1. Introduction
In 1960, the invention of lasers led to the rapid advancements of various scientific and technological fields, giving rise to the laser industry and a multitude of new photonic technologies. These innovations have found widespread applications in optical manipulation, precision measurement, laser communication, laser processing, microscopic imaging, and others [1–3]. However, as laser technology advanced, it encountered technical bottlenecks in various applications, prompting the development of light field control technologies.
Diffraction is an intrinsic property of light, but many practical applications require minimizing or avoiding it. A non-diffracting beam is a localized optical wave packet that remains unchanged during transmission over several Rayleigh lengths. Among these, the well-known Bessel beams were the first to be both theoretically and experimentally validated [4]. In 1987, Durnin introduced the concept of ‘non-diffracting beams,' which, despite initial controversy, quickly gained attention due to their evident anti-diffraction properties. Under ideal conditions, a non-diffracting beam maintains its transverse intensity distribution during propagation and can even recover after encountering obstacles. Compared to the traditional Hermite–Gaussian or Laguerre–Gaussian laser modes, these beams can propagate over several Rayleigh lengths with minimal energy losses. As research progressed, new types of non-diffracting beams, such as Mathieu, parabolic, and Airy beams, have been discovered [5–9], all exhibiting similar non-diffracting and self-accelerating characteristics. In 2007, Siviloglou and Christodoulides theoretically predicted and experimentally introduced non-diffracting Airy beams in optics [10]. The discovery of different types of non-diffracting beams has led to their broad applications [2–11]. For instance, Bessel beams are now used as optical tweezers for particle manipulation, capture, and transport [9], while Airy beams can guide particles along curved trajectories, enabling them to maneuver around obstacles [10, 11].
This article investigates the normalized paraxial diffraction equation and employs the self-similarity method to derive its self-similar solution, described by the Scorer function, which we refer to as the Scorer beam. We analyze linear combinations of counterpropagating Scorer beams, which are controlled by two key parameters: the initial pulse width and the attenuation factor. A key advantage of Scorer beams is that during transmission, adjacent pulses exhibit alternating attenuation and peak value variations, a feature of particular significance for optical communications. By tuning these parameters, we demonstrate that a larger attenuation factor results in a more pronounced beam compression and an increased pulse amplitude. Additionally, reducing the initial pulse width further compresses the pulse and boosts its amplitude.
The rest of this article is organized as follows. In section 2 , we introduce the new non-diffracting Scorer beam–an exact solution to the normalized paraxial diffraction equation with an external potential–characterized by two parameters: the initial pulse width and the attenuation factor. By selecting appropriate values for these parameters, we generate alternating periodic wave solutions. Through linear superposition of counterpropagating Scorer beams, we produce compressed optical wave packets and further investigate their basic properties through computer simulations. The final section provides a summary of our findings.
2. Mathematical model and the pulse compression by counterpropagating Scorer beams
The (1+1)-dimensional electric field envelope $u\left(\xi ,s\right)$ propagating along the $\xi $-direction within an external potential $V\left(\xi ,s\right)$ obeys the normalized paraxial wave equation [12]:1 ) reduces to the familiar free-space (or uniform dielectric medium) Schrödinger equation possessing a myriad of different solutions, among others the well-known Airy function solution [13–15]. We are looking for a new self-similar solution of equation (1 ) by using the self-similarity method [16–19], and to this end we specifically assume that it has the form:
$\begin{eqnarray}{\rm{i}}\frac{\partial u}{\partial \xi }+\frac{1}{2}\frac{{\partial }^{2}u}{\partial {s}^{2}}+\gamma V\left(\xi ,s\right)u=0,\end{eqnarray}$
where $\xi =z/k{x}_{0}^{2}$ is the normalized transmission distance in units of Rayleigh lengths, with $k=2\pi n/\lambda $ denoting the wave number, $\lambda $ the wavelength in vacuum, and $n$ the refractive index of the material. $s=x/{x}_{0}$ is the dimensionless transverse coordinate scaled by ${x}_{0}$. Further to this, $\gamma \ne 0$ is a real constant, controlling the strength of the external potential V. Obviously, when $\gamma =0$ equation ( $\begin{eqnarray}u\left(\xi ,s\right)=\frac{F\left(\theta \right)}{\sqrt{{w}_{0}}}{{\rm{e}}}^{\left(s+\rho {\xi }^{2}\right)\alpha +{\rm{i}}\left[A\left(\xi \right)+B\left(\xi \right)s+C\left(\xi \right){s}^{2}\right]},\end{eqnarray}$
where $F\left(\theta \right)$ is the wave amplitude, $\theta $ is the self-similar variable of a traveling-wave type ${\theta }{=}\frac{{s}{-}{\mu }\left({\xi }\right){+}{\rm{i}}{\omega }\left({\xi }\right)}{{{w}}_{{0}}}$, with $\mu \left(\xi \right)$ and $\omega \left(\xi \right)$ being two undetermined real parameters, and ${w}_{0}$ is the initial beam width. Additionally, $\rho $ is a real constant to be determined, and $\alpha $ ($0\leqslant \alpha \lt \lt 1$) is a real constant, called the attenuation factor. It serves to transform the beam from infinite energy to finite energy, thereby speeding its convergence. The attenuation factor also performs a ‘truncation operation,' a common practice in optics. Other terms in the phase of the wave solution (2) include the spatial phase shift, frequency shift, and chirp function, represented by $A\left(\xi \right)$, $B\left(\xi \right)$, and $C\left(\xi \right)$, respectively. They are all as yet undetermined functions, and they all depend on the transmission distance $\xi $.By substituting the self-similar solution (2) into equation (1 ), collecting the terms of the same power in $\frac{{\partial }^{n}F}{\partial {\theta }^{n}}$ ($n=\mathrm{0,1,2}$), separating the real and imaginary parts and setting them equal to zero, one obtains two partial differential equations:2 ). Thus, in equations (3A ) and (3B ), we choose a chirpless beam with a linear frequency shift in the phase, $C\left(\xi \right)=0$, $B=\frac{\xi }{2{w}_{0}^{3}}$, which leads to the phase shift of the form $A=-\frac{{\xi }^{3}}{12{w}_{0}^{6}}+\frac{{\alpha }^{2}}{2}\xi $. For the quadratic and linear velocity terms in the self-similar variable, we assume $\mu =\frac{{\xi }^{2}}{4{w}_{0}^{3}}$, $\omega =\alpha \xi $, and for the constant ρ in the attenuation factor, $\rho =-\frac{1}{2{w}_{0}^{3}}$. The final choice is to assume that the external potential is inversely proportional to the wave amplitude F in equation (1 ), $V\left(\xi ,s\right)=\frac{1}{2{w}_{0}^{2}F}$, and that $\gamma =\frac{1}{\pi }$, resulting in a specific choice for the amplitude F. At first, this choice may seem strange, but the situation in which the potential depends on the solution is not that uncommon, especially in nonlinear optics. Commonly, V depends on the intensity |u|2. More generally, the change in the index of refraction of the medium may involve different interference patterns between the forward and backward propagating components of the counterpropagating solutions. Such situations are of interest here, as will be demonstrated and utilized below. With the last assumptions, the equation for the amplitude in equation (3A ) simplifies to a second-order ordinary differential equation:1 ):
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{{w}_{0}^{2}F}\displaystyle \frac{{\partial }^{2}F}{\partial {\theta }^{2}}-2{w}_{0}\displaystyle \frac{\partial B}{\partial \xi }\theta +\displaystyle \frac{2}{{w}_{0}F}\displaystyle \frac{\partial F}{\partial \theta }\left(-\displaystyle \frac{\partial \omega }{\partial \xi }+\alpha \right)-2\displaystyle \frac{\partial A}{\partial \xi }\\ \,-2\displaystyle \frac{\partial B}{\partial \xi }\mu -2{s}^{2}\displaystyle \frac{\partial C}{\partial \xi }+{\alpha }^{2}-{\left(B+2{Cs}\right)}^{2}+2\gamma V=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\displaystyle \begin{array}{c}\frac{2}{{w}_{0}F}\frac{\partial F}{\partial \theta }\left(-\frac{\partial \mu }{\partial \xi }+B+2{Cs}\right)+4\xi \rho \alpha \\ +\,2\omega \frac{\partial B}{\partial \xi }+2\alpha \left(B+2{Cs}\right)+2C=0.\end{array}\end{eqnarray}$
To simplify these equations, we make specific choices for the unknown parameters and functions in equation ( $\begin{eqnarray}\frac{{\partial }^{2}F}{\partial {\theta }^{2}}-\theta F=-\frac{1}{\pi },\end{eqnarray}$
which is the standard Scorer equation [20–22]. The general solution of the Scorer equation can be expressed as a combination of the Scorer functions, F(x) = c1Hi(x)+c2Gi(x). However, we take only Gi, $\begin{eqnarray}F\left(\theta \right)={\rm{Gi}}\left(\theta \right),\end{eqnarray}$
as the desired solution, defined as [20–22]: ${\rm{Gi}}\left(\theta \right)=\frac{1}{\pi }{\int }_{0}^{\infty }\sin \left(\theta t+\frac{1}{3}{t}^{3}\right){\rm{d}}t$. Finally, we obtain the special self-similar solution of equation ( $\begin{eqnarray}\begin{array}{rcl}u\left(\xi ,s\right) & = & \frac{1}{\sqrt{{w}_{0}}}{\rm{Gi}}\left(\frac{4{w}_{0}^{3}s-{\xi }^{2}+4{w}_{0}^{3}\alpha i\xi }{4{w}_{0}^{4}}\right)\\ & & \times \,{{\rm{e}}}^{\alpha s-\frac{\alpha }{2{w}_{0}^{3}}{\xi }^{2}+{\rm{i}}\left(-\frac{{\xi }^{3}}{12{w}_{0}^{6}}-\frac{{\xi }^{2}}{8{w}_{0}^{3}}+\frac{1}{2}{\alpha }^{2}\xi +\frac{\xi s}{2{w}_{0}^{3}}\right)},\end{array}\end{eqnarray}$
expressed in terms of only two free parameters: the attenuation factor $\alpha $ and the beam width w0. Since solution (6) includes the Scorer function, we refer to it as the Scorer beam.Next, we pay attention to the dynamic behavior of the finite-energy Scorer beam. We consider the input and initial conditions at the central position ($\xi =0$), which take the form $u\left(\xi =0,s\right)=\frac{1}{\sqrt{{w}_{0}}}{\rm{Gi}}\left(\frac{\sigma s}{{w}_{0}}\right){{\rm{e}}}^{\sigma \lambda s}$, where $\sigma =\pm 1$. The sign of $\sigma $ determines the propagation direction of the Scorer beam. For $\sigma =+1$, we define the beam propagating from $-\infty $ to $+\infty $ as ${u}_{+}$; the form of the solution comes from equation (6 ), namely:1 ) is linear, both $u{}_{+}\left(\xi ,s\right)$ from equation (7A ) and $u{}_{-}\left(\xi ,s\right)$ from equation (7B ) are solutions, and their linear combination $u\left(\xi ,s\right)={c}_{1}{u}_{+}\left(\xi ,s\right)+{c}_{2}{u}_{-}\left(\xi ,s\right)$ is also a solution, where ${c}_{1}$ and ${c}_{2}$ are arbitrary constants. The most obvious choice is ${c}_{1}={c}_{2}=1$, providing an equal weight to the superposition of both solutions. However, the goal of this paper is to investigate the compression of Scorer beams, preferably with equal weights, but with an eye on the wave superposition that would result in a beam distribution symmetric under reflection. Therefore, to make the solution $u\left(\xi ,s\right)$ complying with these requirements, we choose ${c}_{1}=1$ and ${c}_{2}={\rm{i}}$, yielding the combination:8 ) describes the linear combination of two counterpropagating Scorer beams, where a phase shift of π/2 is added to the second beam. Clearly, the first term ${u}_{+}\left(\xi ,s\right)$ represents the beam transmitted along the $\xi $-axis from $-\infty $ to $+\infty $, while the second term ${u}_{-}\left(\xi ,s\right)$ represents the Scorer beam propagating in the opposite direction. Such a superposition leads to a diffractionless propagation of superposed beams, symmetric across the propagation direction. For a more detailed discussion, please refer to [23–25]. In the following, we will explore the effects of the attenuation factor and the initial pulse width on the dynamic compression of Scorer beams.
$\begin{eqnarray}\begin{array}{rcl}{u}_{+}\left(\xi ,s\right) & = & \frac{1}{\sqrt{{w}_{0}}}{\rm{Gi}}\left(\frac{4{w}_{0}^{3}s-{\xi }^{2}+4{w}_{0}^{3}\alpha i\xi }{4{w}_{0}^{4}}\right)\\ & & {\rm{\times }}\,{{\rm{e}}}^{\alpha s-\frac{\alpha }{2{w}_{0}^{3}}{\xi }^{2}+{\rm{i}}\left(-\frac{{\xi }^{3}}{12{w}_{0}^{6}}-\frac{{\xi }^{2}}{8{w}_{0}^{3}}+\frac{1}{2}{\alpha }^{2}\xi +\frac{\xi s}{2{w}_{0}^{3}}\right)}.\end{array}\end{eqnarray}$
In the opposite case, $\sigma =-1$, we define the beam propagating from $+\infty $ to $-\infty $ as ${u}_{-}$, which can be expressed as: $\begin{eqnarray}\begin{array}{rcl}{u}_{-}\left(\xi ,s\right) & = & \frac{1}{\sqrt{{w}_{0}}}{\rm{Gi}}\left(\frac{-4{w}_{0}^{3}s-{\xi }^{2}+4{w}_{0}^{3}\alpha i\xi }{4{w}_{0}^{4}}\right)\\ & & {\rm{\times }}\,{{\rm{e}}}^{-\alpha s-\frac{\alpha }{2{w}_{0}^{3}}{\xi }^{2}+{\rm{i}}\left(-\frac{{\xi }^{3}}{12{w}_{0}^{6}}-\frac{{\xi }^{2}}{8{w}_{0}^{3}}+\frac{1}{2}{\alpha }^{2}\xi -\frac{\xi s}{2{w}_{0}^{3}}\right)}.\end{array}\end{eqnarray}$
Since equation ( $\begin{eqnarray}u\left(\xi ,s\right)={u}_{+}\left(\xi ,s\right)+{\rm{i}}{u}_{-}\left(\xi ,s\right),\end{eqnarray}$
which still retains equal weights but greatly suppresses transverse wings of the superposed beams and makes the intensity distribution mirror-symmetric across the propagation direction. Thus, equation (2.1. The influence of attenuation factor ${\boldsymbol{\alpha }}$ on the Scorer beam compression
To investigate the effect of attenuation factor ${\boldsymbol{\alpha }}$ on the Scorer beam compression, we fix the initial pulse width ${{\rm{w}}}_{{\rm{0}}}={\rm{1}}$, and calculate the spatiotemporal intensity distributions for different attenuation factors ${\boldsymbol{\alpha }}$, while keeping ${{\boldsymbol{w}}}_{{\boldsymbol{0}}}$ constant. We look at the combined intensity $I\left(\xi ,s\right)={\left|u\left(\xi ,s\right)\right|}^{2}=u\left(\xi ,s\right){u}^{* }\left(\xi ,s\right)$ of the counterpropagating Scorer beams. From equation (8 ), it follows that $I=\left({u}_{+}+{\rm{i}}{u}_{-}\right)\left({u}_{+}^{* }-{\rm{i}}{u}_{-}^{* }\right)$ = ${\left|{u}_{+}\left(\xi ,s\right)\right|}^{2}$ + ${\rm{i}}\left[{u}_{-}\left(\xi ,s\right){u}_{+}^{* }\left(\xi ,s\right)-{u}_{+}\left(\xi ,s\right){u}_{-}^{* }\left(\xi ,s\right)\right]$ + ${\left|{u}_{-}\left(\xi ,s\right)\right|}^{2}$, where the asterisk denotes complex conjugation. Therefore,
$\begin{eqnarray}\begin{array}{ccc}I\left(\xi ,s\right) & = & {I}_{+}\left(\xi ,s\right)+{\rm{i}}\left[{u}_{-}\left(\xi ,s\right){u}_{+}^{* }\left(\xi ,s\right)\right.\\ & & \left.-{u}_{+}\left(\xi ,s\right){u}_{-}^{* }\left(\xi ,s\right)\right]+{I}_{-}\left(\xi ,s\right),\end{array}\end{eqnarray}$
where the brackets enclose interesting interference terms, while ${I}_{+}\left(\xi ,s\right)={\left|{u}_{+}\left(\xi ,s\right)\right|}^{2}$ and ${I}_{-}\left(\xi ,s\right)={\left|{u}_{-}\left(\xi ,s\right)\right|}^{2}$.Under the condition $0\leqslant \alpha \lt \lt 1$, we gradually increase the attenuation factor ${\boldsymbol{\alpha }}$, for example, ${\rm{\alpha }}=0.01$, $0.03$, etc. The numerical simulation results, shown in figures 1 and 2, illustrate the effects of intensity compression with respect to the attenuation factor ${\boldsymbol{\alpha }}$. Figure 1 depicts the intensity distribution of Scorer beams for ${\rm{\alpha }}=0.01$ and initial pulse width ${{\rm{w}}}_{0}=1$, in the opposite propagation directions. The top row shows the Scorer beam transmitted in the positive $\xi $-axis direction, with its spatiotemporal distribution displayed on the left. As seen, the beam follows a parabolic trajectory. To better visualize the variation of intensity with transmission distance, we present a cross-section at ${\rm{s}}=0$ on the right. Before reaching the peak intensity, the intensity on the negative $\xi $-axis increases with the transmission distance $\xi $, but at the same time secondary peaks appear between the adjacent primary peaks. After the beam reaches maximum intensity, a secondary peak appears again, before the intensity decays to zero. The Scorer beam propagating in the opposite direction behaves in a similar but opposite fashion, obeying the symmetry I‒(ξ) = I+(‒ξ), as shown in the middle row of figure 1. For I‒, on the negative $\xi $-axis the intensity starts from zero, with the secondary peak appearing first, followed by the main peak on the positive $\xi $-axis. As $\xi $ increases, the intensity decays, alternating peaks emerge, and eventually the intensity decays to zero.
Figure 1. The intensity distributions of the Scorer beams with the attenuation factor $\alpha =0.01$ and the initial pulse width ${{\rm{w}}}_{0}=1$. Left column: intensity distributions of the individual and combined beams. Right column: cross-sectional cuts through the distributions at ${\rm{s}}=0$. |
Figure 2. The left column shows the intensity distributions of the individual and combined beams. The right column shows the cross-sectional cuts through the distributions at s = 0. The settings are the same as in figure 1, except for the attenuation factor $\alpha =0.03$. |
The lower row displays the superposition of the counterpropagating Scorer beams, as described by equation (9 ). The combined beam intensity gradually increases along the negative $\xi $-axis, reaching the main peak, while on the positive $\xi $-axis, the peak intensity gradually decreases. It should be emphasized that the combined peak intensity is greater than the individual peak intensities (${I}_{\max}\gt 0.2$, ${I}_{+\max}={I}_{-\max}\lt 0.2$). Between the two primary peaks, a W-shaped secondary peak appears, as can be seen on the right side of the low row in figure 1. Clearly, the combination of the two oppositely propagating Scorer beams reveals a significant beam compression effect. Comparing the three plots in the right column of figure 1, the Scorer beams propagating along the $\xi $-positive or $\xi $-negative directions show only three pulse peaks from $-5$ to $5$ on the $\xi $-axis, but their combination exhibits six peaks. In other words, by superposing the Scorer beams propagating in opposite directions, not only is the combined beam compressed, but its intensity is also significantly enhanced.
By keeping the initial pulse width ${{\rm{w}}}_{0}=1$ unchanged and increasing the attenuation factor from ${\rm{\alpha }}=0.01$ to ${\rm{\alpha }}=0.03$, the numerical simulation results are presented in figure 2. While the transmission characteristics of the Scorer beam remain similar to those in figure 1, several notable differences emerge: (1) The amplitude of the Scorer beam propagating in the positive direction has greatly increased, with the maximum peak value increasing from less than $0.20$ to $0.37$, and the corresponding $\xi $ position shifting from $-3$ to $0$, as seen in the right column of the top rows of figures 1 and 2. Additionally, the number of pulses along the $\xi $-axis from $-15$ to $5$ increased from $13$ to $16$. (2) The amplitude of the Scorer beam propagating in the negative direction has also greatly increased, with the peak maximum increasing from less than 0.2 to 0.37, and the corresponding $\xi $ position shifting from 3 to 0, as shown in the right column of the middle row of figures 1 and 2. (3) For the combined beam, within the range from -5 to 5 along the $\xi $-axis, the number of pulses increased from 6 to 8, and the maximum peak nearly doubled, rising from 0.17 to almost 0.34. The W-shaped pulse at the center has disappeared, due to beam compression. Notably, the interference pattern in figure 2 appears slanted and exhibits a significantly more complex structure than the one in figure 1.
By further increasing the attenuation factor ${\boldsymbol{\alpha }}$ to 0.06, we observe that the combined Scorer beam becomes even more compressed, with each pulse's peak value increasing accordingly (not shown). We conclude that as long as the attenuation factor ${\boldsymbol{\alpha }}$ satisfies the condition $0\leqslant \alpha \lt \lt 1$, the larger the attenuation factor, the more pronounced the beam compression and the more complex the interference pattern will be.
2.2. The influence of the initial pulse width on the infinite-energy Scorer beam compression
In equations (7A ) and (7B ), when the attenuation factor is zero ($\alpha =0$), the beams simplify into infinite-energy Scorer beams. In this case, equation (8 ) becomes an exact solution of the normalized paraxial diffraction equation (1 ), allowing it to describe the dynamic behavior of the combined beam. However, due to the infinite energy of this Scorer beam, its intensity distribution can only be observed through numerical simulation over a finite region.
Below, we explore equation (9 ) with $\alpha =0$, focusing on the effect of the initial pulse width on the compression of Scorer beams. For convenience, the initial pulse width ${w}_{0}=1$ is selected in equation (9 ), resulting in a compressed combined Scorer beam, as shown in the top row of figure 3. The left side displays its spatiotemporal distribution, while the right shows its cross-sectional view at $s=0$. From this graph, it is evident that the combination of counterpropagating Scorer beams exhibits slow attenuation in intensity along both the positive and negative $\xi $-axis, presumably with an infinite tail. This suggests that the Scorer beam possesses infinite energy and an infinite wavefront, making it an ideal beam that cannot be generated in the laboratory.
Figure 3. Intensity profiles of the counterpropagating the Scorer beams with infinite energy. The settings are the same as in figure 1 except for the attenuation factor $\alpha =0$. The top row is the initial pulse width ${w}_{0}=1$ and the bottom row is the initial pulse width ${w}_{0}=1.2$. |
If we increase the initial pulse width from ${w}_{0}=1$ to w0 = 1.2, the intensity distribution is shown in the lower row of figure 3. Comparing the two graphs, the top row along the $\xi $-axis from -15 to 15 displays 26 pulses, whereas the bottom row shows 20 pulses. In addition, the intensity in the lower row decays faster and the tail expands. Therefore, the smaller the initial pulse width, the more significant the compression of the counterpropagating Scorer beams. In fact, when calculating the energy of the beam at $\xi $ = 0, it is found that the integral does not converge, that is
$\begin{eqnarray}E\left(\xi =0,s\right)={\int }_{-{\rm{\infty }}}^{+{\rm{\infty }}}{\left|u\left(\xi =0,s\right)\right|}^{2}{\rm{d}}s\to {\rm{\infty }}.\end{eqnarray}$
The above expression indicates that the energy carried by the beam is infinite. As previously mentioned, an infinite-energy Scorer beam cannot be realized in the laboratory. While this type of beam has no direct practical significance, as an ideal beam it can be used in theoretical arguments, much like plane waves or infinite-energy Airy beams.2.3. The influence of the initial pulse width on the finite-energy Scorer beam compression
Now, we explore the effect of the initial pulse width on the compression of Scorer beams in the presence of the attenuation factor. Selecting a non-zero attenuation factor ($0\lt \alpha \lt \lt 1$) guarantees that the combined Scorer beam described in equation (9 ) can rapidly decay. Moreover, the convergence of equation (10 ) mentioned above ensures that such Scorer beams can be realized in the laboratory. The expression for the energy of the combined Scorer beam with the attenuation factor present can be calculated, to yield11 ). As expected, their energy is related to both the attenuation factor and the initial pulse width.
$\begin{eqnarray}E\left(\xi =0,s\right)={\int }_{-{\rm{\infty }}}^{+{\rm{\infty }}}{\left|u\left(\xi =0,s\right)\right|}^{2}{\rm{d}}s=\sqrt{\frac{1}{8\pi {w}_{0}\alpha }}{{\rm{e}}}^{\frac{2}{3}{\alpha }^{2}}.\end{eqnarray}$
Hence, the combination of the truncated counterpropagating Scorer beams converges, confirming that they have finite energy, according to equation (Next, we fix the attenuation factor to $\alpha =0.01$ and observe the compression effect on the pulse by varying the initial pulse width. First, we select an initial pulse width of ${w}_{0}=1.1$, with the results shown in the top row of figure 4. On the negative half of the $\xi $-axis, ranging from -15 to 0, the amplitude of each pulse gradually increases until it reaches $\xi =-3$, where a peak forms, followed by a decrease. A secondary peak appears at $\xi =0$. On the positive half of the axis, a secondary peak emerges at $\xi =0$, followed by a peak at $\xi =3$, and the amplitude of the final pulse gradually decays to zero. This in broad sketches describes a combination of the counterpropagating Scorer beams with finite-energy.
Figure 4. Intensity profiles of the counterpropagating Scorer beams with the attenuation factor $\alpha =0.01$. The settings are the same as in figure 1 except for the values of w0. The top row is with the initial pulse width ${w}_{0}=1.1$ and the bottom row is with the initial pulse width ${w}_{0}=0.8$. |
Keeping the attenuation factor fixed and reducing the initial pulse width to ${w}_{0}=0.8$, results in the distributions shown in the bottom row of figure 4. The basic features are similar to the top row of figure 4, but the beam is further compressed and the amplitude is further increased. Specifically, between $\xi =-5$ and $\xi =5$, there are four pulses in the top row and six pulses in the bottom row, while the maximum peak amplitude increases from 0.28 to 0.37. This demonstrates that reducing the initial pulse width further compresses the pulses and increases their amplitudes.
3. Conclusions
We employed the self-similarity method to analyze the normalized paraxial diffractive equation with an external field and derived its exact solution, described by the Scorer function, known as the Scorer beam. To accelerate the convergence of this beam, we introduced two parameters—the attenuation factor and the initial pulse width—to control its dynamic behavior. We explored the linear combination of counterpropagating Scorer beams and examined, through numerical simulations, the effects of varying the attenuation factor and the initial pulse width on the compression of superposed beams. Our findings indicate that, as long as the attenuation factor satisfies the required condition, increasing it leads to more pronounced beam compression and higher pulse amplitude. Additionally, reducing the initial pulse width further compresses the pulse and increases its amplitude. Simultaneously, as parameters change, the interference patterns develop significantly more complex structures, unlike anything previously observed.