The Advantages and Disadvantages of Using Structured High-Order but Single Laguerre–Gauss LG_p0 Laser Beams

Abstract

Most laser applications are based on the focusing of a Gaussian laser beam (GLB). When the latter is subject to a phase aberration such as the optical Kerr effect (OKE) or spherical aberration (SA), it is recognised that the focusing performance of the GLB is degraded. In this paper, it is demonstrated that high-order radial Laguerre–Gauss LG_p0 beams are more resilient than the GLB when subject to the OKE or SA. This opens up opportunities to replace with advantages the usual GLB with a high-order LG_p0 beam for some applications.

1. Introduction

Since the invention of the laser, most people have seen the usual Gaussian laser beam (GLB) as a kind of perfect beam for well-founded reasons based on some parameters characterising its propagation (divergence, M² factor, brightness). Generally, the laser cavities are inherently multimode, and a laser oscillation occurring exclusively on the GB is achieved as a result of various strategies. The latter involves either the matching of the pump size for end-pumped solid-state lasers to the size of the desired Gaussian beam in order to maximise the modal overlap [1] or, more simply, by inserting a diaphragm inside the cavity, allowing only the Gaussian mode to oscillate as the fundamental mode [2,3,4]. The high-order transverse modes have been ignored over a long period of time in such a way that, today, the standard laser beam in most commercial lasers is the GLB. However, we have observed a revived interest in higher-order azimuthal Laguerre–Gaussian modes since the discovery of their orbital angular momentum along the optical axis [5]. High-order azimuthal Laguerre–Gaussian modes have been experimented with in various configurations of optical resonators [6,7,8,9,10,11]. In contrast, high-order radial Laguerre–Gaussian modes have received little attention of late [12,13]. In the next sections, our study will only deal with high-order radial Laguerre–Gauss LG_p0 beams made up of a central peak surrounded by p rings of light. The electric field associated with a collimated LG_p0 beam is given by

$$E_{in}(\rho )=E_0{L}_p(2{\rho }^2/W^2)\exp (-{\rho }^2/W^2)$$

(1)

where L_p is the Laguerre polynomial of order p; W = 1 mm, the width of the collimated Gaussian LG₀₀ beam; and $\rho$ is the radial coordinate.

The beam propagation factor of LG_p0 beams is $M^2=(2p+1)$, excluding their use for applications requiring high brightness. However, what can be an important drawback can also be an advantage for certain applications of lasers. Indeed, it has been recently shown that LG_p0 beams are outperforming the Gaussian beam in at least two applications, which are 3-dimensional microfabrication [14] by two-photon polymerisation and in optical tweezers subject to spherical aberration [15]. More precisely, the two above applications involve a rectification of the LG_p0 beam, which involves transforming the negative rings of the LG_p0 beam into a positive one by using a binary diffractive optical element (BDOE) made up of annular zones introducing a phase shift equal to 0 or π, giving rise to a transmittance equal to +1 or −1. The positions of the phase jump from 0 to π or from π to 0, following exactly the zeros of the Laguerre polynomial L_p. The main feature of rectified LG_p0 beams is to give rise to a quasi-Gaussian intensity profile in the focal plane of a focusing lens. The rectification makes a transfer of the energy carried by the rings toward the central part of the focused pattern while keeping the beam propagation factor equal to (2p + 1) [14].

In Section 2, we will examine the effect of particular aberrations (spherical aberration and Kerr effect) on LG_p0 beams. We will be very surprised by the behaviour of high-order beams. In Section 3, we will consider the spatial properties of “rectified” LG_p0 beams. In Section 4, we will show that optical tweezers enlightened by an LG_p0 beam ($p\ge 1$) subject to spherical aberration allow for several increases in the longitudinal and transverse trapping effects compared to the trap enlightened by a pure Gaussian beam. In Section 5, it will be demonstrated that a phase-only binary diffractive optical element is able to transform an LG_p0 beam into a flat-top or optical bottle beam in the plane of a focusing lens.

2. LG_p0 Beams Subject to an Aberration

Before proceeding further, it is useful to recall the propagation properties of a pure LG_p0 beam characterised by an electric field given by Equation (1). Its intensity profile, shown in Figure 1, is made up of a central peak surrounded by p rings of light. It is important to view the wavefront of a collimated radial LG_p0 beam as alternately concentric out-of-phase rings.

Three fundamental quantities characterising LG_p0 beams are the Rayleigh range $z_R$, the beam propagation factor $M_p^2$, and the longitudinal distribution $W_p(z)$ of the beam width based on the second-order intensity moment [16].

$$z_R=\frac{\pi W_0^2}{\lambda M^2}$$

(2)

$$M_p^2=(2p+1)$$

(3)

$$W_p^2(z)=W_0^2\left[1+{\left(\frac{\lambda M_p^{2}z}{\pi W_0^2}\right)}^2\right]$$

(4)

where the origin z = 0 of the longitudinal coordinate is in the beam waist plane of the focused beam and $W_0$ is the beam waist radius of the $L{G}_{00}$ beam. It is important to note that the beam width W appearing in Equation (1) has a simple physical meaning only for the fundamental mode $L{G}_{00}$, i.e., the width of the Gaussian beam. The beam spreading increases with mode order p so that the beam width $W_p$ in the far-field ($z>>z_R$) is given by

$$W_p=W\sqrt{2p+1}$$

(5)

Most uses of laser beams are based on their focusing, as shown in Figure 2, where (PO) stands for a phase object which can be an optical Kerr effect (OKE), spherical aberration (SA) or a binary diffractive optical element (BDOE). These phase objects will be further defined. It is important to keep in mind a particularity of focused LG_p0 beams giving rise to the same on-axis intensity distribution.

In the presence of the PO, the incident collimated LG_p0 beam diffracts, allowing for its restructuring. The diffracted beam through the ensemble (PO + lens) in the plane z is characterised by the electric field $E_d(r,z)$, given by the well-known Fresnel–Kirchhoff formula [17]:

$$E_d(r,z)=\frac{2\pi }{\lambda z}{\displaystyle \underset{0}{\overset{\infty }{\int }}\tau (\rho )E_{in}(\rho )\exp [-i\Delta \varphi (\rho )]}\exp \left[-\frac{i\pi {\rho }^2}{\lambda }\left(\frac{1}{z}-\frac{1}{f_L}\right)\right]J_0\left[\frac{2\pi }{\lambda z}r.\rho \right]\rho d\rho$$

(6)

where r is the radial coordinate in plane z, $k=2\pi /\lambda$ is the wave number, $J_0$ is the zero-order Bessel function of the first kind, and $\tau (\rho )$ stands for the transmittance of the phase object (PO). The focusing lens of focal length f_L = 125 mm is assumed to be set quite close to the PO. The diffraction integral given by Equation (6) is calculated numerically by using a FORTRAN 77 routine based on the numerical integrator dqdag from the International Mathematics and Statistical Library (IMSL). The main characteristics of the diffracted field are as follows:

(i)

The intensity distribution $I_d(r,z)={\left|E_d(r,z)\right|}^2$, which can be split into a longitudinal distribution $I_d(0,z)$ and a transversal distribution $I_d(r,f_L)$ in the plane $z=f_L$.

(ii)

The beam propagation factor $M_D^2$ of the diffracted LG_p0. To determine the $M_D^2$ factor, we need to determine numerically the longitudinal distribution of the width $W_D(z)$ of the diffracted beam based on the second-order intensity moment [18].

$$W_D^2(z)=\frac{2{\displaystyle \underset{0}{\overset{\infty }{\int }}{I}_d(r,z)r^{3}dr}}{{\displaystyle \underset{0}{\overset{\infty }{\int }}{I}_d(r,z)rdr}}$$

(7)

The calculation of $W_D$ was conducted for 60 values of coordinate z on either side of plane z = f_L. From the longitudinal distribution of $W_D$, we deduced two features. Firstly, the minimum value of $W_D$, denoted as $W_{D\min }$, and, secondly, the longitudinal position, denoted as $z_{\min }$, where the beam focuses, that is, $W_D=W_{D\min }$. Note that $z_{\min }$ is almost equal to $f_L$ whatever the value of p. The value of the beam propagation factor $M_D^2$ is defined from a fit of the plot $W_D$ versus z with the following parabola:

$$W_D^2(z)=W_{D\min }^2\left[1+{\left(\frac{M_D^2\lambda (z-z_{\min })}{\pi W_{2\min }}\right)}^2\right]$$

(8)

Note that the infinite bound in the integral appearing in the numerator of Equation (7) should have a finite value when conducting its numerical evaluation. The choice of the upper integration bound is based on the value of the radial coordinate r, beyond which the intensity $I_d(r)$ could be decreed negligible. Unfortunately, the cubic term $r^3$ enhances strongly the contribution of the beam wings, although the associated intensity is weak. As a result, the $M_D^2$ value is very dependent on the upper integration bound and could be overestimated. This is the reason why we preferred a different way of calculating the $M^2$ factor of the $L{G}_{p0}$ beam diffracted through a BDOE. This method was based on the decomposition of the diffracted field $E_d(r,z)$ upon a complete basis made up of Laguerre–Gauss functions.

Now let us define the phase aberrations characterising the phase object (PO). The first one is the optical Kerr effect (OKE), for which the complex (PO) transmittance is noted ${\tau }_1(\rho )$, and the second one is the primary spherical aberration (SA), characterised by its complex transmittance ${\tau }_2(\rho )$.

2.1. OKE Aberration

The phase object is made up of a nonlinear material having a thickness d and a refractive index $n(\rho )=n_1+n_2{I}_{in}(\rho )$, where $I_{in}(\rho )={\left|E_{in}(\rho )\right|}^2$ is the incident intensity distribution and $n_1$($n_2)$ is the linear (nonlinear) refractive index. As a result, the beam incident on the lens is subject to a phase shift profile $\Delta \varphi (\rho )$, given by

$$\Delta \varphi (\rho )\approx k{n}_{2}d{I}_{in}(\rho )$$

(9)

where $k=2 \pi /\lambda$ and $\lambda =1064 \mathrm{n}\mathrm{m}$. The incident collimated beam has a power P and an on-axis intensity $I_0=2 P/(\pi W^2)$, which is unchanged whatever the mode order p. The Kerr phase shift $\Delta \varphi (\rho )$ then takes the following form:

$$\Delta \varphi (\rho )={\varphi }_0\times {\left[L_p\left(\frac{2{\rho }^2}{W^2}\right)\right]}^2\times \exp \left[-\frac{2{\rho }^2}{W^2}\right]$$

(10)

$$\mathrm{with} {\varphi }_0=\frac{4n_{2}Pd}{\lambda W^2}$$

(11)

${\varphi }_0$ is the nonlinear on-axis phase shift also called the “breakup integral” or “B-integral” by the community of high-power lasers [19,20,21,22,23,24]. The complex transmittance of the Kerr phase object is written as ${\tau }_1(\rho )=\exp [-i\Delta \varphi (\rho )]$. The nonlinear phase shift $\Delta \varphi (\rho )$ has to be viewed as a phase aberration consisting of defocus and high-order spherical aberrations, which are given in [25] and will not be repeated here. As a result, the defocus term will be responsible for a shift of the best focus toward the lens. In addition to this focus shift, the beam pattern in the geometric focal plane of the linear lens is additionally widened by the spherical aberrations. All these phenomena contribute to the decrease in the on-axis intensity in the focal plane of the focusing lens. Since the nonlinear phase aberration $\Delta \varphi (\rho )$ is intensity-dependent, the distortion suffered by the focal spot is also intensity-dependent. Several authors [25,26,27,28,29,30] have considered this issue by numerical modelling of the diffracted intensity distribution in the geometrical focal plane of the focusing lens when the incident beam is Gaussian, i.e., p = 0. In this case, it is observed that the OKE is responsible for a reduction in the focused intensity in the plane $z=f_L$. It is worth noting that since the laser intensity is obviously time-dependent for pulses, light will focus at different locations and, thus, the best focus will “zoom”, giving rise to the so-called “focal zoom”, which is well described in [31]. As a result, the temporal pulse shape in the focal plane $z=f_L$ is distorted so that it shows a dip near the time corresponding to the peak of the input pulse [19,32]. What was just described in terms of spatiotemporal distortion corresponds to a Gaussian incident beam. The situation is very different for higher-order $L{G}_{p0}$ beams, as shown in Figure 3, which displays the variations in the on-axis intensity in plane $z=f_L$ versus ${\varphi }_0$ [33]. It is seen that the effect of OKE is deleterious when focusing a Gaussian beam since the collapse of $I_d(0,z=f_L)$, the intensity at the centre of the focusing plane, increases greatly with ${\varphi }_0$, the nonlinear on-axis phase shift. The plots in Figure 3 show that higher-order $L{G}_{p0}$ beams are highly resistant to the OKE focal shift so that the intensity $I_d(0,z=f_L)$ remains high despite the nonlinear phase shift. We can see in Figure 3 that the $L{G}_{10}$ beam is superior to the other $L{G}_{p0}$ beams in its resistance to the OKE. This is why, in the following sections, particular emphasis will be placed on the $L{G}_{10}$ beam.

Concerning the temporal distortion of the pulse intensity $I_d(0,z=f_L)$ at the beam centre in plane $z=f_L$, due to the OKE, it is much less pronounced for the $L{G}_{10}$ than for the $L{G}_{00}$ beam, as shown in Figure 4. The time dependence of the incident pulse has a Gaussian form, i.e., the electric field $E_{in}(\rho )$ has to be multiplied by $\exp [-t^2/{\tau }^2]$.

To have a good overview of the degradation of the focused intensity distribution due to the OKE, it is useful to compare the longitudinal distribution of the on-axis intensity $I_d(0,z)$, as shown in Figure 5. The plots in Figure 4 and Figure 5 show that the $L{G}_{10}$ beam is more resilient than the $L{G}_{00}$ when both are subject to the optical Kerr effect. This resilience has been discussed in detail in [34]. The superiority, in terms of resilience in space and time, of the $L{G}_{10}$ beam over the Gaussian beam when subject to the OKE suggests at least two implications: first, it is possible to reduce the protective capacity of optical limiters based on the OKE by enlightening with a $L{G}_{10}$ beam in place of the usual $L{G}_{00}$ beam [30] and, second, it could be possible to solve the problem of producing ultra-intense pulses, avoiding the destruction of amplifiers by the phenomenon of beam collapse observed with a Gaussian beam.

In order to clarify expectations on optical power limiters (OPLs) [34,35,36,37,38,39,40,41,42,43,44,45,46,47], the optical limiting device in its simplest form intended to protect some optronic systems is described in Figure 6. The basic elements of the limiting device are a nonlinear Kerr medium of thickness d characterised by a nonlinear refractive index $n_2$, a linear lens of focal length $f_L$, and a diaphragm of radius $R_D$ set at a distance L from the ensemble (Kerr medium + lens). Note that it has recently [48] been shown that the position L of the diaphragm is relatively critical since the ensemble (Kerr medium + lens + aperture) acts (i) like a saturable absorber if $L<f_L$, and (ii) is an optical limiting device if $L\ge f_L$.

It is worth noting that the transmission of a saturable absorber (optical limiter) increases (decreases) with increasing input power. The operating principle of the OPL based on the Kerr effect can be easily understood by considering the Kerr lensing effect, which shifts the best focal point at position $z_{\max }$ at high power, while the best focus is located at position z = L at low power. As a result, the longitudinal focal shift from z = L to $z=z_{\max }$ enlarges the beam incident on the diaphragm, leading to a reduction in its transmittivity. From the plot in Figure 7, we can then expect that at a high power level, the $L{G}_{10}$ beam will be less attenuated than the $L{G}_{00}$ beam [33]. In the case of a pulsed incident laser beam, the transmissivity T of the diaphragm is defined as the ratio of output and input energies. Figure 7 shows the variations of T versus ${\varphi }_0$ for $L{G}_{00}$ and $L{G}_{10}$ beams. The conclusion is that enlightening an OPL based on the Kerr effect by a $L{G}_{10}$ laser beam instead of the usual Gaussian beam is a countermeasure since the OPL is almost inefficient in protecting the optical system set after the OPL. Note that the vertical scale in Figure 7 is logarithmic.

The second application concerns the difficulty of laser pulse amplification at a high intensity level in the presence of the Kerr effect occurring in the amplifying medium constituting the amplifier chain. It is worth recalling that, usually, high-power laser chains are made up of a fractioned amplifying medium in order to avoid beam collapse. As seen previously and discussed in detail in [21], the focusing shift due to the Kerr lensing effect is very much less pronounced for the $L{G}_{10}$ beam than for the $L{G}_{00}$ beam, and that has an important consequence of laser beam collapse occurring when an intense laser pulse propagates in a bulk optical amplifier. Indeed, it could be expected that the replacement of the usual Gaussian beam by a $L{G}_{10}$ beam should extend the upper limit of the laser intensity in a high-power laser system, especially for nanosecond pulses. Indeed, for such relatively large pulses, the technique called CPA (chirped pulse amplification) is not as effective as that for femtosecond pulses. For the sake of completeness, it is worth noting that replacing the usual $L{G}_{00}$ with an $L{G}_{10}$ beam in the concept of high-power laser chains in the nanosecond regime should require an evaluation of both phenomena known as small- and large-scale self-focusing. This question deserves a close examination that has not yet been conducted in the literature, but which is outside the scope of this paper.

It is known that the beam propagation factor $M_D^2$ of the beam emerging from the phase object could resume the beam distortion. The variations of $M_D^2$ versus ${\varphi }_0$ are shown in Figure 8 for $L{G}_{00}$ and $L{G}_{10}$ beams, and suggest that both beam divergences increase when the nonlinear phase shift ${\varphi }_0$ is increased.

For the $L{G}_{00}$ beam, unsurprisingly, this results in a decrease in the on-axis intensity (see Figure 3) as the beam divergence increases, i.e., ${\varphi }_0$ increases. However, for the $L{G}_{10}$ beam, one observes an unusual behaviour since the on-axis intensity in the plane $z=f_L$ remains high (see Figure 3) when ${\varphi }_0$ is increased, while its divergence is increased (see Figure 8).

This unexpected behaviour can be understood in terms of transverse correlation vanishing (TCV) due to the presence of a phase aberration. The transverse correlation concept applied to a laser beam means that there is a relation between its centre and its wings: if the beam spreads (shrinks), this results in a decrease (increase) in its on-axis intensity.

Let us first consider the Gaussian beam, viewed as a reference beam in the absence of the focusing lens, in the far-field, namely, at distance z, which is very large compared to its Rayleigh range $z_0=\pi W_0^2/\lambda$, where $W_0$ is the beam waist radius. It is easy to show that the relation between the on-axis intensity $I_d(0,z)$ in the far-field and the far-field divergence $\theta$ is given by

$$I_d(0,z)\times \theta =K$$

(12)

$$\mathrm{With} K=2P/z^2$$

(13)

where P is the beam power. Equation (12) describes the basic property of a Gaussian beam, which can be formulated as follows.

For a pure Gaussian beam, i.e., without aberration, the far-field on-axis intensity necessarily decreases if the beam divergence is increased, i.e., if the beam is enlarged. Note that this is also a basic property of many other laser beams. However, in the presence of aberration, which is here a Kerr aberration described by Equation (9), this basic property can be changed drastically, and this phenomenon can be described as transverse correlation vanishing. It is interesting to compare the TCV associated with the $L{G}_{00}$ and $L{G}_{10}$ beams by the use of the figure of merit noted FM and defined by

$$FM=\frac{{\left[K\right]}_{\varphi \ne 0}}{{\left[K\right]}_{\varphi =0}}$$

(14)

Note that a value of FM close or equal to unity means that the beam fulfils Equation (12), i.e., the far-field on-axis intensity varies relative to the inverse of its divergence angle. The variations of FM versus the nonlinear phase shift ${\varphi }_0$ for the $L{G}_{00}$ and $L{G}_{10}$ beams are shown in Figure 9. After some algebra, we can express the far-field on-axis intensity ${\left[I_d(0,z)\right]}_{\varphi \ne 0}$ of the beam subject to the OKE in terms of the on-axis intensity ${\left[I_d(0,z)\right]}_{\varphi =0}$ without aberration as follows:

$${\left[I_d(0,z)\right]}_{\varphi \ne 0}=FM\times \frac{(2p+1)}{M_D^2}\times {\left[I_d(0,z)\right]}_{\varphi =0}$$

(15)

For both $L{G}_{00}$ and $L{G}_{10}$ beams, the beam propagation factor $M_D^2$ increases with ${\varphi }_0$, as shown in Figure 8, but the increase of FM with ${\varphi }_0$ is more important for the $L{G}_{10}$ beam than for the $L{G}_{00}$ beam, as shown in Figure 9, so that the on-axis intensity remains high for the $L{G}_{10}$ beam while it decreases rapidly with ${\varphi }_0$ for the $L{G}_{00}$ beam (see Figure 3). In fact, the diffraction occurring upon the Kerr phase shift produces a transfer of energy from the ring of the $L{G}_{10}$ beam toward the centre of the beam, maintaining a high intensity in the centre of the focal spot despite the phase aberration. A first conclusion can be drawn from the results of this question, which concerns the quality of laser beam focusing. For a Gaussian beam, the presence of the Kerr effect degrades systematically the focusing quality, i.e., there is intensity reduction in the focal plane. In contrast, it is found that a higher-order transverse Laguerre–Gauss beam, in particular, the $L{G}_{10}$ beam, is highly resistant to the Kerr effect, i.e., the intensity in the focal plane remains unchanged until there is at least a nonlinear phase shift of about ten radians.

2.2. Spherical Aberration

The effect of a spherical aberration (SA) on the focusing of a Gaussian beam is well documented in the literature [49,50,51,52] while the case of a $L{G}_{p0}$ beam is less considered or even not studied. The latter is addressed in this Section. Before proceeding, let us define the spherical aberration characterising the PO introduced on the path of the laser beam to be focused, as shown in Figure 2. The complex transmittance τ₂ of the phase object causing the SA is given by

$${\tau }_2(\rho )=\exp [-i{\varphi }_{SA}(\rho )]$$

(16)

$$\mathrm{with} {\varphi }_{SA}(\rho )=k{W}_{40}\frac{{\rho }^4}{{\rho }_0^4}$$

(17)

where k = 2π/λ. $W_{40}$ is the SA coefficient and ${\rho }_0$ is the radius of the unit circle, which contains 99% of the incident power. The variations of ${\rho }_0$ with the mode order p are given in

Table 1. Variations of the radius ${\rho }_0$ of the unit circle with the mode order p.

p	0	1	2	3
ρ0	1.5 W	2.13 W	2.58 W	2.96 W

.

It is well known that the presence of SA when focusing a Gaussian laser beam degrades the focusing performance, i.e., the intensity in the focal plane is reduced [49,50,51,52]. However, for the higher-order Laguerre–Gauss beams (p ≥ 1), we observe the opposite effect in a similar way to what has been observed with the OKE in the previous section. Indeed, the influence of the SA on the on-axis intensity distribution for p = 2, for instance, is shown in Figure 10. It is seen that the SA presence produces an axial shift of the best focus (maximum of $I_d(0,z)$) toward the lens (beyond the plane $z=f_L$) when $W_{40}$ is positive (negative). In addition, one observes a substantial strengthening of the best focus intensity in the presence of SA.

In order to have a synthetic view of the SA influence on the maximum of the axial intensity, it is convenient to introduce a dimensionless factor Y defined by

$$Y=\frac{{\left[I_d(0,z)\right]}_{\max }^{W_{40}\ne 0}}{{\left[I_d(0,z)\right]}_{\max }^{W_{40}=0}}$$

(18)

The meaning of factor Y is that Y > 1 (Y < 1) indicates that the presence of spherical aberration increases (decreases) the focused intensity. The variations of Y versus $W_{40}$, for several values of mode order p, are shown in Figure 11. It is worth observing in Figure 11 that Y is less than unity for the $L{G}_{00}$ beam and greater than one for $p\ge 1$. One can conclude that, in the presence of SA, the focusing of a $L{G}_{p0}$ beam for p > 0 is more efficient than for the GLB.

Another important feature characterising the restructured laser beam is its longitudinal and radial gradient distributions, particularly for the implementation of optical traps (see Appendix A). This point will be expanded on later, but it is useful to consider the influence of $W_{40}$ on the longitudinal intensity gradient for a high-order Laguerre–Gauss beam (p = 2 for instance). This is illustrated in Figure 12, which shows clearly that the presence of the spherical aberration enhances the longitudinal intensity gradient.

3. The Rectifed LG_p0 Beams

The concept of “rectification” has been already presented in [14]. The rectification of an $L{G}_{p0}$ beam for p ≥ 1 involves introducing a phase object (Figure 2) on its path with the function to convert the alternately out-of-phase rings into a unified phase. The phase object used for achieving the rectification of an $L{G}_{p0}$ beam is a binary diffractive optical element (BDOE) made up of annular zones introducing a phase shift equal to 0 or π, giving rise to a transmittance equal to +1 or −1 characterised by the transmittance ${\tau }_3(\rho )$, given by

$${\tau }_3(\rho )=\left\{\begin{array}{l}-1 \mathrm{for} 0<\rho \le {\rho }_1\\ {(-1)}^{i+1} \mathrm{for} {\rho }_i<\rho \le {\rho }_{i+1} \mathrm{and} (i+1)<p\\ {(-1)}^{p+1} \mathrm{for} \rho >{\rho }_p\end{array}\right.$$

(19)

The position of the phase jumps from 0 to π or from π to 0, following exactly the zeros of the Laguerre polynomial L_p given in

Table 2. Roots of Laguerre–Gauss polynomials: $L_p({\rho }_i/W)=0$.

p	Values of Ratio ρi/W for the Zeros of the Intensity of the LGp0 Mode
1	0.707106
2	0.541195	1.306562
3	0.455946	1.071046	1.773407

, where the ${\rho }_i’s$ are the p radial positions for which the intensity of the incident beam is null.

The peculiarities of the rectified $L{G}_{p0}$ beams are that (i) the intensity distribution in plane $z=f_L$ is single-lobed, i.e., the rectification achieves a transfer of energy contained in the rings toward the central peak [15], and (ii) the beam propagation factor is unchanged [14] and remains equal to $M^2=(2p+1)$.

The beam propagation factor of rectified LG_p0 beams is $M^2=(2p+1)$, excluding their use for applications requiring high brightness. However, what can be an important drawback can also be an advantage for certain applications of lasers. Indeed, it has been recently shown that LG_p0 beams are outperforming the Gaussian beam in at least two applications, which are 3-dimensional microfabrication [14] by two-photon polymerisation and in optical tweezers subject to third-order aberration [15]. The main feature of the rectified LG_p0 beam is to give rise to a quasi-Gaussian intensity profile in the focal plane of a focusing lens, as shown in Figure 13. The rectification makes a transfer of the energy carried by the rings toward the central part of the focused pattern while keeping the beam propagation factor equal to (2p + 1) [14].

Table 3. Power content ${\alpha }_{LG}$ (α_R) in the central peak of a pure (rectified) Laguerre–Gauss $L{G}_{p0}$ beam for p varying from 1 to 3. The ratio $\eta =V_G/V_R$ represents the focal volume enhancement factor.

p	αLG (%)	αR (%)	η
1	26.4	82.6	17
2	15.6	78.5	68
3	11.2	76.3	172

allows us to compare the fraction of power ${\alpha }_{LG}$(${\alpha }_R$) in the central peak of an LG_p0 beam (rectified LG_p0).

In order to better understand the benefits of a rectified LG_p0 beam, there is a need to introduce the concept of focal volume, which defines the volume associated with the focused beam where the intensity can be considered as high. First, let us define the focal volume $V_G$ associated with the Gaussian beam (p = 0). It is worth recalling that the focusing of a collimated Gaussian beam of width W incident on a converging lens of focal length f yields a Gaussian focal spot of width $W_0$, given by

$$W_0\approx \frac{\lambda f}{\pi W}$$

(20)

Note that in Equation (20), it is assumed that the incident Rayleigh distance $(\pi W^2/\lambda )$ is larger than the lens focal length f. Two fundamental quantities characterising the Gaussian beam are necessary. The first one is the Rayleigh range $z_R$ and the second is the longitudinal distribution $W(z)$ of the beam width based on the second-order intensity moment, given by

$$z_R=\frac{\pi W_0^2}{\lambda }$$

(21)

$$W^2(z)=W_0^2\left[1+{\left(\frac{\lambda z}{\pi W_0^2}\right)}^2\right]$$

(22)

where the origin z = 0 of the longitudinal coordinate is in the beam waist plane of the focused beam. The focal volume $V_G$ is defined by the following integral:

$$V_G={\displaystyle \underset{-z_R}{\overset{+z_R}{\int }}\pi W^2(z)dz}$$

(23)

It is worth recalling that the focusing of a collimated Gaussian beam of width W incident on a converging lens of focal length f yields a Gaussian focal spot of width $W_0$, given by

$$W_0\approx \frac{\lambda f}{\pi W}$$

(24)

The associated focal volume is obtained by applying Equation (23):

$$V_G=\frac{{\lambda }^3{f}^4}{{\pi }^2{W}^4}$$

(25)

In the same way, it is possible to define a focal volume $V_R$ for the rectified $L{G}_{p0}$ beam, for which it should be noted that (i) its beam waist size, noted W_f in the plane $z=f_L$, is smaller than that of the focused $L{G}_{00}$ beam, as shown in Figure 13 for p = 3, and (ii) its reduced Rayleigh range is $\pi W_f^2/(\lambda M^2)$. Finally, the reduction in the focal volume by using a rectified $L{G}_{p0}$ beam in place of the usual $L{G}_{00}$ beam can be characterised by the following factor:

$$\eta =V_G/V_R$$

(26)

The calculation of factor $\eta$ can be found in [53] and its value is given in Table 3. As pointed out above, in some applications, such as 3-dimensional microfabrication [14] by two-photon polymerisation, for instance, the size of the voxel is of primary importance. A small size allows for the fabrication of small 3-D structures with a high resolution. It may be accepted that the voxel volume is proportional to the focal volume. Consequently, in the case of an incoming Gaussian laser beam, it is seen from Equation (25) that the reduction of $V_G$ requires a short focal length f and a large beam width W. However, tight focusing potentially involves a spherical aberration with the effect of reducing the focused intensity, i.e., increasing the spot size $W_0$. It is then clear that the use of a laser beam having a poor quality ($M^2>1$) can give, after reshaping, rise to a focused quasi-Gaussian pattern. This is a possible solution for reducing the voxel size. For instance, let us consider the case p = 3 for which $\eta =172$ (see Table 3). This means that one can expect an increase in the transverse and longitudinal resolution by a factor of about ${(172)}^{1/3}\approx 5.5$.

4. Optical Tweezers Enlightened by a Structured LG_p0 Beam

Usually, optical tweezers are based on the focusing of a Gaussian beam. The essential features of the theory of optical tweezers are recalled in Appendix A, in which two figures of merit are introduced. The first one, noted $Z_L$, concerns the improvement of the longitudinal trap stability, and the second one, noted $Z_R$, describes the improvement of the radial gradient of intensity. The objective is to examine if the restructuring of the incident beam in the set-up shown in Figure 2 and the introduction of an adequate phase object (PO) are able to make $Z_L$ and $Z_R$ larger than unity, i.e., improve the performances of the trap.

Before proceeding, it is important to note that all LG_p0 beams for $p\ge 0$ have the same on-axis intensity distribution. Consequently, replacing the usual Gaussian beam by an LG_p0 beam cannot improve the longitudinal scattering and gradient forces and thus cannot enhance the longitudinal stability of the trap. However, replacing the usual Gaussian beam by an LG_p0 beam improves slightly the radial gradient force, as shown in

Table 4. Size of the central lobe of an $L{G}_{p0}$ beam for p = 1 to 3 and the radial enhancement factor.

p	Normalised Radius (ρ/W) of the Central Peak of a Pure LGp0 Beam	${}^{\mathit{p}}\mathit{Y}_{\mathit{R}}$
1	0.7	1.9
2	0.53	2.4
3	0.45	2.9

. These values of ^pY_R without a PO are identical to those given in Ref. [54]. Now, let us address the crux of the matter by identifying the appropriate phase object (PO) to make $Z_L$ and $Z_R$ larger than one. In fact, this objective is fulfilled by the two phase objects that have been considered above and that are described by Equations (10), (17) and (19).

4.1. The Optical Trap Enlightened by a Rectified LG_p0 Beam

It is very easy to see that a rectified LG_p0 beam allows for the improvement of the ratio $Z_L$. Indeed, intensity and longitudinal gradient are enhanced due to the strong rise in the maximum on-axis intensity owing to the energy transfer from the rings to the central peak. The latter effect is inherent to the rectification process. It has been found that intensity in plane $z=f_L$ of a rectified LG_p0 beam is enhanced compared with the focusing of a Gaussian beam of the same power by a ratio approximately equal to p [15].

Table 5. Variations of ratio $Z_L$ for a rectified $L{G}_{p0}$ beam.

p	1	2	3
$Z_L$	2.06	3.37	4.77

shows that factor $Z_L$ is improved, particularly as p is increased [52]. It has been found that the use of a rectified LG_p0 beam does not improve the radial force of the optical tweezers since ${}^{p}Y_R<1$ for p = 1 to 3 [54].

4.2. The Optical Trap Enlightened by an LG_p0 Beam Subject to SA

In Section 2.2, it was recalled that the presence of a positive spherical aberration degrades the focusing performance of a Gaussian beam and enhances the focused intensity of an $L{G}_{p0}$ beam. Here, we will envisage a spherical aberration that can be positive or negative and consider its influence on ratios $Z_L$ and $Z_R$. Details of this study can be found in [55], and it will be enough to examine the final results depicted in Figure 14, Figure 15 and Figure 16. The presence of a spherical aberration, particularly when it is negative, can significantly improve the longitudinal stability of optical tweezers, as shown in Figure 14. Figure 15 confirms that the presence of a positive or negative SA degrades the longitudinal performance of the trap when it is enlightened by a GLB. Concerning the radial force, Figure 16 shows that it is substantially improved, especially for a negative SA, which can be obtained in different ways [55].

5. Restructuring an LG_p0 Beam into a Flat-Top or Optical Bottle Beam

In Section 3, we introduced the concept of “beam rectification”, which allows us to generate a quasi-Gaussian intensity profile in the focal plane of a lens focusing a high-order $L{G}_{p0}$ beam crossing a BDOE defined by Equation (19). Now, in the following section, we will demonstrate that the “partial rectification” of an $L{G}_{p0}$ beam by using a particular BDOE—defined hereafter—can give rise to a flat-top or optical bottle beam in the vicinity of the focal plane of a focusing lens.

Before proceeding, let us define the three BDOEs that are considered in the next section. These three devices are a diaphragm, a circular π-plate (CPP), and an annular π-plate (APP), characterised by their transmittances (field ratios) ${\tau }_D(\rho )$, ${\tau }_C(\rho )$, and ${\tau }_A(\rho )$, respectively.

The diaphragm of radius ${\rho }_0$ has a transmittance given by

$${\tau }_D(\rho )=\left\{\begin{array}{l}1 for \rho \le {\rho }_0\\ 0 for \rho >{\rho }_0\end{array}\right.$$

(27)

where $\rho$ is the radial coordinate.

The CPP is a BDOE that introduces a π-phase shift in the central part of the incident beam. It is characterised by the transmittance ${\tau }_C(\rho )$, expressed as

$${\tau }_C(\rho )=\left\{\begin{array}{l}-1 for \rho \le {\rho }_0\\ +1 for \rho >{\rho }_0\end{array}\right.$$

(28)

The APP is made up of a ring introducing a π-phase shift and its transmittance is

$${\tau }_A(\rho )=\left\{\begin{array}{l}-1 for {\rho }_1\le \rho \le {\rho }_2\\ +1 for \rho <{\rho }_1 and \rho >{\rho }_2\end{array}\right.$$

(29)

The CPP and APP are assumed to be made of a transparent material on which is etched a relief of height (or depth) $h=\lambda /[2(n-1)]$, where n is the refractive index of the material and $\lambda =1064 \mathrm{nm}$ the wavelength. The height (or depth) of the relief introduces a π-phase shift. Note that, as will be seen later, the BDOE can include two or more dephasing rings.

5.1. Transformation of a Radial Laguerre–Gaussian LG_p0 Beam into a Flat-Top

As pointed out above, a pure LG_p0 beam is characterised for p > 0 by an electric field given by Equation (1) involving radially a series of p oscillations alternatively positive and negative. For convenience, let us introduce $P^{+}$ ($P^{-}$), the optical power carried by the positive (negative) part of $E_{in}(\rho )$, and the total power $P_{tot}=P^{+}+P^{-}$. An important parameter characterising the incident beam is ${\eta }^{+}$, a dimensionless quantity defined as

$${\eta }^{+}=\frac{P^{+}}{P_{tot}}$$

(30)

The pertinence of parameter ${\eta }^{+}$ has been recently demonstrated [15,56]. Indeed, it was found that when ${\eta }^{+}$ is close to 0.96, the far-field distribution is made up of a flat-top intensity profile [56]. In contrast, when ${\eta }^{+}$ is in the range of 67% to 74%, the far-field is a hollow beam or, more precisely, an optical bottle beam [57].

In the next section, we will demonstrate that the implementation of the ${\eta }^{+}$ concept allows for the reshaping of the LG_p0 beam, for $p\ge 1$, into a flat-top (FT) or optical bottle beam (OBB). Before proceeding, let us consider the value of ${\eta }^{+}$ associated with a pure LG_p0 beam given in

Table 6. Fraction of power carried by the negative rings potentially subject to rectification and ratio ${\eta }^{+}$ for a pure LG_p0 beam.

p	0	1	2	3	4	5
Central peak (positive)	100%	26.4%	15.6%	11.1%	8.6%	7.1%
Ring #1 (negative)		73.6%	23.5%	15.3%	11.5%	9.24%
Ring #3 (negative)				53.7%	17.5%	11.8%
Ring #5 (negative)						45.7%
η+ (pure LGp0)	100%	26.4%	75.9%	30.9%	70.8%	33.2%

.

Recently, it has been shown [56] that an $L{G}_{10}$ beam can be transformed into an FT profile in the focal plane of a focusing lens by increasing its ${\eta }^{+}$ value from 26.4% to 93.4% just by partially truncating the negative ring by using a diaphragm that gives rise to a loss of about 72%. In the next section, it will be shown that a $L{G}_{10}$ beam can be transformed into an FT intensity profile in the far-field region by converting a part of the negative ring into a positive value without loss. This is achieved by a transparent phase-only diffractive optical element such as CPP or APP described above. In addition, we will examine if the ${\eta }^{+}$ concept could be applied to higher values of p in order to generate a flat-top intensity profile after focusing. For the study of the partial rectification of an LG_p0 beam in order to generate an FT profile in the focal plane of a converging lens, we will consider the optical layout given in Figure 2, where the phase object (PO) is replaced by one of the BDOEs defined in Equations (27)–(29). In the following, the calculation of the diffracted field through the ensemble (BDOE + lens) in plane z is obtained by using the well-known Fresnel–Kirchhoff formula given by Equation (6).

Before proceeding, it is important to note that the ability of transforming an $L{G}_{p0}$ beam into a flat-top intensity profile by using a phase-only BDOE achieving a partial rectification is very different depending on the parity of the beam order p. As said above, the position of the phase jumps from 0 to π and from π to 0, characterising the phase-only BDOE, in order to achieve the partial rectification of a ring light, with some connected to the position of the zeros of the Laguerre polynomials $L_p$. The latter are given in Table 2.

5.1.1. Reshaping of Odd $L{G}_{p0}$ Beams: $L{G}_{10}$, $L{G}_{30}$, $L{G}_{50}$

(a)

LG₁₀ beam

As given in Table 6, the value of ${\eta }^{+}$ for the free $L{G}_{10}$ beam is equal to 26.4% and must be modified to about 90% (with reference to [56]) in order to obtain a flat-top profile in the focal plane of a converging lens. For this, we planned to partially rectify the negative ring by using as a BDOE an annular π-plate (APP) defined by Equation (29), where (ρ₁/W) = Y_A and (ρ₂/W) = Y_B. The value of Y_A was fixed to 0.707106 (zero of LG₁₀ beam intensity) and Y_B was variable. To find the right value of Y_B, allowing for the transformation of the LG₁₀ beam into an FT intensity profile, we needed to “rectify” (90−26.4) = 63.6% of the ring power content. As a consequence, the circle of radius Y_BW contained 90% of the total power $P_{tot}$, and the value of Y_B could be deduced from the variations of the normalised power $P(\rho )/P_{tot}$ contained in the circle of normalised radius $Y=\rho /W$, plotted in Figure 17. The extrapolated value of Y for which $\left[P(\rho )/P_{tot}\right]=90\%$ was Y_B = 1.7. Then, we needed to check that the intensity distribution $I_d(r,z=f_L)$ was effectively a flat-top.

The result is shown in Figure 18, which confirms the achievement of an FT profile. The diffracted beam had a beam propagation factor found to be $M_D^2=4.6$. We could also determine as previously the value of Y_B giving rise to an optical bottle beam (OBB). For this, as said previously, the parameter ${\eta }^{+}$ had to be in the range of 67% to 74%. As a result, the partial rectification of the ring needed to be in the range of 67−26.4% to 74−26.4%, namely, 40.6% to 47.6%. From the plot in Figure 17 for p = 1, we extrapolated the value of Y_B, which was in the range of 1.4 to 1.47.

The result shown in Figure 19 displays the expected hollow intensity profile with a beam propagation factor of $M_D^2=5.4$. However, in order to confirm that we are dealing with an OBB, we have to plot the on-axis intensity distribution, namely, $I_d(0,z)$ versus z. The result is shown in Figure 20, which effectively confirms that the reshaping of the LG₁₀ beam generates for Y_B = 1.4 an OBB “closed” by the two longitudinal peaks of intensity playing the symbolic role of the “cork” and “bottom” of the light bottle. It is worth remembering that an OBB consists of a dark (minimal intensity) region surrounded in all directions by regions of higher intensity. Such OBBs are very useful for the trapping of cold neutral atoms [58,59,60] or metal nanoparticles [61,62,63,64,65,66,67]. It is interesting to compare the OBB length Δz = 4.3 mm, which represents the distance between the two intensity peaks appearing in Figure 20, to the Rayleigh range $z_R$ of the pure focused LG₁₀ beam (without a BDOE). The distance $z_R$ (for p = 1) is expressed as a function of the beam waist size $W_0$ of the focused collimated Gaussian beam of width W = 1 mm:

$$z_R=\frac{\pi W_0^2}{(2p+1)\lambda }=5.3 \mathrm{mm}$$

(31)

With W₀ = λf_L/(πW) = 42 μm

(32)

As a result, the OBB length was found to be $\Delta z=0.8 z_R$, while the same reshaping of the LG₁₀ beam achieved by using a diaphragm gave an OBB length of $\Delta z=6.6 z_R$ [57].

(b)

LG₃₀ beam

A close look at Table 6 shows that the rectification of a single ring was not enough to increase ${\eta }^{+}$ from 30.9% (pure LG₃₀ beam) to the range of 67% (OBB) to 90% (FT). Consequently, it was necessary to rectify fully Ring #1 and partially Ring #3 so that the adequate BDOE was a double annular π-plate (APP) made up of two π-dephasing rings, bounded by the circles of normalised radius Y_A = 0.455 and Y_B = 1.071 for the first ring and the circles of normalised radius Y_C = 1.77 and Y_D = variable for the second ring. In order to reshape the LG₃₀ beam into an OBB, ${\eta }^{+}$ needed to be close to 67%, and this could be achieved if 20.8% of Ring #3 were rectified. As conducted previously for the LG₁₀ beam, we deduced Y_D = 2.26 from Figure 17 for p = 3. The value expected for Y_D in order to obtain an FT profile was Y_D = 2.58. The intensity profile $I_d(r,z=f_L)$ showed that the best value of Y_D to obtain the best OBB (FT) was 2.1 (2.45), as shown in Figure 21a,b. Both beams had a beam propagation factor of $M_D^2=10.8$ instead of a beam propagation factor of $M_D^2=7$ associated with the incident beam.

(c)

LG₅₀ beam

The LG₅₀ beam had three positive and three negative zones and was characterised by ${\eta }^{+}=33.2\%$. Consequently, it is seen from Table 6 that the generation of an OBB was possible with a single APP rectifying partially Ring #5. The single APP was bounded by the circles of normalised radius Y_A = 2.51 (fifth zero of intensity) and Y_B = 3.18, determined from Figure 17. By doing this, the BDOE rectified 67−33.2% = 33.8% of the last ring, and the best hollow intensity profile shown in Figure 22 was obtained for Y_B = 2.95 instead of the expected value Y_B = 3.18. The plot in Figure 22 shows that the quality of the hollow profile was bad because of the existence of numerous rings. It was found that the partial rectification of Ring #5 did not allow the generation of an FT profile, whatever the value of Y_B.

The second possibility allowing for the generation of an OBB or FT was the use of a double APP rectifying totally Ring #3 and partially Ring #5. In accordance with Table 6 and Figure 17, the needed BDOE was made up of two π-dephasing rings bounded by the circles of normalised radius Y_A = 1.34 and Y_B = 1.88 for Ring #3 and the circles of radius Y_C = 2.51 and Y_D for the second ring, having different values depending on the desired intensity profile, i.e., OBB or FT. Without repeating what has been explained above, we expected an OBB if $Y_D=3$ and an FT profile if Y_D = 3.5. Just as before, we found from the numerical calculation that the OBB and FT profile were obtained for different values of Y_D. In Figure 23a,b, it is seen that the intensity profile was an OBB (FT) for Y_D = 2.6 (Y_D = 3.05). The beam propagation factor was found to be $M_D^2=14.5$ ($M_D^2=15.5$) for the OBB (FT) profile instead of the beam propagation factor $M_D^2=11$ associated with the incident beam.

5.1.2. Reshaping of Even $L{G}_{p0}$ Beams: $L{G}_{20}$, $L{G}_{40}$

By following the procedure detailed above for odd LG_p0 beams, we modified the value of ${\eta }^{+}$ of $L{G}_{20}$ and $L{G}_{40}$ beams by using a single APP or double APP without reaching the expected FT and OBB profiles in plane $z=f_L$. In addition, no conspicuous beam reshaping was observed when the geometry of the single or double APP was varied. It must be concluded, without identifying the reasons for such behaviour, that the reshaping of an LG_p0 beam into an FT or OBB profile by using a binary phase-only DOE is only possible for odd order p.

In summary, we have demonstrated that the intensity profile in the focal plane can be a flat-top (FT) or an optical bottle beam (OBB), provided that ${\eta }^{+}$ has an adequate value 96% or 67 to 74%. The adjustment of parameter ${\eta }^{+}$ was made using a phase-only binary diffractive optical element (BDOE) introducing a π-phase shift realising an operation of partial rectification of the negative part of the $L{G}_{p0}$ beam. The wording rectification means that the negative electric field is transformed into a positive electric field. The resulting restructuring of the $L{G}_{p0}$ beam by the partial rectification led to an FT or an OBB, but only for odd order p. The phase-only BDOE could be made of a transparent material on which was etched (one etching level) a relief introducing the π-phase shift over the adequate circular areas calculated to obtain the desired ${\eta }^{+}$. The use of such a device has at least two main advantages, which are, firstly, low cost compared to a spatial light modulator [68] or deformable mirror [69] and, secondly, the ability to sustain high light intensities without damage.

6. Conclusions

It is important to note that the high-order transverse mode has been disqualified from the applications of laser beams since the 1960s because it is more divergent and less bright that the Gaussian laser beam ($L{G}_{00}$ beam). Since that time, the Gaussian laser beam has been put on a pedestal, but, from the 1990s, high-order modes have gained renewed interest. We can mention the higher-order azimuthal Laguerre–Gauss modes since the discovery that they possess well-defined orbital angular momentum along the optical axis for non-zero values of the azimuthal order [5]. In contrast, high-order radial Laguerre–Gaussian beams have received little attention as of late [70]. In this review, we wanted to emphasize some very promising properties of restructured high-order radial Laguerre–Gauss beams. It would be desirable to experiment with optical trapping with restructured $L{G}_{p0}$ beams (by using OKE, SA, or rectification) in order to benefit from improved trapping performances. However, the first step should be to produce a $L{G}_{p0}$ beam since commercial lasers deliver mainly Gaussian beams. This can be achieved by inserting inside the laser cavity a mask made up of absorbing rings [13] or a BDOE such as that described by Equation (19) [9,10,70]. Another possibility could be to transform a Gaussian beam into a $L{G}_{p0}$ beam by using an SLM and then carrying out efficient amplification [71]. It will be interesting to extend the study of intensity and gradient distributions with other structured laser beams such as Bessel–Gauss beams and generalised Laguerre–Gauss beams, which have attractive properties [72]. Finally, it would be interesting to examine if restructured $L{G}_{p0}$ beams can be more performant than GLBs for applications other than those considered in this paper [73].