1. Introduction

The ability of light to exert a mechanical force on matter has been well documented since the pioneering studies of Arthur Ashkin [1, 2, 3, 4]. In particular, at the size scale of cells right down to the size of a single atom, light may trap and move such objects at will in a non-contact and sterile fashion [5, 6]. This insight has spawned a range of techniques that have made major inroads into various disciplines including the biological, physical and chemical sciences as well as microfluidic engineering. Examples include optical manipulation of single molecules [7], colloidal particles [8] and cells [9] as well as fundamental studies of the momentum of light [10, 11]. The two optical forces acting on matter are the optical gradient force and the scattering force [6]. The former results from refraction and draws particles of appropriate index of refraction into the intensity maximums of a light field while the latter may act to propel particles along the beam propagation direction.

The particular concept of “Optically mediated particle clearing” has only recently enriched the field of optical manipulation. This concept facilitates clearance of colloidal particles and cells from a region in a sample chamber by propelling these objects in an arc to a different sample region [12, 13] akin to a snowblower clearing roads from snow. The particle clearing effect is based on the unusual properties of Airy wavepackets which were realized in the optical domain as so-called Airy beams in 2007 [14, 15], although originally discovered in quantum mechanics over 30 years ago [16]. Airy beams do not spread while propagating. Moreover, the beam profile exhibits transverse spatial acceleration meaning the beam intuitively bends while propagating. More precisely, Airy beams exhibit an overall triangularly shaped transverse beam profile featuring a main spot at the triangle’s apex and a series of side lobes whose intensity decreases with distance from the main lobe. Given these particular properties an Airy beam is capable of dragging objects across the side lobes into the intense main spot due to the gradient force followed by object transport along the curved trajectory of the main spot due to the scattering force.

In this paper we report a novel concept termed “Optical path clearing (OPC)” which crucially aims to clear a sample volume in contrast to sample area clearing as elucidated above. The key aspect of OPC is the general prospect of enhanced signal transmission over extended distances in dynamic turbid media. A particular example refers to the challenge of sending a laser beam through a colloidal suspension or an aerosol whilst retaining the beam profile by avoiding massive light scattering on water droplets. Our basic OPC studies featured both theoretical modeling and experimental realization. The theoretical model for beam propagation was based on the paraxial equation of diffraction for misaligned optical systems [17] where each particle is modeled as a misaligned aperture and lens. Optical forces were determined via a Maxwell stress tensor model. We have considered both annular-shaped Laguerre-Gaussian (LG) beams and assemblies of multiple Airy beams which are all particular solutions of the paraxial equation of diffraction.We have performed proof-of-principle experiments using dense colloidal suspensions which were irradiated with the respective beams. Overall, we find that the successful application of OPC will require a dedicated quest for the best beam profile which provides high performance clearing and which, at the same time, avoids particle jamming and thermal convection mediated by high beam peak intensities. We provide a suggestion for such optimized beam configurations including a successful experimental realization.

This paper is organized as follows: Section 2 provides an outline description of our theoretical model of OPC. We then describe our experimental results in Section 3. Crucially, we demonstrate that (rotating) multiple Airy beams are capable of clearing regions in a sample in a synchronized effort. This section also presents OPC experiments based on so-called “inverted axicon beams” which are engineered for efficient clearing in highly dense media. We conclude the paper in Section 5 including a discussion on advanced Airy-beam based patterns for OPC.

2. Theoretical model

2.1. Paraxial equation and Huygens integral

In our studies the beam propagates over large distances and has a large beam waist compared to the beam wavelength. Therefore, we can model the beam propagation through a colloidal suspension by the monochromatic paraxial equation

with z the direction of propagation and k the wavevector. The scalar field u(x,y,z) modulates the carrier wave exp(i(ωt − kz)) and defines the electromagnetic vector fields via the magnetic vector potetials A = u(x,y,z)exp(−ikz)(1,0,0) and the Lorentz gauge. In the case of free space propagation the paraxial equation (1) is equivalent to the Huygens integral

which gives the scalar potential u in any transversal plane at a distance z from the initial plane z ₀ with the transverse coordinates x′ and y′. The Huygens integral allows one to determine the final beam profile after its propagation through a series of ABCD elements, a formalism used to describe lenses, apertures or curved mirrors. To model beam propagation through colloidal suspensions we must consider beam propagation through laterally offset paraxial elements

We denote the start position of the A₁B₁C₁D₁ element as z ₁, the end position as z′₁, the optical length as L ₁, and the lateral offset as (x ₁,y ₁). The general and distance dependent solution then reads

As shown in Ref. [17], the challenge to find a general solution for a particular beam is mastered by constructing a suitable set of parameters which allow one to treat beam propagation as a transformation of these parameters.

In our study, we have focused on single-ringed LG beams [10] and finite Airy beams [14] with the respective initial beam profiles

for the LG beam and

for the Airy beam. z_r is the Rayleigh range of the LG beam. Ai is the Airy function, s ₀ the characteristic transverse length scale and a the aperture coefficient which determines the propagation distance of the finite Airy beam [18].

2.2. Description of particles

The transparent colloidal particles are modeled as non-absorbing, apertured ball lenses; that is the beam is conceptually decomposed into an obstructed beam and a part that is focused by the particle in question. The obstruction is implemented as a destructively interfering Gaussian beamlet originating from a soft aperture defined by the matrix

where ω_a is the waist of the aperture which we treat to be proportional to the particle radius. The aperture is equivalent to a position dependent intensity transmission coefficient given by exp(−((x−x_a)² + (y−y_a)²)/w ² _a) where x_a and y_a are the lateral position of the aperture. The focusing part of the beam corresponds to the propagation of the apertured paraxial beam through a ball lens that can be described as a thick lens with the associated matrix:

where R, n ₀ and n ₁ are the radius of the particle, the outer and the inner indexes of refraction, respectively [19]. In total, after passing through the particle the single beam is decomposed into the superposition of three paraxial beams: the unchanged incident beam, the destructively interfering shadow beam and the focused apertured beam. Overall, the model ensures conservation of the total intensity of the incident beam.

2.3. Paraxial Maxwell’s stress tensor and optical forces

Our determination of the optical forces is based on the conservation of the electromagnetic moment which is given by Maxwell’s stress tensor

Using the definition of the electromagnetic field and its link to the scalar field u we can deduce the paraxial stress tensor

where we neglect the higher order terms ∣∂u/∂x∣² and ∣∂u/∂y∣² similar to the method used when calculating the paraxial Poynting vector [20]. To obtain the optical force acting on the particle we need to integrate the projected stress tensor onto the normal of a surface surrounding the particle:

where we have chosen two infinite transverse planes situated at z = z ₁ just before the particle and at z = z ₂ just after the particle. It is not necessary to integrate on the sides of the particle as the boundary there is at infinity, where the field is zero. The integral (11) can be expressed analytically in the case of both LG beams and Airy beams, the latter, however, requiring a Taylor expansion. Finally, the decomposition of the incident beam into a superposition of multiple paraxial beams must be taken into account. In the most general case the beam can be written as

which implies the total optical force

where the F _ij are determined according to Equation (11) using the respective combinations of field components u_i and u_j.

2.4. OPC for a Laguerre-Gaussian beam

To elucidate the physics underlying OPC we consider here the explicit example of a single-ringed LG beam [10]. This will allow us to highlight the key physics while also pointing to some of the issues that arise in selecting an appropriate beam profile for the clearing beam.

In our application we have considered 50 particles of 2 µm in diameter with a refractive index n = 1.45 dispersed in water in a volume of 8 · 10⁻¹⁵ m³. The incident clearing beam is a single-ringed LG beam of azimuthal index l = 1 [10]. The beam features a wavelength of 532 nm, a waist of 2 µm and a total power of 0.1 W. The beam intensity profile is shown in Fig. 1(a). To indicate the clearing efficiency we introduced a Gaussian probe beam propagating along the dark core of the LG beam; the idea was that successful clearing will manifest itself in an unperturbed Gaussian profile of the probe beam since light scattering particles were removed from the beam path by the LG beam. For each time step iteration we let the particle propagate a distance proportional to the optical force plus a small normally distributed random displacement corresponding to the Brownian motion of the particle. When the particles reach the top of the simulation box we place it back at the bottom of the box such that a steady state of swarm motion is achieved simulating a large reservoir of particles. The clearing process is shown in Fig. 1(b)–(d). In Fig. (b) we see that all particles have been dragged to the bright ring of the LG beam. The probe beam profile is highly distorted before the LG beam is switched on as shown in Fig. (c) and fully recovered 100 time steps after the LG beam has been switched on as demonstrated in Fig. (d).

 

Fig. 1. OPC using a single-ringed LG beam of azimuthal index l = 1. (a) Transverse LG beam intensity profile in the absence of particles. The dashed black circle indicates the region of maximum intensity and is also included in Figs. (b)–(d). (b) Transverse particle position distribution after 100 time steps. (c) Probe beam intensity profile before LG beam was switched on. (d) Probe beam intensity profile 100 time steps after LG beam has been switched on.

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3. Experimental setup

We have used a standard optical-tweezers setup as, for instance, described in [6, 21]. A spatial light modulator (Hamamatsu LCOS X10468) allowed us to shape beam amplitudes and phases at will including beam multiplexing which is explained in detail in Reference [22]. For our OPC studies we have created single ringed LG beams and multiple Airy beams which were finally incident onto a sample chamber with a cross section of 25π mm² and a height of 1 mm. Experiments were conducted using both non-density matched and density matched colloidal suspensions using H₂O–D₂O mixtures in the latter case. The non-density matched suspensions allowed us to fundamentally study the novel beam configurations under the well-established area clearing conditions [12, 13] before proceeding to OPC using the density matched suspensions. We used aqueous suspensions of both solid polystyrene and silica spheres of various diameters σ, ranging from σ = 1.5 µm to σ = 5.7 µm in order to realize different strength ratios of the gradient and the scattering force which exhibit a cubic and square dependence on the particle diameter, respectively [6]. We have used a CW diode pumped solid state laser (P _max = 6 W, λ = 532 nm) as a light source. Typical laser powers in the sample plane were P Ȥ 1 W for the LG beam and P Ȥ 25 mW per Airy beam. The LG laser powers were significantly higher than the ones considered in theory in order to reduce the time required for successful clearing. This allowed us to successfully demonstrate clearing in proof-of-principle experiments which are not concerned about engineering tasks such as the reduction of convection flows which are potentially mediated by high laser powers. A CCD camera served to observe the colloidal suspension illuminated by a halogen light source. Successful clearing manifested itself in an enhanced transmission of the sample illumination.

4. Results

4.1. Laguerre-Gaussian beam

We have observed OPC using LG beams in qualitative agreement with the theoretical predictions as presented in Section 2.4 that is particles were dragged onto the bright ring of the LG beam used in our experiment. However, we found that the efficiency is very low due to the following effects:

Problems (i)–(iii) are amplified by the fact that LG beams do not offer flexible handling of beam parameters including core size, ring size and beam spreading during propagation. Therefore, different beams as well as more advanced OPC strategies are required as discussed in the following section.

4.2. Multiple-Airy beams

The Airy beam is a strong candidate to resolve problems (i)–(iv) denoted above due to the characteristic transverse beam profile as shown in Fig. 2(a). Problem (i) is addressed by arranging multiple Airy beams in a ring configuration with the bright main lobe on the outside; the inner region is then covered by the beam side lobes which drag particles out of the central region into the main spots due to the increasing beam intensity towards the main lobes. The multiple Airy beam configuration also tackles problem (ii); due to the transverse beam acceleration the main lobes move away from the central region and so cleared particles are propelled by scattering forces away from the cleared central region. As a consequence, particle jamming is a less severe problem since particles are not re-forced directly into the central region. Moreover, particles are efficiently re-collected due the asymmetric beam profile which persists during beam propagation. Finally, Airy beams exhibit self-healing during propagation [12, 23]; that is the beam intensity profile is restored after a certain propagation distance. As a consequence, Airy beams are less sensitive to distortions mediated by cleared particles which resolves problem (iii).

 

Fig. 2. Multiple Airy beam clearing. (a) Transverse beam intensity profile of an Airy beam. (b)–(d) Micrograph of colloidal sample (b) before, (c) shortly after and (d) ≈ 10 min after the clearing beams were switched on. The two squares indicate the initial position and size of the Airy beams, the lower on being oriented as shown in Fig. (a) and the upper one being flipped by 180°. The two arrows indicate the clockwise relocation of the two beams. Moreover the distance between the beams has been persistently increased throughout the course of the experiment yielding “hollowing” of the sample as shown in Fig. (d).

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We have realized the idea of a ring of multiple Airy beams by rotating two opposed Airy beams. Although clearing is only achieved on long time scales (several minutes) this approach allowed us to apply simple SLM encoding techniques which are the adequate choice for the proof-of-principle demonstration of multiple Airy beam based particle clearing. The results are shown in Fig. 2(b)–(d). Two beams are rotated accompanied by a persistent increase of the distance between the beams. After approximately 10 minutes, the two beams have hollowed a dense sample of colloidal particles as shown in Fig. (d).

4.3. Inverted-axicon beam

Both, the LG mode and the Airy beam based OPC partially relies on a gradient force component that pulls the particle into high intensity regions and channels them away from the optical axis. In highly concentrated aerosols or colloidal ensembles the fine spatial features of these beams can be ruined by phase randomization and light scattering as the beam propagates deeper into the medium where even the non-diffracting beams might lose their self-reconstructing property. A possible way around this might bring a specially engineered so-called “inverted axicon beam” discussed in the remainder of this subsection.

 

Fig. 3. Clearing beam created with an inverted axicon. (a) Phase mask corresponding to an inverted axicon. (b) Schematic beam propagation.

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The concept of an inverted axicon beam is visualized in Fig. 3. As opposed to a regular cone-shaped axicon, the inverted axicon tilts the incident wave front outwards where the magnitude of the tilt depends on the axicon angle. The created beam has a annular-shaped intensity profile akin to the LG beam intensity profiles. As the beam propagates the Poynting vector of the field leads away from the axial region and so any particle exposed to the beam in its high intensity region is drawn away from the axis under the influence of scattering force. Thus the optical clearing does not rely on gradient force component and clearing might be extended to longer pathways. Particles located near the optical axis are not influenced by the beam directly but as they are randomly moving under the influence of thermal diffusion, they are cleared once they reach the off axis high intensity region. Crucially, the required phase shift can be imposed using a computer controlled SLM and the phase mask shown in Fig. 3(a) and so the beam spreading can be controlled in situ by changing the imposed inverted axicon angle. To create an inverted axicon beam the optical setup was rebuilt in a manner that the SLM plane is directly projected into the sample plane area of 50 µm × 50 µm. Applying spatial modulation similar to that shown in Fig. 3(a), we formed a conical wavefront with an apex angle of 160°.

 

Fig. 4. OPC in a dense bulk aqueous colloidal suspension using an inverted axicon beam. The beam propagates perpendicularly to the image plane. Particles are propelled and gathered on the beam’s bright ring as revealed by the dark ring-shaped shade occurring in the right-hand side figure.

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At the laser output power of P = 1 W we have achieved a full successful experimental realization of OPC in a dense bulk colloidal sample as shown in Fig. 4. From left to right, the inverted axicon beam gradually hollows the sample by dragging particles into the bright ring which coincides with the dark ring-shaped shade occurring in the right-hand side figure. This dark shade is caused by particles gathering on the beam’s bright ring after having been cleared from the center.

5. Conclusion and outlook

We conclude this paper with a rating based on clearing performance parameters as presented in Table 1.

Table 1. Rating of clearing beams. “MA”= Multiple Airy, “IA”= Inverted axicon, “−B”=Beam. “+”= good, “〇”= satisfactory, “−”= not satisfactory. The item “Analytical description” refers to the question whether the relevant beam propagation can be described on the basis of the paraxial approximation.

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Clearly, both the multiple Airy beam configuration and the inverted axicon beam are superior to the LG beam which allows one to achieve OPC but with a very poor efficiency. The multiple Airy beam configuration retains coverage of the central region with side lobes over the entire propagation distance and is therefore the best choice in terms of clearing performance. The inverted axicon beam features highly flexible adjustment of the beam spread and therefore performs satisfactorily in handling particle jamming. However, the beam consists of a dark core and a bright ring akin to the LG beam and therefore shares the relatively poor clearing performance after a certain propagation distance when a pronounced extended dark core has evolved. On a final note, all beams mediate convection flows which lower the clearing performance. The convection currents are mainly a problem in terms of laser absorption within the sample, and accordingly the convection problem must be tackled by avoiding or at least minimizing laser absorption.

 

Fig. 5. Multiple Airy beams arranged on a triangular pattern. The red circle highlights the inner ring of beams which clear the central region from particles. The red arrows indicate the direction of transverse deflection during propagation of the respective beams.

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The successful application of OPC requires further improvements of both clearing performance and efficiency. Based on our present results, we suggest these crucial advances may be achieved by utilizing dynamic multiple-step clearing approaches based on either multiple Airy or inverted axicon beams. The idea is to use a configuration of multiple Airy beams as shown in Fig. 5 which is operated according to a time-sharing protocol; that is the individual beams are switched on and off in manner that the most powerful and efficient clearing is achieved. For instance, the problem of particle jamming is expected to be significantly reduced since the clearing effort is distributed to many beams, the outer ones supporting the inner ones which are the primary clearing beams. This distribution of the clearing effort proves also promising in terms of a significant reduction of convection flows; the individual beams have a relatively low intensity and can flexibly be operated in time-shared modes which proves promising for an effective suppression of synchronized actuation of convection currents. Given the flexibility of inverted axicon beams, an alternative approach would be to use a time-shared protocol based on inverted axicon beams of various spread; at a certain propagation distance, the beams then have ring-shaped intensity patterns of different diameters which mutually assist in a shared clearing effort. The inverted axicon beam, however, cannot be described analytically within the paraxial approximation, to date. As a consequence, the powerful analytical formalism comprised in Section 2 cannot be applied to the inverted axicon beam as opposed to the multiple Airy beam configuration.