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 could be decreed negligible. Unfortunately, the cubic term enhances strongly the contribution of the beam wings, although the associated intensity is weak. As a result, the 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 factor of the beam diffracted through a BDOE. This method was based on the decomposition of the diffracted field 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 , and the second one is the primary spherical aberration (SA), characterised by its complex transmittance .
2.1. OKE Aberration
The phase object is made up of a nonlinear material having a thickness
d and a refractive index
, where
is the incident intensity distribution and
(
is the linear (nonlinear) refractive index. As a result, the beam incident on the lens is subject to a phase shift profile
, given by
where
and
. The incident collimated beam has a power
P and an on-axis intensity
, which is unchanged whatever the mode order
p. The Kerr phase shift
then takes the following form:
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
. The nonlinear phase shift
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
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
. 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
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
beams, as shown in
Figure 3, which displays the variations in the on-axis intensity in plane
versus
[
33]. It is seen that the effect of OKE is deleterious when focusing a Gaussian beam since the collapse of
, the intensity at the centre of the focusing plane, increases greatly with
, the nonlinear on-axis phase shift. The plots in
Figure 3 show that higher-order
beams are highly resistant to the OKE focal shift so that the intensity
remains high despite the nonlinear phase shift. We can see in
Figure 3 that the
beam is superior to the other
beams in its resistance to the OKE. This is why, in the following sections, particular emphasis will be placed on the
beam.
Concerning the temporal distortion of the pulse intensity
at the beam centre in plane
, due to the OKE, it is much less pronounced for the
than for the
beam, as shown in
Figure 4. The time dependence of the incident pulse has a Gaussian form, i.e., the electric field
has to be multiplied by
.
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
, as shown in
Figure 5. The plots in
Figure 4 and
Figure 5 show that the
beam is more resilient than the
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
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
beam in place of the usual
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
, a linear lens of focal length
, and a diaphragm of radius
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
, and (ii) is an optical limiting device if
.
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
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
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
beam will be less attenuated than the
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
for
and
beams. The conclusion is that enlightening an OPL based on the Kerr effect by a
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
beam than for the
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
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
with an
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
of the beam emerging from the phase object could resume the beam distortion. The variations of
versus
are shown in
Figure 8 for
and
beams, and suggest that both beam divergences increase when the nonlinear phase shift
is increased.
For the
beam, unsurprisingly, this results in a decrease in the on-axis intensity (see
Figure 3) as the beam divergence increases, i.e.,
increases. However, for the
beam, one observes an unusual behaviour since the on-axis intensity in the plane
remains high (see
Figure 3) when
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
, where
is the beam waist radius. It is easy to show that the relation between the on-axis intensity
in the far-field and the far-field divergence
is given by
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
and
beams by the use of the figure of merit noted
FM and defined by
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
for the
and
beams are shown in
Figure 9. After some algebra, we can express the far-field on-axis intensity
of the beam subject to the OKE in terms of the on-axis intensity
without aberration as follows:
For both
and
beams, the beam propagation factor
increases with
, as shown in
Figure 8, but the increase of FM with
is more important for the
beam than for the
beam, as shown in
Figure 9, so that the on-axis intensity remains high for the
beam while it decreases rapidly with
for the
beam (see
Figure 3). In fact, the diffraction occurring upon the Kerr phase shift produces a transfer of energy from the ring of the
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
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
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
where
k = 2π/λ.
is the SA coefficient and
is the radius of the unit circle, which contains 99% of the incident power. The variations of
with the mode order
p are given in
Table 1.
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
) toward the lens (beyond the plane
) when
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
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
, 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
beam and greater than one for
. One can conclude that, in the presence of SA, the focusing of a
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
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.