Light beats the spread: “non-diffracting” beams

$Page 1: Light beats the spread: “non-diffracting” beams$
Laser Photonics Rev. 4, No. 4, 529–547 (2010) / DOI 10.1002/lpor.200910019 529

Abstract “Non-diffracting” beams do not spread as they propa-

gate. This property is useful in many areas. Here, the theory, gen-

eration, properties, and applications of various “non-diffracting”

beams, including the Bessel beam, Mathieu beam, and Airy

beam is reviewed. Applications include imaging, micromanipu-

lation, nonlinear optics, and optical transfection.

“Non-diffracting” finite energy Airy beam clearing a chamber

of micro-particles.

© 2010 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

Light beats the spread: “non-diffracting” beams

Michael Mazilu1,*, David James Stevenson1,2, Frank Gunn-Moore2, and Kishan Dholakia1,**

1 SUPA, School of Physics and Astronomy, University of St Andrews, North Haugh, Fife, Scotland KY16 9SS, UK2 School of Biology, University of St Andrews, Fife, Scotland KY16 9TS, UK

Received: 31 March 2009, Revised: 6 July 2009, Accepted: 28 July 2009

Published online: 28 September 2009

Key words: Non-diffracting beam, optical micromanipulation, optical injection, Bessel beam, Airy beam, Mathieu beam, Diffraction,nonlinear optics.

PACS: 03.50.-z, 42.25.Bs, 42.25.Fx, 42.40.Jv

1. IntroductionThe last fifty years have witnessed a revolution in interdis-ciplinary science. Photonics has played a central role inthe advances that have been made. This has been largelyfuelled by the use of the laser. Laser technology has quitesimply changed major areas of science beyond recognition.In the remit of the biosciences the term “biophotonics” isnow an ubiquitous one and the laser has played a majorrole in terms of clinical surgery, cell and molecular biology,spectroscopy and indeed therapy. The very ability to shapethe beam form of a laser light field has profound conse-quences on many scientific applications. Colloidal physicsis another area where the laser has made such an impact:optical forces can influence the motion and interaction be-tween dispersed colloids and create interesting physical testsystems where an optically imposed pattern or “landscape”can influence inter-colloidal interactions and aggregation.In this paper, we describe the use of advanced laser beamshaping or sculpting to avoid the ubiquitous phenomena ofdiffraction. “Non-diffracting” light modes namely Bessel,Mathieu and Airy modes are described and their appli-

cations are discussed including those in the biomedicalsciences. We commence with a brief introduction to thesefields where we concentrate on their scalar representationand then expand upon applications that directly benefitfrom such “diffraction-free” light field shaping.

2. Description of “non-diffracting”light modes

When we consider waves of any form, the topic of diffrac-tion needs special attention. This is a consideration for alloptical arrangements and in the case of laser beams diffrac-tion corresponds to the expansion or convergence of thebeam as it propagates. Indeed, laser beams offer coherentand monochromatic light outputs that have low divergenceover a long propagation distance (meters). However, inmany instances encountered in biosciences, such as con-focal laser scanning microscopy for example, we need touse high numerical aperture optics to tightly focus thesefields thus obtaining the smallest possible beam spot sizes.

Corresponding authors: e-mail: * [email protected], ** [email protected]

© 2010 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim

530 M. Mazilu, D. J. Stevenson, et al.: “Non-diffracting” beams

The widely used Rayleigh range �� characterises the ex-pansion or spread of a Gaussian laser beam. It denotes thedistance over which the area of the laser field doubles. Thisdistance is given as ��

�� where � is the wave-length and �� is the beam waist. One can immediatelysee that when using high numerical aperture microscopeobjectives, the Rayleigh range is of the order of a few mi-crons as the beam waist is so small (of the order of theincident wavelength). This results in the diffraction of thelaser beam, which quickly converges and diverges on eithersides of the focal position. The consequence is a reducedintensity per unit area as one moves away from the beamfocus position. This is naturally factored into the design andapplication of all optical systems and even exploited. Anexample is two-photon excitation or imaging [1], where thetwo-photon effect is restricted to the focal region, and thuswe may obtain optical sectioning of images without the useof pinholes to discriminate against out of focus light.However, diffraction is a limiting issue in many ap-

plications: Ideally one may wish to have long interactionor propagation lengths where the transverse dimensionsof the light field do not expand appreciably, such as laserrange finding. More general laser physics applications innonlinear optics, colloidal science and biophysics may toobenefit from light fields that avoid appreciable spreading,and this is the topic of this paper. How do we achieve thisthough when we appear constrained by the Gaussian beam?If we look at the Helmholtz equation, which governs thepropagation of monochromatic light, we can start to seethat there are in fact solutions whose transversal intensityshape is propagation invariant: indeed a “perfect” infiniteplane wave is the simplest such solution and thus these maybe termed “non-diffracting”.

The most intuitive way to understand any light field is tofirstly consider its decomposition into a set of plane waves:Plane waves provide us with a basis set by which one mayexpand any given light field. When we sum up the planewaves moving over a distance ��, each one accrues a phaseshift that may be denoted by �� where �� denotes theaxial wavevector component of the plane wave. In the nor-mal course of propagation such waves each accrue with anindividual phase shift and the beam at any given point is thesummation (interference) of these waves. However, thereare geometries of plane waves where each wave accruesexactly the same phase shift as each other. This is the caseof all straight propagating “non-diffracting” beams. Suchlight fields appear exactly the same (profile and size scaleis unaltered) at any given plane in their propagation direc-tion and are termed propagation invariant. This leads to theproperty of immunity to free space diffraction as the formof the superposition does not change at all on propagation.Beyond the plane wave the most popular “non-diffracting”mode is the now famous Bessel light mode. This mode hasall its constituent waves with equal �� and equal amplitude.We can have all the plane wave components inclined withrespect to the optic axis by equivalent angles. Although theBessel beam is the most popular “non-diffracting” modewe describe in this paper it is not the only one: the Airy

beam and Mathieu beam have also come to prominence inrecent years. All three of these beams are described in thenext sections where we distinguish between straight andaccelerating “non-diffracting” beams.

2.1. Straight “non-diffracting” beams

First, we consider the family of “non-diffracting” beamsthat propagate in a straight line and are solutions of thescalar Helmholtz monochromatic wave equation

��

��

��

��

��

��

��

� � � � (1)

where � is the angular frequency of the monochromaticwave, the speed of light and one electromagnetic fieldcomponent i.e. for linearly polarised light along the x di-rection � ��. Taking � as the direction of propaga-tion and �� as the axial wavevector component of the“non-diffracting” beam, we can write any straight “non-diffracting” beam using Whittaker’s integral

��

�

��

(2)where � is the transversal wave vector amplitude fulfillingthe relationship �� . The function�� can be any complex function that gives the ampli-tude and phase of the plane waves composing the “non-diffracting” beam, where each of these plane waves isidentified by its azimuthal angle � . Basically, this gen-eral solution can be understood in the following way. Any“non-diffracting” beam is composed of any superpositionof many plane waves whose wavevectors lie on a circle,defined by �, in the plane perpendicular to the directionof propagation. The relative phase and amplitudes betweenthe composing plane waves is given by the function ��and for particular functions the integral in Eq. (2) can beexpressed analytically. A higher order Bessel beam is givenby �� where � is theorder of the beam, while a higher order Mathieu beamby �� where �� and�� are the Mathieu cosine and sine functions and� the separation variable that will be discussed later. TheMathieu and the Bessel beams can also be deduced directlyfrom Helmholtz equations using the variable separationmethod (see [2] for example).A sculpted light field termed “the Bessel light beam”,

where diffraction is “suppressed” during free space propa-gation in theory and experiments, was first proposed in the1980s [3,4] and is the most well known of the straight “non-diffracting” light fields. The solution consisted of a planewave in �, multiplied by the zeroth-order Bessel function.The origin though predates this work: Durnin interpretedthe original work of Whittaker on the Helmholtz equa-tion [5, 6] and realised his solutions were independent ofpropagation distance � – i.e. “non-diffracting”. Diffraction

© 2010 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim www.lpr-journal.org

Laser Photonics Rev. 4, No. 4 (2010) 531

Figure 1 (online color at: www.lpr-

journal.org) (a) Transversal and (b) longitu-

dinal cross-sections of a zeroth order Bessel

beam. The gray scale represents its intensity

and the thin red line the position of the cross

section. (c) Transversal and (d) longitudinal

cross-sections of a first order Bessel beam.

The colouring corresponds to the phase pro-

file of the field and the brightness to its in-

tensity.

is restricted over a specific finite distance – were one to usean ideal version of this beam an infinite input aperture (andthus infinite power) would be required which is unfeasible.In 1987 Durnin et al. experimentally demonstrated a Besselbeam [3] that retained its key propagation invariant featuresover a limited range. An important point is that when wecompare the propagation properties of a Bessel beam toa Gaussian beam the appropriate comparison is betweenthe central core size of the Bessel beam and the spot sizeof the Gaussian beam [7]. In this case, power transport isequivalent between both forms of the beam [7].

The so-termed zeroth order Bessel beam is the mostwidely known and used “non-diffracting” light field [8].Fig. 1 shows this beam’s transverse profile as well as across section showing the relative amplitudes of the con-stituent rings (see Figs. 1a,b). Higher order Bessel functionshave dark cores (singularities) on axis (see Figs. 1c,d) andmay be generated holographically [9] or by illuminating anaxicon with a Laguerre-Gaussian (LG) beam [10]. Analo-gously with the LG modes with non zero azimuthal index� [11], higher order Bessel beams, � � �, have orbitalangular momentum (OAM) [12]. The orbital angular mo-mentum can be seen as a vortex of light where the flow oflight along a closed path is non-zero. This orbital angularmomentum originates in the relative phase between the con-stituent plane waves forming the higher order Bessel beam.In effect, the azimuthal phase change of the beam given by�� is equivalent to atilted phase front making the optical energy flow in a spiralmanner as the Bessel beam propagates [13]. Depending onthe order of the beam, at the beam profile centre there is ei-

ther a bright spot (index � � �) or a hollow core l (� � �) ormore accurately a “vortex”. The Bessel beam transverse pro-file requires light to be distributed equally amongst its con-stituent rings: essentially the longer the beam propagationdistance the more rings the beam possesses. For a Besselmode of a given power, this has power evenly distributedand equally shared between its constituent rings [7, 14]. Anincreased number of rings implies a longer “diffraction-free” propagation distance; we are thus trading this featureagainst an absolute power in the central maximum of azeroth order Bessel beam.One may also consider the “non-diffracting” Mathieu

beams [15]: these represent a set of closed-form expres-sions that are solutions of the Helmholtz equation and maybe considered as elliptical generalisations of the more wellknown Bessel beams. Mathieu beams are described by the“ellipticity” or “separation” parameter �, and the integer�, which denotes the order of the mode. Analogously totheir Bessel counterparts these beams may possess orbitalangular momentum due to an azimuthal component of thePoynting vector for � � � [16]. Fig. 2 depicts the trans-verse cross sections of a third order (� � �) Mathieu modeclearly showing the phase profile. Experiments in opticaltrapping (see later) exploit this OAM to induce controlledparticle rotation.Another interesting class of straight “non-diffracting”

beam are parabolic beams. These beams are derived fromHelmholtz equations in the same way as the Mathieu beams.For the parabolic beams the separation of variables is donein a parabolic cylindric coordinates [17,18] as opposed toelliptic cylindrical coordinates.

www.lpr-journal.org © 2010 by WILEY-VCH Verlag GmbH & Co.KGaA, Weinheim



journal.org) Transversal cross-sections of

a third order Mathieu beam. (a) The gray-

scale represents the intensity of the beam.

(b) The colour hue represents the phase of

the field (as shown in the legend of Fig. 1).

The black contours and points correspond

to positions where the field intensity is

zero and the phase is not defined.


journal.org) (a) Transversal and (b) lon-

gitudinal cross-sections of an Airy beam.

The gray-scale corresponds to the inten-

sity of the Airy beam and the thin red lines

indicate the positions of the cross sections.

2.2. Accelerating “non-diffracting” beams

Accelerating “non-diffracting” beams refer to light beamsthat maintain their shape during propagation but do notpropagate in a straight line. The simplest curve on whichthese beams can travel is a parabolic path. One way todetermine the shape and properties of these beams is tostudy the paraxial Helmholtz equations which correspondsto equation (1) where only slowly varying solutions thatpropagate in a single direction are considered. Again takingz as the direction of propagation we have

��

��

��

��

��

�� (3)

where � is the magnitude of the wavevector as definedabove and �� the slowly varying envelope of thecarrier wave �� . Taking the �� plane asthe plane in which the parabolic path lies we can again writethe most general “non-diffracting” solution in an integralform as:

��

��

��

��

��

��

��

�

��

�

��

�� (4)

where �� is the Airy function, �� a characteristic lengthscale giving the opening of the parabola and the function

�� can be any complex function. Choosing this functionto be a Dirac distribution �� gives the standardAiry beam while other choices lead to more complicatedbeams: all of these are “non-diffracting” while moving on aparabolic path. These beams can be understood in the sameway as the straight propagating “non-diffracting” beamswhere the Airy beam replaces the plane waves. Indeed, theintegral in Eq. (4) corresponds to a superposition of Airybeams that are laterally displaced by ��

�� and

where �� gives their relative phase and ampli-

tudes. We observe that this phase term includes a modu-lation in the direction perpendicular to the parabolic pathplane. This is equivalent to the transversal wavevector nec-essary for straight propagating “non-diffracting” beams andcorresponds to a tilt in the �-direction of the Airy beamsuch that the displaced Airy beam has the same parabolicopening without incurring any additional phase change inthe direction of propagation. This is a necessary conditionfor the Airy beams, under the integral, to remain lockedin relative phase as they propagate maintaining thus theintensity profile along the parabolic path. Fig. 3 shows thetransversal and longitudinal cross sections of an Airy beamthat illustrates the beam’s parabolic trajectory.

The origin of the Airy beam stems from the work ofBerry and Balazs [13]. They described a wavepacket so-lution to the free space Schrodinger equation that is also“non-diffracting” and in addition exhibits a transverse ac-



celeration and thus avoided the dispersion seen with othersolutions: the main features of the wavepacket follow aparabolic trajectory in free space. Dispersion would seemto be an inherent feature of all solutions and Ehrenfest’stheorem (analogous to Newton’s second law) also forbidsany acceleration – so how does such acceleration arise?Well the issue of acceleration is not strictly applicable hereas the Airy function describing the beam is not square inte-grable and the exact centre of gravity cannot be defined. Inthe case of the finite-energy Airy beam, where the centreof gravity can be defined, we observe that whilst the finiteenergy Airy beam accelerates sideways, the centre of grav-ity maintains its rectilinear motion. This is only possiblebecause the finite energy Airy beam slowly loses its shapeand the fringe contrast decreases as the beam propagateson its curved trajectory.Interestingly “non-diffracting” beams can also be con-

structed that propagate along different paths to a parabolicone. The simplest such a case corresponds to the imagingof the Airy beam through a slightly misaligned telescopewhere the beam performs a hyperbolic motion [19]. Duringthis hyperbolic propagation, the amplitude of the Airy beamchanges while its intensity profile remains constant.

2.3. Self-healing properties and infinite energy

The self-healing property for all forms of “non-diffracting”beams refers to their reconstruction in beam profile afterbeing scattered by a finite sized object [20, 21]. In practice,this means that even though an object obstructs a “non-diffracting” beam and disturbs its field locally, some dis-tance after the object, the beam recovers its original shape.This distance can be determined via simple geometric argu-ments. This distance is termed the healing distance and ispresent in both straight and accelerating “non-diffracting”beams [22, 23]. Intuitively, this can be understood whenconsidering the shadow in the wake of the finite sized ob-ject. This shadow is equivalent to a destructively interferingbeam originating at the object. But this shadow beam is ingeneral a diffracting beam and as such after its diffractionlength we remain only with the incident “non-diffracting”beam. Fig. 4 shows this self-healing effect in action for anAiry beam.

Analogously with the concept of perfect plane waves,perfect “non-diffracting” beams are not physically possibleas such beams would have an infinite amount of energy andwould need their simultaneous generation over an infinitetransversal plane. In practice, it is possible to create a finiteenergy beam that exhibits behaviour akin to the idealisedversion but only over a limited propagation distance. Thisdistance depends on their transversal size and beam param-eters [24]. We now progress to explore the three main formsof propagation invariant beams in more detail.

2.4. Vectorial form

Vector “non-diffracting” beams corresponds to the sim-plest generalisation of the scalar “non-diffracting” beamsdescribed above. These vector beams represent vectorialsolutions of Maxwell’s equations fully describing the elec-tric � and magnetic� fields associated with the beam. Themost straight forward way to introduce these vector fieldsusing their scalar counterpart � is through the use of thecylindrical Debye potential [25]. Within this representationwe have:

� � �� (5)

� ��

�� (6)

where �� represents the unit vector in the direction of prop-agation, � the vector product and �� the vacuum perme-ability. Here we chose SI units and the scalar field � hasthe units of the scalar electric potential. Using the samescalar “non-diffracting” field it is possible to construct analternative vectorial solution through the use of the du-ality transformation. This transformation corresponds toexchanging the electric and magnetic fields while changingthe sign of one.

3. Generating “non-diffracting” light fields

3.1. Bessel and Mathieu beam generation

From Fourier optics theory one finds that an annulus (in-finitely thin) is the Fourier conjugate for a perfect ideal

Figure 4 (online color at: www.lpr-journal.org) Self-

healing properties of the Airy beam. (a) Ray trac-

ing picture where black lines represent rays that are

blocked by the obstacle (green horizontal line) and the

red coloured lines to rays that continue to propagated

and reconstruct the Airy beam after some distance.

(b) Electromagnetic field representation where the

gray scale represents the intensity profile of the recon-

structing Airy beam after an obstacle (green line) is

placed in front of the first lobe.




journal.org) (a) Schematic representation of

the Bessel beam generation using the first

order diffraction of a spatial light modula-

tor (SLM) and a lens. The amplitude mask

corresponds to a ring and its Fourier trans-

form in the focal plane of the lens gives the

finite energy Bessel beam. (b) Schematic rep-

resentation of the Bessel beam generation

using an axicon. The blue lines represent the

phase front of the beam and the curve on the

right to the intensity profile of the generated

Bessel beam. (c) The inset shows the axial

intensity variation (normalised) of the finite

Bessel beam generated by the SLM and axi-

con method.

Bessel beam. This leads to the most intuitive way of cre-ating a Bessel beam, although this is probably the leastefficient. An annular aperture may be positioned in theback focal plane of a lens [4,8] creating a Bessel mode [26].The on-axis intensity here can suffer from rapid oscilla-tory behaviour. The zeroth order mode appears as a centralmaximum with a set of concentric rings that have equalpower. Essentially the more rings one has, the further thismode propagates whilst retaining its non-spreading charac-terisitics. Fig. 5 shows the generation of a Bessel mode withthe annular aperture in this case generated upon a spatiallight modulator (SLM). This is now a popular device con-sisting of an array of optically or electrically addressableliquid crystal elements. Notably Durnin and colleagues [7]confirmed that whilst such a distribution of power acrossthe beam profile might seem detrimental in fact Besseland Gaussian modes transmitted power equally well over agiven distance. A Fabry-Perot cavity in combination withan annular slit may also be used for Bessel mode gener-ation [27, 28]. This has a somewhat smoother intensityprofile than the annulus and lens combination describedabove. For the production of higher-order Bessel beamsit is possible to use interferometric techniques [29] whichdecompose the Bessel beam into an odd and even part anduses a Mach-Zehnder interferometer to recombine these.

In practical terms, Bessel light modes may be efficientlygenerated by use of a glass lens termed an axicon [28,30].The axicon is a conically shaped transparent optical elementthat readily imposes a phase shift �� ontoan incident Gaussian light beam where � is the refractiveindex of the axicon material, � the internal angle of theaxicon, � and � are the cylindrical coordinates on the frontface of the axicon and � the wavevector. A key feature ofthe Bessel modes is that all the waves are inclined at thesame angle to the optical axis: the beam is a set of wavespropagating on a cone that all interfere to yield the Besselmode profile. Careful illumination of such a device is re-

quired to avoid any aberrations, notably astigmatism, thatcan be detrimental to the output mode. Normal incidenceillumination of axicon lenses is key to avoid aberrationssuch as astigmatism, which result in non-circularly sym-metric patterns [31–33]. If the axicon is illuminated witha converging or diverging beam, one can also change thebaseline of the beam profile and obtain an offset or tiltacross the beam profile, resulting in a type of biased opticalpotential energy landscape: such a washboard-like opticalpotential can be used for optical micromanipulation as weshall see later [34]. Illuminating the axicon with a Laguerre-Gaussian beam having an azimuthal index �, one can alsogenerate higher-order Bessel beam [35] of order �. Recentwork has also explored how to remove any issues related toimperfections or the rounding of the axicon tip that in itselfcan be a source of spherical waves [36].Microfabricated axicons have also been generated

which is important for applications in microfluidics andoptical trapping [37]. In this domain, we have seen thegeneration of variable axicon elements in microfluidic envi-ronments enabling the generation of reconfigurable Besselbeams [38].The holographic generation of all forms of “non-

diffracting” modes is also now established as a powerfuland efficient method. Indeed, using this method even asuperposition of multiple “non-diffracting” modes can begenerated, each having any lateral or axial position [39].Such holograms can be in the form of static glass elementsthat are appropriately etched [9]; or fully reconfigurablewhen using SLMs.

Generating Mathieu beams is very similar to generatingBessel beams. Indeed, the spatial spectral decomposition ofboth beam shapes gives an infinitely thin circle (see Eq. 2)except for the amplitude and phase along this circle. It isthus possible to create the Mathieu beam by illuminating athin annular aperture with an elongated Gaussian beam andpropagating the apertured beam through a Fourier lens [40].



Higher-order Mathieu beams can also be generated holo-graphically [41]. These higher order Mathieu beams carryorbital angular momentum, just like their higher orderBessel beam counterparts [42]. Indeed, Mathieu beamscan be seen as a superposition of Bessel beams [41].

3.2. Airy beam generation

Following the work of Berry and Balazs [13] lookingat dispersion free solutions to the Schrodinger equation,thirty years passed before Airy beams were realised in thelaboratory. For generating Airy beams in the laboratory,Siviloglou and colleagues [24] cleverly exploited the corre-spondence between the quantum mechanical Schrodingerequation and the paraxial wave equation. Siviloglou et al.used an SLM to impose the cubic phase profile upon aGaussian beam. Fig. 6 shows the phase profile used. TheAiry beam is the Fourier transform of this function andthis can be implemented experimentally using a lens. Theresulting beam profile fits well to that of a finite energyAiry beam. The Poynting vector and angular momentum ofthe Airy beam was explored by Sztul and Alfano [43]. Theyshowed that the Poynting vector follows the tangent line ofthe direction of beam propagation. Further considerationsshowed that whilst the Airy beam has zero total angular mo-mentum, there is angular momentum associated with boththe main intensity peak and the ”tail” of the Airy profile.

Figure 6 Cubic phase mask used on the Spatial Light Modulator(SLM) to generate the Airy beam in (Fig. 3). The gray scale gives

the phase profile on the SLM where white corresponds to a �

and black to a �� phase shift. The Airy beam is generated by

diffracting the incident light on the SLM and optically Fourier

transforming the field by using a lens.

It is interesting to remark that the distinguishing cubicphase term that is applied here was already well known forapplication in imaging optics, where researchers are look-ing to improve depth of focus and imaging characteristics.It was observed that by using the concept of “wavefrontencoding”, image quality may be traded against depth offield. In such studies, a cubic phase mask is used in theimaging system to render the point-spread function insensi-tive to misfocus i.e. regardless of its depth a source pointis observed having the same spread. The observed imagecontains all the depth information of the sample with equalblur and an object viewed through such a mask was seento be essentially “out of focus” over a wide range. Theextended depth of field can be achieved by digitally pro-cessing the observed intermediary image to retrieve a sharpimage of the sample at different depths. This feature showsagain some interesting links between “focus-free” opticsand “diffraction-free” propagation [44].

4. Experiments with “non-diffracting”light modes

We have described a range of “non-diffracting” light modesand their generation. We now progress to the applicationsof “non-diffracting” light modes. As we shall see they havealready made an impact in a diverse range of topics andcontinue to generate major interest.

4.1. Nonlinear optics

It is worth remarking here that we concentrate upon freespace light fields that require no additional nonlinearity tosustain their “non-diffracting” or propagation invariant be-haviour: these modes we describe are propagation invariantin free space and all linear media. Naturally in parallel withthe topics presented here there is the area of optical solitonswhere an inherent nonlinearity in the propagating mediumis necessary to sustain a diffraction-free behaviour. Thereare two main kinds of such solitons. Firstly spatial solitonswhere the nonlinear interaction in the medium can coun-teract the effects of diffraction. The incident optical fieldmay alter the refractive index whilst propagating creatingsomething akin to a graded index optical fiber. If the fieldis itself a propagating mode of the guide it has generated,then it will sustain itself along the length of the guide in adiffraction-free manner. Temporal solitons also exist wherefor fields that are already defined are sustained as dispersionis balanced through nonlinear interactions. Key reviews inthis interesting topic already exist and we refer the inter-ested reader to these [45, 46] as a detailed discussion ofsuch features is outside the scope of the present work.Nonlinear optics is an intriguing and important area

particularly for optical field generation at difficult to accesswavelengths and where non-diffractive beams may poten-tially provide benefits. Bessel light modes have featured in a



variety of nonlinear wave-mixing studies: examples includethe enhancement of photorefractive two-wave mixing, thecreation of Bessel beams within the coherent anti-StokesRaman scattering process, stimulated Raman scattering,higher order stimulated Brillouin scattering (SBS) and thirdharmonic generation [47–50]. The most basic non-linearprocess to consider is that of second harmonic generation(SHG) in a single pass geometry. This was studied usingBessel beams by Wulle and Herminghaus in 1993 [51]. Thenonlinear conversion efficiency, which is equivalent to thephase-matching efficiency, was observed to be dependentupon the Bessel mode’s longitudinal wavevector. This pa-rameter could be tuned by appropriately focusing the beam.Bessel beams act as light beams with a tuneable wavelengthin such experiments. Wulle and Herminghaus put forwardthe notion that Bessel beams could be used in conditionswhere typically phase matching was difficult (eg wherenormal temperature and angle phase matching conditionsdo not apply).For the purposes of our discussion of nonlinear fre-

quency conversion we may consider an ideal plane waveand ignore depletion. SHG efficiency in this instance is pro-portional to the square of the incident beam intensity andto the length of the nonlinear crystal. For a tightly focusedGaussian beam the SHG efficiency is naturally enhanced.However, if the focus is too tight, the Rayleigh range of thebeam divergence is short and this leads to a decrease in theconversion efficiency away from the beam focus position.The Boyd-Kleinman condition determines an “optimal” bal-ance between these two factors. In this respect one can seethat a “non-diffracting” beam appears to offer an interest-ing alternative within SHG. To this end, the conversionefficiency of both Gaussian and Bessel beams in bulk crys-tals with similar crystal interaction lengths was recorded byShinozaki et al. [52]. Their calculations appeared to showthat the Bessel beam was 48% more efficient than the cor-responding Gaussian beam. However, the simplified modelused for this study did not take into account the intensityvariation along the Bessel beam axis of an experimentallyrealised beam nor the full cross section of the beam includ-ing all of the rings. Subsequent studies by Arlt et al. [53]took these important issues into account. They performedan experimental comparison between Bessel and Gaussianbeam second harmonic generation. A holographically gen-erated zeroth order Bessel beam and an Lithium Triborate(LBO) crystal for the second harmonic generation wereused. Boyd and Kleinman had examined the optimal Gaus-sian beam parameters for second harmonic generation andArlt et al. found that the Bessel mode conversion efficiencywas in fact less than for a Boyd-Kleinman focused Gaussianbeam [53].Indeed their studies confirmed that Bessel beam ef-

ficiency can never exceed the conversion efficiency of aBoyd-Kleinman focused Gaussian of similar power- thereason for this is due to the fact that the power of the Besselbeam is distributed equally among its rings. Magni’s the-oretical study determined [54] that the optimal beam forsecond harmonic generation was in fact a beam made of

a combination of Laguerre-Gaussian beams. In fact theoptimum produced was a mere 2% increase in efficiencyover an optimised Gaussian beam. Further second harmonicexperiments explored non-colinear phase matching withBessel modes in a peridoically poled crystal [55]. Jarutiset al. [56] studied second harmonic generation with higherorder Bessel beams where the Bessel mode was generatedby illuminating an axicon with the appropriate Laguerre-Gaussian mode [10].Optical parametric oscillators (OPO) are powerful non-

linear systems that convert the pump field into longer wave-length signal and idler fields. They are often used for wave-length generation over large bandwidths. Belyi and col-leagues [57] explored parametric generation of light bypumping with Bessel light beams. They showed an advan-tage of Bessel beam pumping over Gaussian light beams. Atheoretical study of azimuth-matched nonlinear interactionof Bessel light beams was also presented which showed apotential increase in the energy conversion from an inputBessel beam compared to that of a Gaussian beam. In theirexperiment, Piskarskas et al. demonstrated the first para-metric oscillator that was pumped using a Bessel beam [58].The nonlinear crystal was KTP and it produced a character-istic output beam due to the non co-linear phase matchingthat was in the form of an annulus with a central brightregion.Two related studies by Binks and King used a Bessel

mode to pump a periodically poled material based OPO togenerate a Gaussian signal beam. These studies concludedthat the Bessel mode was not superior to the use of Gaussianmode [59, 60].Recent experiments show also the usefulness of the

Airy beam in the context of optical nonlinear interactions.Indeed, the Airy beam can be generated using three-wavemixing processes in asymmetric nonlinear photonic crys-tals [61]. Further, the propagation of intense femtosecondAiry beams in air leads to the generation of curved plasmachannels [62]. This enables the study of the nonlinear linkbetween the filamentation process and the filamentationgenerating beam.

4.2. Microscopy, coherence and imaging

In standard two-photon microscopy the microscope objec-tive and its numerical aperture play the leading role in thedetermination of the lateral resolution and the depth ofimaging obtained. As has already been alluded to, in con-ventional microscopy we may increase the depth of fieldbut this is naturally traded against a lowering of the lateralresolution of the system. Using a Bessel beam, two photonexcitation may be demonstrated along a line due to theextended central maximum. The experiments of Dholakiaet al. (2004) showed the Bessel beam excitation of fluo-rescein within a cuvette [63]. Here we retain good lateralresolution but we gain in depth. Separately, Dufour andcolleagues used the axicon lens to increase the depth offield in two-photon fluorescence microscopy. This study



Figure 7 Gaussian vs. Bessel beam imaging. (a) Conventional confocal or multiphoton microscopy extractsinformation from a sample only at the focus of a Gaussian beam, and excludes out of focus information by

virtue of a pinhole (in confocal microscopy) or by virtue of the fact that the two photon effect only occurs at the

high fluences at focus (in multiphoton microscopy). (b) By raster scanning in multiple Z-planes, this leads to a

series of 2D stacks which may be combined using software into an average or maximum projection if desired.

(c) By a single raster scan, the Bessel beam extracts information across an extended axial range, (d) generating

a similar type of image projection without the need for software or multiple raster scans. (e) 15 μm fluorescent

beads embedded in a 1mm thick agar matrix imaged using a Bessel beam. See Fig. 8 for more details. Images

reprinted with permission (Optical Society of America) from Dufour et al. (2006) [64].

used 100 femtosecond pulses at 800 nm and average powerof 300mW. They used a 1mm sample consisting of agarwithin which 15 micron diameter microspheres were dis-persed. High resolution projection images were obtainedwith a 2D scan with a depth of field � 1mm whilst main-taining the lateral resolution of� � microns. Fig. 7 shows acomparison of Gaussian versus Bessel imaging and showsthe imaging at depth of fluorescent beads. Fig. 8 showsa comparison of continuous wave Gaussian, mutiphotonGaussian, and multiphoton Bessel beam excitation clearlyshowing how we may now achieve two-photon excitationalong a line using this “non-diffracting” mode.

It is worth remarking at this stage on the role of coher-ence for Bessel light modes which can play a role in itsimaging and propagation characteristics. Coherence plays acentral role in optics and is key to the effects of interferenceand speckle that may be observed in an experiment. If weconsider the Bessel light mode with particular referenceto its generation via an axicon we see that if the incidentlight field is spatially coherent then temporal coherence issomewhat less of an issue to create the “non-diffracting”rod-like centre as all the light fields traverse the same dis-tance (equivalent optical path length). In this manner Fis-cher et al. created white light “non-diffracting” beams [66]with a variety of sources. They showed that one could gen-erate white light Bessel modes from either a laser diodeoperating below threshold, a supercontinuum laser, or evena halogen bulb with a suitable pinhole (to improve its spa-tial coherence properties). Further work by Leach showedhow a spatial light modulator could remove the dispersionof colours in the rings and create white light Bessel modeswith a defined radial wavevector [67].

Microscopy is not the only form of imaging to benefitfrom axicon lenses and the Bessel light mode. Optical co-herence tomography (OCT) is an interferometric techniquethat has gained major prominence in the last decade forgenerating data that is akin to an optical biopsy. It uses lowcoherent light sources to perform optical sectioning usingthe concept of selective interference in a given plane in acell or tissue sample. In particular it has potential benefits inophthalmology or imaging within the body and is compati-ble with endoscopy. This is an area where optical sculptingto enhance the depth and the use Bessel-type modes is ofinterest. Ding et al. [68] used an axicon lens to obtain anenhanced focussing range (6mm) with 10 μm lateral res-olution. Lee and Roland (2008) [69] extended this studyand showed the use of a micro-optic axicon in the imag-ing of an African frog (Xenopus laevis) tadpole (Fig. 9). Inthis study, an 8 μm lateral resolution was obtained with an4mm depth of focus. Tan et al. used a fibre-based micro-fabricated axicon for common path OCT [70]. This offersa major advantage in endoscopy based imaging systems,where holding the imaging lens a defined distance from thesample of interest is technically challenging.

It is appropriate to comment here on the generation ofBessel “non-diffracting” modes using pulsed laser beams.Indeed here the spatial and temporal characteristics of themode play a role. For pulsed “broadband” light we obtain avariety of beams such as focus wave modes, Bessel-Gausspulses and notably the so-termed X waves. This later modegets its name from what this mode appears as when a snap-shot is taken: the pulse is akin to the letter X that is rotatedaround the horizontal axis. The intensity distribution in thisform is due to the interference of wavepackets of only a



Figure 8 (online color at: www.lpr-journal.org) Comparison

of single photon Gaussian (top), multi-photon Bessel (middle),

and multi-photon Gaussian (bottom) beams. The Bessel beam

has a central maximum with a high degree of lateral confinement

that propagates over a long axial distance. It therefore relaxes

focussing requirements in imaging (Fig. 7), optical coherence

tomography (Fig. 9), and optical injection (Fig. 16). The question

naturally arises as to why only the central spot of the Bessel beam

(“Ring 0”), but the not outer rings (“Ring 1, 2, . . . ”), is visible in

this image. The table demonstrates the relative peak intensity and

peak second harmonic generation (i.e. the probability of multi-

photon fluorescence) as a function of ring number. Note how

rapidly peak second harmonic generation drops off with increasing

ring numbers. Hence, only the central maximum is visible. Image

reprinted with permission from Brown et al. (2008) [65].

few optical cycles in duration [71]. Pulsed Bessel modesof longer duration have also been instrumental in the non-linear excitation as already mentioned to visualise beampropagation [63]. Ultrashort pulse Bessel modes have alsobeen considered in studies by Grunwald and colleagues: in2000 they explored the generation of arrays of femtosecondBessel beams using thin film microaxicon arrays [72]. Inthis study they recorded time integrated intensity distribu-tions for Bessel modes of 12.5 fs and 26 fs duration andcompared this to the continuous wave case. Here, space-time coupling has an effect on the transverse profile ob-served for the Bessel mode. Bessel modes are not the onlyones that have been considered in the ultrashort domain.

Figure 9 Gaussian vs. Bessel beam optical coherence tomogra-phy. (a) Spectral-domain OCT images of an African frog (Xenopuslaevis) tadpole at various axial distances from the sample. TheBessel beam images have an 8 μm invariant lateral resolution over

a distance of 4mm, whereas the Gaussian images are out of fo-

cus after only 1mm. This could allow an endoscope based OCT

probe that does not have strict focussing requirements. (b) A pro-

totype conical tipped optical fibre developed for endoscopic OCT

produces images with � 16 μm lateral resolution over a distance

of 0.6mm. Although this conical tipped fibre does not, strictly

speaking, produce a Bessel beam, the concept is similar. Images

(a) and (b) reprinted with permission (Optical Society of Amer-

ica) from Lee and Rolland (2008) [69] and Tan et al. (2009) [70]

respectively.

Saari explored via numerical simulation three forms ofpulsed Airy modes. The three types showed differing de-pendencies upon frequency parameters and showed varyingshapes and propagation features. A lateral acceleration wasseen in the highest-intensity lobes of the pulses [73].

4.3. Optical micromanipulation

We now progress to review the experiments that have used“non-diffracting” light modes within optical micromanipu-lation that benefit from the key attributes of these modes.Particle manipulation with light is a diverse topic that mayimpact upon a large number of areas: these include sin-gle molecule biophysics, non-equilibrium studies, colloidaldynamics and interactions and fundamental aspects of themomentum of light amongst others. The manner in whichoptical forces may arise on microscopic and nanoscopicparticles is dealt with in more detail elsewhere [74, 75], butin summary can be thought of in the Mie regime as aris-ing from the reflection and refraction of light and invokingNewton’s second law and applying this to momentum con-servation. At the nanoscopic scale it may be thought of asdue to the trapped object acting as a dipole and aligning it-self with the field maximum to minimise its free energy [76].Sculpted light fields that offer an immunity to diffraction



Figure 10 A typical Bessel beam setup

for optical trapping. Prior to entering

the axicon, the beam is expanded using

�� and ��. The amount of expansion de-

termines the number of rings the Bessel

beam contains; a larger incoming beam

results in a greater number of rings, a

longer propagation distance, but also a

decreased amount of power available

in the central core. In fact, to obtain

an infinitely “non-diffracting” Bessel

beam, the axicon would have to be in-

finitely large. After exiting the axicon,

�� and �� serve to reduce the beam

even further to the micron size regime

necessary for micromanipulation, imag-

ing, optical transfection, etc. Reprinted

from Optics Communications 197(4–6), J. Arlt, V. Garces-Chavez, W. Sib-

bett, and K. Dholakia, Optical micro-

manipulation using a Bessel light beam,

239–245, (2001) [77], with permission

from Elsevier.

have been of major interest to this field since 2001 [77] asthey potentially increase interaction distances for opticalforces and permit transport and delivery of objects overlonger ranges [63,78,79]. Much of the thinking here sharessynergy with that in the field of atom optics.A Bessel beam was used by Arlt et al. [80] to perform

two dimensional optical trapping. The experimental setupmay be seen in Fig. 10. The output laser field was expandedwith a telescope and then passed through an axicon. Therefraction through this element created a finite generatedBessel beam which was then downsized with another tele-scope and thus projected into a thin (� 100 μm deep) sam-ple chamber to interact with dispersed colloidal solutionsin order to perform micromanipulation experiments. TheBessel mode from the axicon has an axial intensity varia-tion but this is relatively slow over the propagation length:no strong gradient is created in the axial direction so thetrapping is restricted to two dimensions in the lateral plane.The authors proved the feasibility of using a Bessel modeto trap microspheres and a glass rod.

Subsequently, the Bessel mode’s self healing propertieswere exploited in trapping by Garces-Chavez et al. [81].They employed multiple thin (� 100 μm deep) samplechambers in their experiments, stacked on top of one an-other and separated by cover slips. The aim was to showthat a trapped object in one chamber might act as an ob-struction but that the conical wavevectors and reformationability meant the beam would self-heal in time to trap ob-jects in the next chamber. Fig. 11a illustrates one of themain experiments. The two sample chambers (labeled Iand II) used were 3mm apart, and each 100 μm deep. Thetrapped object which constituted the obstacle in this casewas a trapped hollow microsphere in the beam centre andthis deformed the incident trapping light field (chamber 1,

Figure 11 The self-reconstruction properties of a Bessel beamused in optical trapping. In the bottom chamber (I), (a) a single

hollow 5 μm sphere is trapped between the central maximum

and the first ring of the Bessel beam. (b) Immediately above the

beam, it is still visibly distorted but (c) a short distance later the

beam has reformed without distortion. Chamber II is 3mm above

chamber I. (d) Three solid 5 μm beads are tweezed in a stack

by the same beam, (e,f) which subsequently reforms once again.

Reprinted by permission from Macmillan Publishers Ltd: Nature

(Garces-Chavez et al. [81]), copyright (2002).

Fig. 11). The distortion and subsequent self-healing of theBessel mode were clearly recorded in the region betweenthe two chambers. Thus the beam was able to trap in bothchambers simultaneously. Further variations in this workshowed a low refractive index particle held between the



central maximum and first bright ring, and in a separatechamber a birefringent calcite particle set into rotation inthe Bessel beam centre – the beam here was circularly po-larised to impart spin angular momentum to the calciteparticle [82]. Finally, the authors used a higher particledensity in the chamber: now particles were trapped in notonly the central maximum but also in some of the outerconcentric rings of the zeroth order Bessel mode. Thoughrather obscured, the beam was seen to self heal throughsuch sample and trap more dispersed colloids in anotherchamber further downstream, again in the centre and inthe beam’s concentric ring patterns. Another study soughtto directly visualise this self-healing effect in a samplechamber with a fluorescein stained liquid (a dyed samplemedium). Two photon excitation and fluorescence of theliquid was used to actually map out the light distribution insitu. The Bessel beam’s central maximum pulls particlesinwards due to the gradient force and generates long rangeguiding of microparticles. The St Andrews group recordedthe two-photon fluorescence signal from this dyed samplemedium [63]. Particles 5 μm in diameter were opticallyconfined and propelled with a femtosecond Bessel beam.Experimentally this constituted the direct observation of thebeam propagation as it reformed around a guided particle.In this case, beam reformation occurred � 90 μm in frontof the trapped fluorescent sphere. This provided a powerfulvisualisation technique for the optical field around a givenmicroparticle and indeed has been extended to explore theprocess of optical binding [83].The studies described here so far provide guides for

trapped objects: it would be interesting to extend this toactual three dimensional confinement and thus the trappingof particles along a Bessel mode. In 2005 Cizmar et al. [84]explored the use of two counter-propagating zeroth orderBessel beam modes. This created a very long (millime-ter) interference pattern along the propagation axis and theantinodes in the pattern created three dimensional trappingsites. The interference pattern presented a one dimensionaloptical potential energy landscape and thus acted as an op-tical “conveyor belt” (Fig. 12). Particles could be deliveredover a distance of hundreds of micrometers (Fig. 12). Thedelivery aspect of the process was achieved by shiftingthe relative phase between the two Bessel beams (eg bymoving a mirror). In turn this resulted in a motion of theinterference pattern in space which thus moved trapped col-loidal particles to specified regions. The authors looked attheoretical and experimental aspects of the problem. Theirinvestigation showed that certain sizes of polystyrene parti-cles jumped between neighbouring axial traps with a higherprobability than other sizes. How good are such traps? Sub-sequent experimental data (by video tracking) of the trapstiffness at the antinodes [85] showed that the ratios of lon-gitudinal and lateral optical trap stiffnesses in such standingwave configurations were around one order of magnitudehigher than those one might expect in the classical singlebeam optical tweezers. Thus interferometric optical trapsmay have possible new applications as novel calibratedforce transducers within biological systems. Taking this

Figure 12 An optical microconveyor belt, comprised of twocounter-propagating zeroth order Bessel beams, moving two

410 nm diameter polystyrene spheres over a distance of 250 μm.

Reprinted with permission from Cizmar et al. [84]. Copyright

2005, American Institute of Physics.

system but in fact avoiding interference shows a new basisfor understanding optical binding: binding refers to inter-particle interactions due to light scattering that in turn dom-inates the final equilibria positions for neighbouring parti-cles. Karasek and colleagues created long bound particlechains using two counter-propagating Bessel modes [86].The extended propagation range created up to 200 μm longchains of organised microparticles. The experiment showedshort-range multistability within a single chain and long-range multistability between several chains.Using the Bessel beam standing waves created from

two independent counter-propagating Bessel beams, theywere able to confine polystyrene particles of radius 100nanometers, and arranged them into a one-dimensionalchain over a length of one millimeter [87].Alternatively, the use of co-propagating Bessel beams

may result in a periodic oscillations of the on- axial inten-



Figure 13 (a) The Mathieu beam confines particles to the bright region of the elipse. Transfer of orbital angular momentum onto 3 μmparticles by scattering forces (b) allows them to rotate in an elliptical orbital motion. (c) Their velocity as they travel around the ellipse is

nonlinear. Images reprinted with permission (Optical Society of America) from Lopez-Mariscal et al., 2006 [16].

sity maxima – the so-termed self-imaging effect. This canbe altered by the use of two Bessel beams of different con-ical angle (and thus wave-vector) that interfere with eachother [88–90]. As before careful variation of the path dif-ference of one of the Bessel beam moves the pattern: thusparticles can be selectively held and controllably moved.Numerical simulation by the researchers showed that in-creasing the number of interfering beams allows one to tunewhich particles are held. The experimental results showedtrapped particles with radii from 100 nanometers up to 300nanometers, but it is to be noted that it did require the assis-tance of fluid flow against the beam propagation direction.A domino-like effect of particle hopping from one trap tothe neighboring-occupied traps was also observed [89]. In-terfering Laguerre-Gaussian beams may be incident uponan axicon creating novel Bessel traps for particles [91].

We have so far concentrated on work where the centralmaximum of the beam has taken prominence and so farinferred that the outer concentric rings or features of any“non-diffracting” field are not very important. Many opticalmanipulation experiments show that all features of the trans-verse profile of any “non-diffracting” mode may be used.Studies by Tatarkova et al. [34] used a modified version ofthe whole Bessel profile: the profile itself resulted from a di-vergent Gaussian beam incident upon an axicon. In turn thiscreated an asymmetry in the potential wells created by anyone ring of the Bessel profile. This asymmetry permits hop-ping of particles in the radial direction that brings particlehopping towards the beam centre. Milne et al. [92] exploredthis in more depth: theory and experiment were comparedfor the motion of silica microspheres on the tilted Besseloptical landscape. Computational studies used two numer-ical models to understand the particle dynamics, namelythe Mie scattering and geometrical ray optics models. Bothgave similar results and predicted the existence of a distinctsize-dependence of the particle behaviour and equilibriapositions for particles within a Bessel beam.

High order Bessel beams have an azimuthal phase termand a vortex at beam centre. The azimuthal phase gives rise(as described earlier) to OAM akin to Laguerre-Gaussianmodes [11]. The “non-diffracting” nature of these Bessel

modes and their characteristic feature of a set of concen-tric rings however lends itself to complementary studies ofthis intriguing facet of optical momentum. Garces-Chavezet al. [93] used a higher order Bessel beam to trap particleswhich were then seen to be set into rotation by scatteringfrom the inclined phase fronts: the measured particle rota-tion rates were compared to the predictions of a ray opticstheoretical model and in particular a linear relationship be-tween trap laser power and rotation rate was seen. Lowindex particles are also of interest and the Bessel modewas used to set these into rotation but here the particle wastrapped in the dark region between a given pair of rings ofthe high order Bessel mode [82]. This constituted the firsttransfer of OAM to low index particles and indeed showedthat azimuthal intensity variations were not as influentialas in the case of the rotation of high index particles, whichnaturally being located in the bright regions would be moreperturbed by such variations. More detailed experimentsfollowed using a Bessel mode, including the study of the ro-tation of a birefringent particle around the concentric ringsof a circularly polarised higher order Bessel mode. This ele-gantly illustrated the intrinsic and extrinsic nature of OAMand the decoupling of spin and orbital forms of angularmomentum for the same particle. This laid a framework forusing birefringent particles to directly probe the optical an-gular momentum density in a light field. Further theoreticalstudies explored the dynamics and equilibrium positionsof particles in a Bessel beam in more detail [12, 42]. Highorder Mathieu beams [41], described earlier, have also beenrecently used in optical micromanipulation to explore theirOAM properties [16] showing non-uniform velocities ofparticle motion around the elliptic beam profile (Fig. 13).

Accelerating “non-diffracting” beams can also be usedto manipulate microparticles. Baumgartl et al. [22] haveshown that such a light field may be used for the task ofparticle clearing within a sample chamber: this exploits thefeature of transverse acceleration and optical radiation pres-sure to push particles in parabolic trajectories. The finitepropagation distance of the Airy beam means that parti-cles are lifted and transported laterally but then fall backdown under gravity: the beam propagation characteristics



Figure 14 (online color at: www.lpr-journal.org) Optical clear-

ing using Airy wavepackets. An Airy beam, represented by the

white overlay and whose deflection during propagation is rep-

resented by the white arrow, is projected onto a monolayer of

3 μm silica colloids. Over a time scale of minutes, the particles in

the upper right box (green) are pushed in both vertical (against

gravity, into the page) and transverse directions, resulting in the

upper right box emptying and the lower left (purple) box being

filled. This effect has been coined as the “optical snowblower.”

Reprinted by permission from Macmillan Publishers Ltd: Nature

Photonics (Baumgartl et al. [22]), copyright (2008).

can be engineered to transfer material from one region toanother [22]. In the study of Baumgartl et al. [22] the Airybeam profile was incident upon a sample chamber filledwith monodisperse particles of 5 microns or so in diameter.Trajectories of the particles were recorded. The Airy modedragged particles into the central maximum area and thenlevitated and pushed them due to the optical radiation pres-sure. The particles followed the parabolic trajectory of themain feature of the Airy beam: the finite beam smeared outand near the end of the propagation distance the particlesthen left the beam and returned to the base of the cham-ber, having followed a parabolic trajectory. This meant thatparticles were essentially transported from one quadrantof the chamber to another, thus “clearing” a segment ofthe chamber. See Fig. 14. Such “optically mediated par-ticle clearing” has recently been extended to redistributemammalian cells and colloids in specially designed squaremicrowells (100 μm � 100 μm area and 20 μm high) andcould signal a new form of microfluidic relocation of cellphenotypes or populations (for example) without the needfor microfluidic flow [94].

4.4. “Non-diffracting” beams for cell sorting

The targeted selection of certain cells from a populationis an important tool in biomedical science. In particular a

number of schemes have emerged in recent years includingmicrofluidic sorting of cells either with or without the useof fluorescent markers. Paterson et al. [95] explored cellsorting with red and white blood cells with no markersattached, a technique more generally term passive opti-cal sorting. The experiment placed cells in a concentricring pattern (optical potential energy landscape) createdby the zeroth order Bessel beam. This experiment demon-strated that red and white blood cells (lymphocytes anderythrocytes) could be spatially separated by the transversering structure of the Bessel beam. White cells migratedover the periodic ring structure and collected in the centralBessel core and were then subsequently guided upwardsby radiation pressure. On the other hand, red cells (erythro-cytes), depending upon the exact power used and Besselbeam characteristics, were trapped in the concentric ringswhere they re-oriented themselves (flipped) by ninety de-grees. The red blood cell behaviour was similar to theirbehaviour in more traditional optical trapping geometries.This difference in cell response was a result of the phys-ical shape difference between red and white blood cells:lymphocytes are near spherical and have a nucleus whereaserythrocytes appear as a bi-concave disk. The experimentwas recently extended to a variety of other cell lines. Theaim was to observe if such passive sorting could be ap-plied to stem cells and human cancer cells. However theintrinsic differences between given cell types was typicallynot sufficient on its own to initiate sorting. Such passivesorting relies on natural differences in size, shape or refrac-tive index for successful operation. Paterson and colleaguesthen used dielectric tagging of the cells to enhance the sort-ing process: this tagging involved using surface chemistryand antibody-antigen binding to allow certain beads to at-tach preferentially to the desired sub-population of cells.Specifically, CD2� T-lymphocytes were tagged with anti-CD2 silica spheres. These “tagged” cells responded morestrongly to the Bessel mode [96]. Their higher refractiveindex contrast meant they were influenced more strongly bythe light field. Fig. 15 demonstrates the sorting of untaggedmononuclear cells and sphere-tagged T-lymphocytes. Thesphere-tagged cells move towards the centre of the beam,and are subsequently guided up to the top of the samplechamber, many tens of microns away from the base of thesample chamber. The untagged mononuclear cells in theexperiment did not move significantly in response to theapplied optical field. This methodology is readily appliedto any two-cell populations possessing different surfaceantigen chemistry.

4.5. Atom optics

Manipulation by light is not restricted to the domain of cells,colloids or nanoparticles. In fact the very forces of lightact at the atomic and molecular scale where absorption andspontaneous emission – light scattering – is at the core ofthe area of laser cooling. Additionally the optical dipole orgradient force may also be present: this may be understood



Figure 15 Cell sorting using a Bessel beam. A mixed population of cells resting on the bottom of a sample chamber is exposed to aBessel beam. For one cell type, unlabelled mononuclear cells, nothing happens. For the other cell type, which are CD2� T-lymphocytes

tagged with anti-CD2 5.17 μm silica microspheres (arrows), cells are first pushed in the transverse direction towards the central maximum

of the Bessel beam (white arrow, a–e; black arrow, a–c). Upon entering the central maximum, the lymphocytes are guided vertically

(against gravity) (white arrow, e–f; black arrow, b, c, f) until they reach the top of the sample chamber, where they form a vertical

stack (f). In principle, any cell type tagged with an antibody laden sphere may be separated from another population in this manner.

Reprinted with permission from Paterson et al. [95]. Copyright 2005, American Institute of Physics.

as absorption followed by stimulated emission. Shapedlight fields in particular have been used to exert opticaldipole forces upon laser cooled atoms and are now beingapplied to Bose-Einstein condensates and ultracold Fermigases. “Non-diffracting” fields and their Fourier conjugateshave already been explored in these emergent areas.

Conical glass axicons have been used to create annularoptical potentials: this offers steep potential walls for ringtraps [97] and exploits the far field of the Bessel modewhich is an annulus as its Fourier conjugate. In this studyan 800 μm diameter annulus of width 35 μm was generated.The dark inner region had an intensity of only 0.1% of thepeak intensity. In this manner atoms were confined uponsurfaces with the light field “blue-detuned” from resonancethus repelling the atoms from regions of high field intensity.Combinations of axicons have generated hollow light fieldsas all optical guides that do not spread [98]: this offers analternative to the potential use of more standard Laguerre-Gaussian beams for such studies [99].

Numerical studies have shown details of the dipolepotential for a variety of high order Bessel and Laguerre-Gaussian light beams [100]. Other studies exploited theconcept of arrays of tightly confined traps in a standingwave geometry for single atom confinement and delivery:this in fact builds upon the work in the optical conveyor beltfor manipulation of nanoscopic and microscopic objects asdescribed earlier. By using a moving interferometric patterncreated by Bessel laser beam modes, horizontal transportof ultracold atoms over macroscopic distances of up to 20centimeters was achieved while ensuring uniform trappingconditions in each trap site. The stability of the interferencetraps enabled fast atom transport with velocities of up to6m/s [101].

Bessel light modes have been utilised also in the do-main of atomic physics for studies involving Bose-Einsteincondensates [100]. Axicon generated Bessel light beamswere considered by Arlt and colleagues for creating opticaldipole traps for cold atoms and as atomic waveguides forquantum gases. The use of zeroth-order Bessel beams canbe used to produce highly elongated dipole traps where, typ-ically, the ratio between the longitudinal and transverse trapfrequencies is nearly two orders of magnitude smaller thanthat which can be achieved with a Gaussian beam. Suchunusual systems may permit the study of one-dimensionaltrapped gases and the pursuit of a Tonks gas of impenetra-ble bosons. In the “blue-detuned” regime, first-order Besselbeams with an on-axis vortex present excellent hollow atomguides over centimeter distances. Such a Bessel mode trapoffers an interesting alternative to the more typical use ofmagnetic waveguides. Bessel traps provide an all-opticalsystem with good control of the aspect ratio of the trap.They further offer trapping without a material surface, asin magnetic waveguides, which can result in matter-wavedecoherence. Optical dipole traps lend themselves to trap-ping atoms in different magnetic sublevels which wouldnot be possible in other trap implementations using mag-netic fields.

4.6. Optoinjection of cells

The outer plasma membrane of mammalian cells is imper-meable to a range of hydrophilic substances. This is due tothe inner lamellae of the plasma membrane being highly hy-drophobic. While this is advantageous for the cell, allowingit to maintain its various homeostatic processes, it can be



Figure 16 (online color at: www.lpr-journal.org)

Schematic showing three mammalian cells being op-

tically transfected. (a) Multiphoton mediated trans-

fection will occur using a Gaussian beam correctly

focussed on the membrane, (b) but will not occur

using an unfocussed beam. (c) Bessel beam trans-

fection relaxes this focussing requirement. Note that

there is an intensity variation along the Bessel beam

axis, and that the Bessel beam rings are omitted for

clarity. (d) The transfection efficiency as a function

of axial position of a Gaussian and Bessel beam. Im-

age (d) reprinted with permission from Tsampoula

et al. [108]. Copyright 2007, American Institute of

Physics.

frustrating for the researcher attempting to study the livingcell. There are thousands of interesting membrane imperme-able fluorophores available on the market. However theiruse in live cell imaging and tracking studies is stronglydiminished because of the cell’s impermeability [102]. Atypical way to overcome this problem is to alter the chem-istry of the fluorophore. Smaller organic fluorophores canoften be chemically modified to generate an inactive butmembrane permeable product, upon which entering thecell, it undergoes esterase cleavage, activation, and fluo-rescence [102]. However, this is simply not an option formany larger biological constructs, for example plasmidDNA, mRNA, siRNA, and antibodies are notable examples.Whilst a variety of methods exist to transfect cells the useof laser technology has marked a new era of selectivity,sterility and applicability, and this method is still evolvingin many respects. In 1984, Tsukakoshi et al. showed forthe first time that a focussed light beam on the surface ofa cell can alter its permeability enough to allow plasmidDNA to enter and be expressed [103]. We note a number ofphrases have been associated with this procedure; notablythe term “photoporation” which strictly refers to the use oflight to generate a small re-sealable hole in the membrane.We prefer to use the terms “optoinjection”, or sometimes“optical transfection”. Both terms signify the creation ofthe hole and the injection of biological material. However,transfection refers specifically to the injection of nucleicacids into cells, that are subsequently translated, causing achange in the cell’s properties. Since 1984, optoinjectionhas been demonstrated with a variety of laser sources, celltypes, and impermeable species [104–108]. Emerging asa forerunner in the choice of laser source is the femtosec-

ond pulsed Ti:Sapphire, as it allows exact targeting withthe minimal intra- or inter-cellular collateral damage [65].In this technique, the small region in space encompassedby the focus of a Gaussian laser spot creates a low energyfree electron gas that photochemically reacts with the cellmembrane to produce a tiny, (� 1 μm�) zone of destructionon the cell surface. This photopore is transient; cells havedeveloped a healing mechanism for such small membranedisruptions and actively reseal the pore in a calcium me-diated process [109–112]. Producing such a tiny zone ofdestruction using a focussed Gaussian laser has one distinctdisadvantage. Positioning of the highly-focused laser on thecell membrane requires sub-micron accuracy in the verticalaxis due to the multiphoton process that creates the low den-sity plasma that causes the photopore [106]. Consequently,optoinjection efficiency drops off rapidly with out-of-focusdoses [108]. Efforts were therefore placed on the develop-ment of beams with an extended axial focus, obviating theneed for such precise positioning on the cell membrane.One can naturally see from this discussion that the use ofthe Bessel mode would be potentially advantageous.In the first study of its kind with “non-diffracting”

modes, Tsampoula et al. (2007) used an axicon gener-ated Bessel beam as a “rod-of-light” to optoinject DNAefficiently over an axial range of 100 μm [108]. Fig. 16demonstrates this concept schematically. A focussed Gaus-sian beam, correctly positioned on the membrane of a cell(Fig. 16a), results in a transient photopore being generatedon the membrane. An incorrectly (axially) positioned Gaus-sian beam (Fig. 16b) results in no photopore, as there isinsufficient fluence to generate enough electrons to pho-tochemically induce a photopore. In contrast, the Bessel



beam (Fig. 16c) contains a region of sufficiently high flu-ence over a long axial distance (� 100 μm), allowing opticaltransfection to occur regardless of axial position (Fig. 16d).Why is this important? Determining the correct axial posi-tion (i.e. determining if the cell membrane is “in focus”)is challenging under brightfield illumination, and greatlycomplicates any possibility to automate the procedure. Withcomputer vision software, it is fairly routine to determine“cell” vs “non-cell” regions on a brightfield image. One caneasily envision a system where the operator specifies a largeregion of interest over which he or she wishes optical injec-tion to occur, and allows the computer to scan, target, andphotoporate every cell within that region. Moreover, if theultimate goal is to develop a point-and-click system robustenough that it may be widely adopted within the biologicalcommunity, then a Bessel beam based system makes farmore sense. Such a system became a closer reality recentlywhen Cizmar et al. (2008) used an axicon in conjunctionwith an SLM to generate arrays of tunable Bessel beamson-demand [39]. The technique showed successful opticaltransfection. In this study the end-user no longer needed tospend time focussing the beam on the cell of interest; theysimply pointed-and-optoinjected the cell with the click of acomputer mouse.

5. Conclusion

We are approaching the fiftieth anniversary of the laser.Applications of lasers and in particular shaped laser fieldsin interdisciplinary science are expanding and making anever increasing impact as we learn more about the pos-sibilities for tailoring the propagation characteristics oflight. We have reviewed in this paper the emergent area ofoptical beam shaping or sculpting where we have movedaway from the traditional zero-order Gaussian field andhave used a variety of means to create optical fields withthe characteristics of “diffraction-free” propagation over alimited range.We have described and seen key examples of use in-

cluding the Bessel, Mathieu and Airy light fields. Theseinclude the ability to enhance the “toolkit” in optical micro-manipulation for the trapping and transport of biologicalmaterial and long range imaging. Further we have seenhow “non-diffracting” fields may enhance particle sortingand seen how they may be used for controlled ablationand nanosurgery. Light fields such as the Airy beam offernew prospects such as particle clearing and advantages inplasma physics. The field is in many respects evolving andseeing many advances into new areas: light is shaping theway we perform a variety of exciting interdisciplinary sci-ence.

Acknowledgements We thank the UK Engineering and PhysicalSciences Research Council for Funding and the EU Network of

Excellence PHOTONICS4LIFE. KD is a Royal Society-Wolfson

Merit Award Holder.

Michael Mazilu graduated from theUniversity Louis Pasteur, France witha diploma in Condensed Matter andMaterial Physics. He then moved tothe University of St. Andrews to ob-tain a Ph. D. in physics and worked inthe fields of semiconductor physicsand photonic crystal devices. Hejoined the optical trapping group in

January 2006 and is working on optical micromanipula-tion and Raman spectroscopy for biomedical purposes.

David James Stevenson obtained hisB. Sc. in Biochemistry and Molec-ular Biology at the University ofGeorgia, USA in 1999, his Ph. D.in Bioengineering at the Universityof Strathclyde in 2004, and he hasbeen a postdoctoral fellow at theUniversity of St Andrews since 2004.

His current research involves optoinjection, long termcell imaging, laser guided neuron growth, and lasercell micropatterning.

Frank Gunn-Moore is a Reader inNeuroscience in the School of Biol-ogy, University of St Andrews. He ob-tained his degrees from Edinburghand Cambridge. Following postdoc-toral research at the Universities ofBristol and Edinburgh, where he in-vestigated the biochemistry of neu-rotrophic factors and how the cells of

nervous system interact with each other, he initiated hisown research group at the University of StAndrews in2001. He currently has active research in Biophotonics,Alzheimer’s disease and how neurons grow and develop.

Kishan Dholakia is Professorof Physics at the University ofStAndrews Scotland and an hon-orary adjunct Professor at the Centrefor Optical Sciences at the Universityof Arizona, USA. He heads a large

(20+) group working in various aspects of photonicsincluding micromanipulation and biophotonics. He haspublished over 300 journal/conference papers and hisgroup won the European Optics Prize in 2003. He waselected to the position of Fellow of the Royal Societyof Edinburgh in 2007, and of the Optical Society ofAmerica in 2008.



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Light beats the spread: “non-diffracting” beams