Optical tweezers based on nonlinear focusing

Recently, some authors have proposed to add an optical Kerr effect (OKE) while focusing the Gaussian beam (GB) enlightening an optical trap. These authors conclude that the introduction of a nonlinear lensing (NL) is benefit to the optical trapping capacity. The proposed modelling was based on the Gaussian approximation (GA) which assimilates the Kerr lensing effect to a pure lensing effect free from any aberration. In this paper, we evaluate the longitudinal and radial figures of merit of the optical trap based on NL using a diffraction integral taking into account the aberration associated with the OKE. The conclusion is that the GA modelling underestimates (overestimates) the improving of the longitudinal (radial) trapping ability of the optical trap. In summary, what is gained in the longitudinal efficiency of the nonlinear trap is lost in the radial force which decreases, thereby reducing the possibility to keep trapped the particle.


Introduction
Optical tweezers have become an essential tool for manipulating small particles in a non-invasive manner, i.e., without any mechanical contact. The idea of optical tweezers has been implemented since the 1970s by the pioneer Arthur Ashkin [1,2]. This seminal work has been honoured with the Nobel prize in Physics in 2018. Throughout many years, a certain number of tweezers applications have been developed including both physics and biology [3,4]. The modelling of the interaction between a laser beam and a small particle depends on the ratio between the particle size and the light wavelength. In the following, we will limit ourselves to the case of a spherical dielectric particle of radius a smaller than the laser wavelength (a < < λ). In this case (Rayleigh regime), the particle is treated as a point electric dipole in interaction with the electromagnetic field associated with the enlightening laser beam. The theory of optical tweezers is largely documented in the literature [5][6][7], and in the following, we will present only a few highlight relating to nonlinear focusing. In summary, the interaction between light characterised by the intensity distribution I(r, z) , and a small dielectric particle involves a gradient force ⃗ F grad and a scattering force ⃗ F scat given by [5] where m = n p ∕n m is the ratio of the refractive index of the particle n p and the surrounding medium n m . The gradient force has a transverse and a longitudinal components, since the gradient operator can be written as a sum of two terms, i.e., ⃗ ∇ = ⃗ ∇ z + ⃗ ∇ r . The scattering force pushes the particle along the direction of propagation of the laser beam, i.e., ẑ . The longitudinal stability of the trap is observed if F grad > F scat . An interesting case is when the illuminating beam is a Gaussian beam (GB) characterised by the following intensity distribution [8]: where r (z) is the radial (longitudinal) coordinate, W 0 is the width of the beam-waist, and I 0 the on-axis (r = 0) in the beam-waist plane. The origin z = 0 is assumed to be located in the beam-waist plane. The width W(z) of the Gaussian beam changes with the plane z following the well-known relationship [8]: The optical trap is made up of a focusing lens of focal length f L enlighten by a Gaussian beam of width W 1 carrying a power P. The focused beam has a beam waist (focal point) having a width W 0 given by Ref. [8] Taking into account of Eq. (3) we find the minimum value of the longitudinal and transverse gradients where K 1 and K 2 are positive coefficients easy deductible from Eq. (1). From the two above equations, it is easy to deduce that for increasing the performance of the optical trap it is necessary to reduce W 0 the beam waist of the focused Gaussian beam as far as possible. From Eq. (5), it is seen that this objective can be reached by reducing as much as possible the focal length f L . However, doing this it would result also a decrease in the lens diameter, and therefore, it should be difficult to keep high the numerical aperture (NA) of the focusing optic. This drawback has been solved by some authors [9] using a nonlinear focusing. The idea was to set in front of the focusing microscope objective a thin layer of nonlinear material (Kerr medium) in which the incident GB induces a Kerr lensing effect, and that will shift the focal point towards the lens without degrading the NA value. The proposed modelling of the optical forces [9] was based on the following hypothesis: the Kerr lensing effect is assimilated to a pure lensing effect, implying that the beam emerging from the Kerr medium keeps its Gaussian nature, i.e., having a M 2 factor equal to unity. At first sight, this view seems to be incorrect, since the beam emerging from the Kerr lens has a beam propagation factor M 2 greater than unity and its value increases with the induced nonlinear phase shift [10]. Therefore, the Gaussian beam subject to Kerr effect should not be treated in principle as a pure Gaussian beam ( M 2 = 1 ). Subsequently, this type of treatment will be qualified as Gaussian Approximation (GA). Consequently, one can expect that the diffraction occurring when the GB passes through the Kerr medium acting as a phase aberration will disturb the longitudinal and transversal intensity distribution, and thus the associated intensity gradients. A second modelling of the optical trap is also possible by taking into account the diffraction effect upon the induced Kerr phase shift profile, considered as a phase aberration, using a diffraction integral (DI). In the next, this modelling based on a numerical calculation will be considered as a reference probably very close to the experimental reality. The authors of Ref. [9] have found that the nonlinear lensing improve the optical trap efficiency. According to the above arguments, it seemed necessary to us to evaluate if the conclusions of Ref. [9] need to be weighted regarding to peculiar focusing properties of a GB subject to optical Kerr effect. In this paper, the first objective is to model the performances of optical tweezers based on a nonlinear Kerr lensing by considering the nonlinear lens as an aberrated lens through the use of a diffraction integral to evaluate if the nonlinear lensing improves or degrades the trap performances. The second objective is to evaluate the ability of the Gaussian approximation modelling to describe accurately an optical trap based on a nonlinear lensing.

Focusing properties of a Gaussian beam subject to Kerr effect
Before proceeding, let us recall some basic elements on the Optical Kerr Effect (OKE) occurring in a transparent dielectric material subject to a laser beam having an intensity profile I( ) , where is the radial coordinate in the plane of the Kerr medium. The resulting refractive index n( ) is intensity-dependent and is written as follows: where n 2 is the nonlinear refractive index. The light intensity distribution I( ) takes the following form in the hypothesis of a Gaussian beam having a width W 1 : The incident beam is subject to a Kerr phase shift Δ ( ) which is intensity-dependent, and is expressed as follows: where k = 2 ∕ , 0 = k.n 2 .d.I 0 is the on-axis phase shift, and d is the Kerr medium thickness. Note that in Eq. (10), the constant term k.n 1 .d has been omitted, since it does not contribute to the aberration attached to the Kerr medium. If we take into account the relationship between power P, on-axis intensity I 0 and Gaussian beam width W 1 , namely P = ( W 2 1 I 0 ∕2) , the on-axis phase shift 0 takes the following form: (8) n( ) = n 1 + n 2 .I( ), Note that it is convenient for measuring the nonlinearity strength to use the nonlinear on-axis phase shift 0 , because it includes all the parameter contributing to the nonlinearity ( n 2 , P, d, W 1 ) . Indeed, since the famous article written by M. Sheih-Bahae et al. [11], and many other authors too numerous to cite here, the community of optical nonlinearities used overall the on-axis phase shift given by Eq. (11) as the key parameter to depict the scale of the nonlinearity.
The generic setup involving an optical trap based on nonlinear lensing is illustrated in Fig. 1 showing the incident GB of width W 1 = 1 mm, the focusing lens of focal length f L = 50 mm and the Kerr medium in which is induced a Kerr lensing effect. Note that it is seen in Fig. 1 the shift of the focus due to the action of the Kerr lens. Before to proceed, it is important to justify the choice of a focal length equal to 50 mm which clearly does not achieve the tight focusing usually implemented in optical tweezers. There are two reasons for this choice: first, the use of identical parameters as in previous work [12] concerning the influence of spherical aberrations, and second to ensure the legitimacy for the use of the Fresnel-Kirchhoff integral to calculate the diffracted field in the framework of the paraxial approximation. In the next, all the calculations are done for a wavelength = 1064 nm . In any case, we will examine which modelling (GA, DI) is the most suitable for evaluating the performance of the optical tweezers based on a nonlinear focusing. Before comparing between the (GA and DI) modelling, let us examine the "true" longitudinal and transversal intensity distributions I d (r, z) beyond the ensemble (lens + Kerr medium) for several values of the phase shift 0 . The calculation of the diffracted intensity beyond the lens is done using the Fresnel-Kirchhoff integral given by [13] where r ( ) is the radial coordinate in plane z (lens and Kerr medium plane), and J 0 is the zero-order Bessel function of first order. The incident collimated Gaussian beam has an electrical field given by . The integral given in Eq. (12) is calculated numerically using an FORTRAN routine based on the numerical integrator (dqdag) from the International Mathematics and Statistical Library (IMSL). The field E d (r, z) can also be obtained from a series derived from an HANKEL transform as shown in the Appendix. Note that the intensity distribution (12) can be considered as the "true" value in comparison of the GA modelling.
To be convinced that the GA at first sight is not suitable for describing the properties of an optical trap as done in [9], one just has to look at the longitudinal intensity distributions shown in Fig. 2, and the transversal intensity distribution in the reference plane z = f L shown in Fig. 3 for several values of 0 . This is after having examined the plots in Figs. 2 and 3 that it seems doubtful that a Kerr lens behaves as a pure lens having the only effect to shift the best focus towards ] . The focusing linear lens has a focal length f L = 50 mm the lens. Indeed, it is observed that the on-axis intensity distribution for 0 ≠ 0 does not keeps its characteristic bell-shape observed when 0 = 0 , and that will influence the values of the longitudinal gradient.
The first step is for evaluating the quality of the Gaussian Approximation in the context of focusing a Gaussian beam subject to OKE, and we will consider the evolution of the diffracted beam with the longitudinal position z for different values of the nonlinear phase shift 0 . For that, we define the beam width W D based on the second-order intensity moment [14] as follows: In the next, we will consider the beam width W D as the "true" value compared to the beam widths W G (GA modelling) defined hereafter. The peculiarity of the diffraction of a Gaussian beam on a Kerr phase profile is that the M 2 factor of the beam emerging from the Kerr medium depends only on the phase shift 0 for a given wavelength [12] The coefficients A, C 1 , C 2 ,… appearing in Eq. (14) are determined using a fitting. Note that these coefficients do not depend upon W 1 [10]. The M 2 factor variations are plotted in Fig. 4 versus the on-axis nonlinear phase shift, and the plot is a supplementary argument for concluding that the Gaussian approximation used in [9] might not be appropriate to describe the focusing of a Gaussian beam subject to OKE.

Gaussian approximation (GA)
As pointed above, the Kerr lensing effect, in the framework of the GA modelling, is treated as a thin pure lens characterised by a Kerr focal length obtained in the framework of the parabolic approximation and noted f NL . The latter has been well known a very long time ago but has been On-axis intensity (a.u.) Normalised intensity r (mm) recently revisited [10], and the following expression has been obtained: Note that authors in Ref. [9] use an expression for f NL in which the factor (3 ∕2) is not present. It should be mentioned that Eq. (15) allows to predict accurately the position z max of the best focus (maximum of on-axis intensity I d (0, z) ). The position z max of maximum of on-axis intensity is then given by Note that in Eq. (16) it is assumed that the Rayleigh range W 2 1 ∕ of the incident Gaussian beam is larger than the equivalent focal length z max . It is verified for W 1 = 1 mm.
As pointed above, in the GA framework, the Kerr lens is considered as a pure lens, i.e., without any aberrations. Then, in this case, the beam emerging from the ensemble (Lens + Kerr medium) remains Gaussian whatever the value of 0 , and its width W G (z) is given by the usual formula where the beam-waist width W 0G is given by Eq. (5) in which f L is replaced by z max given by Eq. (16).
Since the minimum values of gradient forces of the optical trap depend on the size of the beam focus as shown in Eqs. (6) and (7), it is interesting to compare the minimum value reached by the beam sizes W 0G and W D min obtained in the framework of the modelling GA and DI, respectively. The results are shown in Fig. 5 which displays the variations of beam width minima ( W 0G and W D min ) versus 0 the on-axis phase shift. At this step, the conclusion that can be drawn is that the Gaussian approximation (GA) is not able to predict the beam-waist size of the focused Gaussian beam subject to optical Kerr effect.
For the sake of completeness, we need to compare the beam widths W G and W D for a wider range of the longitudinal position beyond the focusing lens. This is done in Fig. 6 for a nonlinear phase shift 0 = 7rad , and the following should be recognised: (i) The GA modelling is inappropriate to describe the beam width near its best focus, the region of interest for the mechanism of optical trapping. (ii) Far away from the beam focus, the Gaussian approximation gives results in terms of beam width that are not as bad as all particularly in the focal plane z = f L , the geometric focal plane of the focusing lens. We will discuss hereafter this point.
Let us now discuss the crossing point between the widths W G and W D indicated in Fig. 6 by an arrow at position z = f L . First, let us examine the radial intensity distribution in plane z = f L shown in Fig. 3 for several values of 0 the on-axis nonlinear phase shift. As expected, for large values of 0 , the radial intensity distribution shows a central bright spot surrounded by concentric rings which number increases with 0 (i.e., power P) as already observed [15]. In addition, the plots in Fig. 3 show that the beam pattern has a lateral extend Beam width (µm) z (mm) Fig. 6 Longitudinal variations of the two beam widths ( W G ,W D ) for 0 = 7rad determined in the framework of (GA, DI) modelling, respectively. Note that for z = f L (arrow), we observe a perfect equality between W G and W D whatever the value of 0 that is increasing with 0 . This effect is due to the best focus shift towards the lens, and the diffraction on the Kerr aberration having the tendency to enlarge the beam profile. The least we can say is that the beam pattern in plane z = f L is far from being Gaussian in shape, and yet the comparison between W G and W D shown in Table 1 is at first sight very surprising, since we get the equality W G = W D . This equality is observed at least until the seventh decimal. To the best of our knowledge, this particular property in plane z = f L of the Gaussian beam diffraction upon a Gaussian phase profile has not reported in the literature. It is hardly a coincidence that the equality W G = W D as if the laser beam beyond the Kerr medium kept its propagation factor equal to M 2 = 1 which of course was not the case in accordance with Eq. (14). Probably, the property making equal in plane z = f L the widths W G and W D is linked to the mathematical properties of the Hankel transform expressed by Eq. (14) and developed in the Appendix. Unfortunately, we are not able to go further in this direction because we were unable to demonstrate this property. The question that then needs to be raised is the previous equality W G = W D occurring in plane z = f L takes place for another phase aberration profile than Δ ( ) given by Eq. (10). The answer is no since by replacing 2 in Eq. (10) by 3 or 4 for instance, we obtain W G ≠ W D in plane z = f L whatever the phase shift 0 . The remarkable property discussed above, i.e., W G = W D , in plane z = f L should not suggest that the Gaussian approximation (GA) is well adapted to describe the diffraction of a GB upon a Kerr phase shift. Indeed, it is important to note that in fact, the Gaussian approximation is not valid away from plane z = f L where it is found W G ≠ W D as shown in Figs. 5 and 6 especially around the best focus which is the position of interest for an optical trap. Consequently, we can expect that applying the Gaussian approximation to the optical trap using a nonlinear focusing should overestimate or underestimate the improvement of the trap performance. This point will be addressed in Sect. 3.

Longitudinal optical forces
Now, we are going to look a potential improvement or degradation of the trapping stability when a nonlinear focusing based on OKE is used. It is worthwhile to recall that the trapping stability is expressed by the ratio | which has to be larger than unity for ensuring the particle trapping. By introducing two coefficients G 1 and G 2 easily deductible, Eqs. (1) and (2) can be rewritten as follows: Note that the gradient force has a longitudinal and a transverse component as pointed out in Sect. 1. The condition of particle trapping is that the longitudinal gradient force should overcome the scattering force. For a reference purpose, we consider the optical trap without the Kerr medium ( 0 = 0 ) for which the plots in Fig. 7 show the longitudinal variations of the on-axis intensity, and the longitudinal gradient of the intensity. It is seen that the longitudinal gradient force is positive (negative) before (beyond) the focal point.
In the following, z min refers to the axial position where the longitudinal gradient is minimum (arrow in Fig. 7). This position is important, since it is precisely where the axial gradient force may overcome the axial scattering force. Consequently, we can define the following relevant quantity noted R which is proportional to ratio of the backward axial gradient and the forward scattering forces [16]: where ẑ is a unit vector of the incident beam direction of propagation. Note that parameter R can be viewed as a stability efficiency with respect to the case without nonlinear lensing. Let us recall the two items that we desire to solve in this paper. First, does the nonlinear focusing improve or degrade the performance of the optical trap? Second, does the GA modelling overestimate or underestimate the effects of the nonlinear focusing compared to the DI modelling? For that, it is convenient to introduce the dimensionless quantity Table 1 Variations versus 0 , in the plane z = f L , of the "true" spot radius W D , and the Gaussian beam width W G obtained in the framework of the Gaussian approximation noted Z L as outlined below which indicates that the performance of the optical trap is improved (degraded) when it is larger (smaller) than unity Now, the next step is to compare the figure of merit Z L DI given by Eq. (21) to Z L GA the figure of merit of the optical trap determined in the framework of the Gaussian approximation (see Appendix). The results are shown in Fig. 8 which displays the variations of Z L DI and Z L GA versus 0 . The following should be recognised: .

(i) The introduction of a nonlinear focusing based on
Kerr effect can improve slightly the stability efficiency of the optical tweezers for particular ranges of 0 values. It is seen in Fig. 8 that the figure of merit Z L DI is an oscillatory quasi-periodic function versus 0 . (ii) The Gaussian approximation (GA) underestimates the figure of merit Z L and is unable to adequately report on the variations of Z L DI in particular the existence of several optimums designated by arrows in Fig. 8.

Radial optical forces
The radial optical force is proportional to the radial gradient ∇ r of the transverse intensity distribution evaluated in the best focus plane z max , i.e., where the on-axis intensity I d (0, z) is maximum. For a Gaussian beam, the radial gradient is negative all over the cross section of the beam, and the remarkable value of the radial gradient is its minimum noted ∇ r (r, z max ) min .
Let us now define, as previously for the longitudinal forces, a dimensionless factor noted Z rad which will allow us to know whether the nonlinear focusing improves ( Z rad > 1 ) or reduces ( Z rad < 1 ) the radial optical force ) the longitudinal gradient of the on-axis intensity In the framework of the GA modelling, it is easy to express the ratio Z rad as follows in a similar way when calculating [Z L ] GA given in the Appendix: The comparison of the two modelling (GA and DI) for the radial figure of merit Z rad is done in Fig. 9 which allows to conclude that the radial trapping figure of merit [Z rad ] DI is degraded in the presence of nonlinear focusing. In contrast, the Gaussian Approximation (GA) modelling overestimates the effect of the nonlinear focusing on lateral trapping efficiency of the optical tweezers. Note that the fall-off of [Z rad ] DI versus 0 is very similar to the behaviour of an optical trap when the focusing Gaussian beam is subject to a spherical aberration [16].

Conclusions
In this paper, we have considered the use of a nonlinear focusing, based on Optical Kerr Effect (OKE), in the implementation of optical tweezers in the hope of improving its trapping performances suggested recently in [9]. The authors of Ref. [9] have proposed a modelling of optical forces based on a key assumption assimilating the Kerr lensing effect to a pure lensing effect, implying that the beam emerging from the Kerr medium keeps its Gaussian nature. This hypothesis is obviously false according to a recent study which demonstrates that beam emerging from the Kerr lens has a beam propagation factor M 2 greater than unity, and its value increases rapidly with the induced nonlinear phase shift [10]. Therefore, the Gaussian beam (GB) subject to Kerr effect should not be treated as a Gaussian beam with a beam propagation factor M 2 = 1 . Subsequently, this type of treatment is qualified as Gaussian Approximation (GA). In fact, the diffraction occurring when the Gaussian beam (GB) passes through the Kerr medium acting as a phase aberration will disturb the longitudinal and transversal intensity distribution, and thus the associated intensity gradients. As a consequence, a second modelling, probably very close to the experimental reality, of the optical trap is possible by taking into account the diffraction effect upon the induced Kerr phase shift profile, considered as a phase aberration, using a diffraction integral (DI). We have compared the longitudinal and radial performances of the optical trapping based on nonlinear lensing determined in the GA and DI modelling. In summary, we can conclude that the GA modelling underestimates the longitudinal trapping efficiency, and overestimates the radial trapping efficiency. This conclusion was expected, because the GA modelling developed in [9] is unconscionable, since the parabolic approximation leading to the expression of f NL given by Eq. (15) can only give the position of the best focus plane of the GB subject to OKE, and cannot be accounted for the complexity of the beam reshaping caused by the diffraction on a phase aberration. In view of the obtained results, it could be asked to what extent the trapping performance of optical tweezers can be improved by a nonlinear focusing based on OKE. Indeed, the possible improving of the longitudinal stability of the trap by OKE is greatly outweighed by the reduction of the radial gradient which could result to the escape of the trapped particle. As a consequence, the conclusion in [9] claiming that using a nonlinear lensing improves significantly the optical trapping efficiency should seriously reconsidered. However, we have recently showed that it is possible to improve strongly the performances of optical tweezers by replacing the usual enlightening Gaussian beam by a rectified Laguerre-Gaussian LG p0 beam [16]. The latter is made up of a central peak surrounded by p rings. Note that the rectification operation consists to transform the negative rings of the LG p0 beam into a positive one using a Binary Diffractive Optical Element. The particularity of a rectified LG p0 beam is to produce a quasi-Gaussian intensity profile in the focal plane of the focusing lens [17]. For a given power and incident beam width, we have found that the longitudinal force in the trap is multiplied by a factor ranging from (p + 1) to (p + 2). I 0 (rad) Fig. 9 Variations of Z rad = ∇ r (r, z max ) 0 >0 min ∕ ∇ r (r, z = f L ) 0 =0 min versus the nonlinear on-axis phase shift for a focal length f L = 50 mm , an incident Gaussian beam of width W 1 = 1 mm of wavelength = 1064 nm determined using the Gaussian approximation (GA, dashed line) or the Diffraction Integral (DI, solid line). Note that Z rad > 1 ( Z rad < 1 ) means that the nonlinear focusing has improved (degraded) the lateral trapping efficiency of the optical trap 1 2 2m+1

The trap figure of merit [Z L ] GA for Gaussian approximation
In this modelling, the Kerr lens is assimilated to a linear lens having a focal length f NL given by Eq. (15).

Focusing without OKE
We start with the Gaussian intensity distribution given by Eq. (3) where W 0 = f L ∕( W 1 ) is the beam-waist radius (best focus), and z = 0 the position of the best focus. The gradient of the on-axis intensity I(0, z) is written as follows: where z 0 = W 2 0 ∕ is the Rayleigh distance of the focused Gaussian beam. It is easy to establish that the longitudinal gradient given above is minimum at position z min = z 0 ∕ √ 3 . At this position, the gradient is minimum and its value is given by The intensity at position z = z min is then given by Funding This research dis not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.