22
IG. 10. Mapping the optical funnel in the camera FOV using particles or air scattering intensities.(a) The reconstructed two dimensional intensity profile of the optical funnel. This profile isgenerated by integrating 5000 frames, each containing long exposure scatted intensity of GVparticles illuminated by a 0.2 watt optical funnel. (b) Normalized axial cross-section of the intensityprofile in (a). The green dashed-line indicates the focal plane of the optical funnel. (c) The opticalfunnel profile generated by time integrating air scattering in the beam path.FIG. 11. Scanning electron microscope image of granulovirus particles suspended on silicon substrate. II. MEASUREMENT PARAMETERS granulovirus 2 µ m polystyrene Mean particle velocity, V z , with laser-off (m/s) 17.4 ± ± × × Particle generation rate(particles/second) 1 . × . × GDVN gas flow rate 15 mg/min 16 mg/minTABLE I. Experimental parameters for the particle beam focusing presented in Fig. 4 in the maintext. granulovirus
Mean particle velocity, V z , with laser-off (m/s) -1.72 ± × Particle generation rate(particles/second) 2 . × GDVN gas flow rate 14 mg/minTABLE II. Experimental parameters for the granulovirus particle dynamics presented in Fig. 3main text and Fig. 8. V. ACKNOWLEDGMENTS
In addition to DESY, this work has been supported by the Clusters of Excellence atUniversit¨at Hamburg, the “Center for Ultrafast Imaging” (CUI, EXC 1074, ID 194651731),and “Advanced Imaging of Matter” (AIM, EXC 2056, ID 390715994) of the DeutscheForschungsgemeinschaft (DFG), by the European Research Council through the ConsolidatorGrant COMOTION (ERC-614507) and by the Australian Research Council’s DiscoveryProjects funding scheme (DP170100131). R.A.K. acknowledges support from an NSF STCAward (1231306).
V. AUTHOR CONTRIBUTIONS
S.A., R.A.K., D.A.H., A.V.R., J.K., and H.N.C. conceived the idea and designed theexperiment. S.A., D.A.H., N.R., R.A.K. and A.V.R. performed the measurements. S.A. andR.A.K. developed the particle data analysis tool and analyzed the data. S.L.V. and A.V.R.performed the optical beam modeling. S.A. wrote the initial version of the manuscript withinputs from R.A.K., S.L.V. and A.V.R. All authors contributed in scientific discussions andmanuscript revisions.
REFERENCES R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, Nature , 752 (2000). H. N. Chapman, P. Fromme, A. Barty, T. A. White, R. A. Kirian, A. Aquila, M. S. Hunter,J. Schulz, D. P. Deponte, U. Weierstall, R. B. Doak, F. R. N. C. Maia, A. V. Martin, I. Schlichting,L. Lomb, N. Coppola, R. L. Shoeman, S. W. Epp, R. Hartmann, D. Rolles, A. Rudenko, L. Foucar,N. Kimmel, G. Weidenspointner, P. Holl, M. Liang, M. Barthelmess, C. Caleman, S. Boutet,M. J. Bogan, J. Krzywinski, C. Bostedt, S. Bajt, L. Gumprecht, B. Rudek, B. Erk, C. Schmidt,A. H¨omke, C. Reich, D. Pietschner, L. Str¨uder, G. Hauser, H. Gorke, J. Ullrich, S. Herrmann,G. Schaller, F. Schopper, H. Soltau, K.-U. K¨uhnel, M. Messerschmidt, J. D. Bozek, S. P.Hau-Riege, M. Frank, C. Y. Hampton, R. G. Sierra, D. Starodub, G. J. Williams, J. Hajdu,N. Timneanu, M. M. Seibert, J. Andreasson, A. Rocker, O. J¨onsson, M. Svenda, S. Stern, K. Nass,R. Andritschke, C.-D. Schr¨oter, F. Krasniqi, M. Bott, K. E. Schmidt, X. Wang, I. Grotjohann, . M. Holton, T. R. M. Barends, R. Neutze, S. Marchesini, R. Fromme, S. Schorb, D. Rupp,M. Adolph, T. Gorkhover, I. Andersson, H. Hirsemann, G. Potdevin, H. Graafsma, B. Nilsson,and J. C. H. Spence, Nature , 73 (2011). H. N. Chapman, Annu. Rev. Biochem. , 35 (2019). J. C. H. Spence, IUCrJ , 322 (2017). I. Schlichting, IUCrJ , 246 (2015). K. Ayyer, P. L. Xavier, J. Bielecki, Z. Shen, B. J. Daurer, A. K. Samanta, S. Awel, R. Bean,A. Barty, M. Bergemann, T. Ekeberg, A. D. Estillore, H. Fangohr, K. Giewekemeyer, M. S.Hunter, M. Karnevskiy, R. A. Kirian, H. Kirkwood, Y. Kim, J. Koliyadu, H. Lange, R. Letrun,J. L¨ubke, T. Michelat, A. J. Morgan, N. Roth, T. Sato, M. Sikorski, F. Schulz, J. C. H. Spence,P. Vagovic, T. Wollweber, L. Worbs, O. Yefanov, Y. Zhuang, F. R. N. C. Maia, D. A. Horke,J. K¨upper, N. D. Loh, A. P. Mancuso, and H. N. Chapman, Optica , 15 (2021). A. K. Samanta, M. Amin, A. D. Estillore, N. Roth, L. Worbs, D. A. Horke, and J. K¨upper,Struct. Dyn. , 024304 (2020). M. J. Bogan, W. H. Benner, S. Boutet, U. Rohner, M. Frank, A. Barty, M. M. Seibert, F. Maia,S. Marchesini, S. Bajt, B. Woods, V. Riot, S. P. Hau-Riege, M. Svenda, E. Marklund, E. Spiller,J. Hajdu, and H. N. Chapman, Nano Lett. , 310 (2008). J. Bielecki, M. F. Hantke, B. J. Daurer, H. K. N. Reddy, D. Hasse, D. S. D. Larsson, L. H. Gunn,M. Svenda, A. Munke, J. A. Sellberg, L. Flueckiger, A. Pietrini, C. Nettelblad, I. Lundholm,G. Carlsson, K. Okamoto, N. Timneanu, D. Westphal, O. Kulyk, A. Higashiura, G. van der Schot,N.-T. D. Loh, T. E. Wysong, C. Bostedt, T. Gorkhover, B. Iwan, M. M. Seibert, T. Osipov,P. Walter, P. Hart, M. Bucher, A. Ulmer, D. Ray, G. Carini, K. R. Ferguson, I. Andersson,J. Andreasson, J. Hajdu, and F. R. N. C. Maia, Sci. Adv. , eaav8801 (2019). N. Roth, S. Awel, D. A. Horke, and J. K¨upper, J. Aerosol Sci. , 17 (2018), arXiv:1712.01795[physics]. S. Awel, R. A. Kirian, N. Eckerskorn, M. Wiedorn, D. A. Horke, A. V. Rode, J. K¨upper, andH. N. Chapman, Opt. Express , 6507 (2016). R. A. Kirian, S. Awel, N. Eckerskorn, H. Fleckenstein, M. Wiedorn, L. Adriano, S. Bajt,M. Barthelmess, R. Bean, K. R. Beyerlein, L. M. G. Chavas, M. Domaracky, M. Heymann, . A. Horke, J. Knoska, M. Metz, A. Morgan, D. Oberthuer, N. Roth, T. Sato, P. L. Xavier,O. Yefanov, A. V. Rode, J. K¨upper, and H. N. Chapman, Struct. Dyn. , 041717 (2015). W. K. Murphy and G. W. Sears, J. Appl. Phys. , 1986 (1964). P. Liu, P. J. Ziemann, D. B. Kittelson, and P. H. McMurry, Aerosol Sci. Techn. , 293 (1995). P. Liu, P. J. Ziemann, D. B. Kittelson, and P. H. McMurry, Aerosol Sci. Techn. , 314 (1995). M. F. Hantke, J. Bielecki, O. Kulyk, D. Westphal, D. S. D. Larsson, M. Svenda, H. K. N. Reddy,R. A. Kirian, J. Andreasson, J. Hajdu, and F. R. N. C. Maia, IUCrJ , 673 (2018). D. McGloin, D. R. Burnham, M. D. Summers, D. Rudd, N. Dewar, and S. Anand, FaradayDiscuss. , 335 (2008). V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S.Kivshar, Phys. Rev. Lett. , 118103 (2010). D. R. Burnham and D. McGloin, J. Opt. Soc. Am. B , 2856 (2011). V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, Opt. Express , 5743 (2009). D. E. Smalley, E. Nygaard, K. Squire, J. V. Wagoner, J. Rasmussen, S. Gneiting, K. Qaderi,J. Goodsell, W. Rogers, M. Lindsey, K. Costner, A. Monk, M. Pearson, B. Haymore, andJ. Peatross, Nature , 486 (2018). N. Eckerskorn, L. Li, R. A. Kirian, J. K¨upper, D. P. DePonte, W. Krolikowski, W. M. Lee, H. N.Chapman, and A. V. Rode, Opt. Express , 30492 (2013). N. Eckerskorn, R. Bowman, R. A. Kirian, S. Awel, M. Wiedorn, J. K¨upper, M. J. Padgett, H. N.Chapman, and A. V. Rode, Phys. Rev. Appl. , 064001 (2015). W. Zhu, N. Eckerskorn, A. Upadhya, L. Li, A. V. Rode, and W. M. Lee, Biomedical OpticsExpress , 2902 (2016). B. E. Saleh and M. C. Teich,
Fundamentals of photonics (Wiley-Interscience, New Jersey, 2007). D. P. DePonte, U. Weierstall, K. Schmidt, J. Warner, D. Starodub, J. C. H. Spence, and R. B.Doak, J. Phys. D , 195505 (2008). C. Gati, D. Oberthuer, O. Yefanov, R. D. Bunker, F. Stellato, E. Chiu, S.-M. Yeh, A. Aquila,S. Basu, R. Bean, K. R. Beyerlein, S. Botha, S. Boutet, D. P. DePonte, R. B. Doak, R. Fromme,L. Galli, I. Grotjohann, D. R. James, C. Kupitz, L. Lomb, M. Messerschmidt, K. Nass, K. Rendek, . L. Shoeman, D. Wang, U. Weierstall, T. A. White, G. J. Williams, N. A. Zatsepin, P. Fromme,J. C. H. Spence, K. N. Goldie, J. A. Jehle, P. Metcalf, A. Barty, and H. N. Chapman, Proc.Natl. Acad. Sci. U.S.A. , 2247 (2017). D. Oberth¨ur, J. Knoska, M. Wiedorn, K. Beyerlein, D. Bushnell, E. Kovaleva, M. Heymann,L. Gumprecht, R. Kirian, A. Barty, V. Mariani, A. Tolstikova, T. White, L. Adriano, S. Awel,M. Barthelmes, K. D¨orner, L. X. Paulraj, O. Yefanov, D. James, J. Chen, G. Nelson, D. Wang,A. Echelmeier, B. Abdallah, A. Ros, G. Calvey, Y. Chen, S. Frielingsdorf, O. Lenz, A. Schmidt,M. Szczepek, E. Snell, P. Robinson, Boˇzidararler, G. Belsak, M. Macek, F. Wilde, A. Aquila,S. Boutet, M. Liang, M. Hunter, P. Scheerer, J. D. Libscomb, U. Weierstall, R. Kornberg,J. Spence, L. Pollack, H. N. Chapman, and S. Bajt, Sci. Rep. , 44628 (2017). M. L. Quillin and B. W. Matthews, Acta Cryst. D , 791 (2000). K. R. Beyerlein, L. Adriano, M. Heymann, R. Kirian, J. Knoska, F. Wilde, H. N. Chapman,and S. Bajt, Rev. Sci. Instrum. , 125104 (2015). S. Awel, R. A. Kirian, M. O. Wiedorn, K. R. Beyerlein, N. Roth, D. A. Horke, D. Oberth¨ur,J. Knoska, V. Mariani, A. Morgan, L. Adriano, A. Tolstikova, P. L. Xavier, O. Yefanov, A. Aquila,A. Barty, S. Roy-Chowdhury, M. S. Hunter, D. James, J. S. Robinson, U. Weierstall, A. V. Rode,S. Bajt, J. K¨upper, and H. N. Chapman, J. Appl. Cryst. , 133 (2018), arXiv:1702.04014. M. Ester, H.-P. Kriegel, J. Sander, X. Xu, et al. , in
Kdd , Vol. 96 (1996) pp. 226–231. R. W. Bowman and M. J. Padgett, Rep. Prog. Phys. , 026401 (2013). J. H. McLeod, J. Opt. Soc. Am. , 592 (1954). O. Brzobohat´y, T. ˇCiˇzm´ar, and P. Zem´anek, Opt. Express , 12688 (2008). V. Jarutis, R. Paˇskauskas, and A. Stabinis, Opt. Commun. , 105 (2000). H. Rohatschek, J. Aerosol Sci. , 717 (1995). H. Horvath, KONA Powder and Particle Journal , 181 (2014). D. K. Hutchins, M. H. Harper, and R. L. Felder, Aerosol Sci. Techn. , 202 (1995)., 202 (1995).