Open Access
Add to BibSonomy
Abstract
The self-healing phenomenon of structured light beams has been comprehensively investigated for its important role in various applications including optical tweezing, superresolution imaging, and optical communication. However, for different structured beams, there are different explanations for the self-healing effect, and a unified theory has not yet been formed. Here we report both theoretically and experimentally a study of the self-healing effect of structured beams in lenslike media, this is, inhomogeneous lenslike media with a quadratic gradient index. By observing the appearance of a number of shadows of obstructed structured wave fields it has been demonstrated that their self-healing in inhomogeneous media are the result of superposition of fundamental traveling waves. We have found that self-healing of structured beams occurs in this medium and, interestingly enough, that the shadows created in the process present sinusoidal propagating characteristics as determined by the geometrical ray theory in lenslike media. This work provides what we believe to be a new inhomogenous environment to explain the self-healing effect and is expected to deepen understanding of the physical mechanism.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Self-healing of the non-diffracting beam is the most remarkable feature, which exhibits outstanding resilience against perturbations and tends to reform during propagation. Typical examples include the Bessel beam and the Airy beam [1–4]. The Bessel beam is non-diffracting and non-accelerating during propagation in space; while the Airy beam is self-accelerating along a parabolic trajectory in space. These structured beams exhibit self-healing; i.e., they can be reconstructed close to their original shapes when they are partially obstructed at some plane along their propagation [5–7]. At some point it was believed that this property was particular only for Bessel beams but it has been demonstrated that practically any structured optical beam can present self-healing in in free space, including Laguerre-Gauss (LG), Hermite–Gauss (HG) and Ince–Gauss (IG) beams [8–12]. Owing to their intriguing self-healing and other properties, structured beams have been applied in particle manipulation [13], super-resolution imgaing [14], quantum entanglement [15], and many other fields [16].
On the other hand, linear and nonlinear propagation dynamics of structured light in lenslike media have been extensively studied [17,18]. Such investigations date back to the early years of the development of the laser in the mid 60’s [19–21]. A lenslike medium is an inhomogeneous medium whose refraction index has a quadratic dependence with respect to the distance to a defined axis, in other words the medium has a quadratic gradient index of refraction (QGRIN). Due to the analogy between the paraxial wave equation and the quantum Schrödinger equation recently there have been new interest in investigating propagation of beams in what are called quadratic potentials [22,23]. It has been demonstrated that the nondiffracting wave packets can accelerate along an arbitrary trajectory centered on an elliptic or circular orbit in the quadratic potentials, which are called pendulum-type light beams [24].
To date, diverse kinds of explanations to the self-healing effect for obstructions, together with corresponding experimental verification, have been extensively investigated, some of them apply within the near field of the dimensions of the aperture which can be explained using diffraction Babinet’s principle [7,25]. The complex amplitude of the field diffracted on the obstacle decreases with factor $1/z$ and it becomes equal to zero for $z \to \infty$. In addition, within the near field region of the dimensions of the obstruction, the reconstruction of non-diffracting beams can be explained by the internal transverse power flow [5] or the geometrical optics and catastrophe optics [26,27]. The alternative description apply for the far field determined by the size of the aperture and the transverse extent of the beam and it is based on the interference of traveling waves. In this case of self-healing large obstructions are used covering several rings or spots of the structured wavefield pattern that is three or more times larger than the obstruction. The later approach has been related to the observations of a Chinese philosopher more than two thousend years ago. The Chinese philosopher Mo Zi, while studying nature, and in particular optics, observed that “When there are two shadows (it is because there are two sources of light). Two (rays of) light grip (i.e. converge) to one light-point. And so you get one shadow from each light-point.” [28]. By observing the creation and evolution of shadows in the process of self-healing of optical beams, and associating optical wavefronts with what philosopher Mo Zi refers to as light sources, it was demonstrated that structured optical wavefields are composed by the interference of fundamental traveling waves [4,9–11,27,29,30].
In this paper, we demonstrate theoretically and experimentally the self-healing of structured beams in lenslike media for the first time. Unlike the previously mentioned self-healing effects of different structured beams corresponding to different mechanisms, the reconstruction is due to the combined effects of the superposition of traveling waves and the proper characteristics of rays propagating in inhomogeneous lenslike media with a quadratic gradient index. By placing an obstruction in front of a structured beam off axis it is observed the formation of a number of shadows that propagate along sinusoidal trajectories as rays do. The self-healing process even occurs when the dynamics of the beam includes orbiting the propagation axis along an elliptic or circular helix trajectory and breathing of the beam. We suggest that the Pendulum-type light beams with self-healing properties have great application prospects in the fields of particle manipulation, superresolution imaging and optical routing.
2. Wave equation in quadratic GRIN media: Gaussian modes
Consider the propagation of optical beams in a lenslike medium whose gradient index of refraction behaves quadratically with respect to the transverse coordinates [21,31,32]
where $n_0$ is the refractive index on axis and $g$ is a constant that characterizes the GRIN medium. The Helmholtz equation describing the propagation of a beam in this medium is [22–24] with $k= 2\pi n_0/\lambda$. Setting $E(x,y,z) = E_0 u(x,y,z) \exp (-ikz)$ the paraxial wave equation is obtainedNormalizing with respect to Gaussian beam parameters $\xi =x/w_0$, $\eta =y/w_0$, $w_0$ the Gaussian beam waist, $\zeta = z/L_D$, in which $L_D=kw_0^2/2$ is the Rayleigh distance, we get the normalized paraxial wave equation
Following the standard procedure [21], we use the ansatz for Gaussian beams $u(\xi,\eta,\zeta )=\exp [-i p(\zeta ) - i(\xi ^2+\eta ^2)/2q(\zeta )]$, where the first term in the argument carries longitudinal information only and the second describes the transverse variations on propagation; beam width and phase front curvature. By substitution in Eq. (4) we get the next set of equations
The first equation for the parameter $q$ is of the Ricatti type. For the purpose of the present study only this equation will be solved and reduced to get the behavior of the beam width. The rest is a simple exercise. From the Ricatti’s equation, when $q'=0$ implies a beam mode whose width does not change on propagation and $q_m=i/\gamma$. If we consider at the entrance of the medium a normalized beam on axis with plane phase front, namely, $u(\xi,\eta,\zeta =0)=\exp [-(\xi ^2+\eta ^2)/2]$, then the stationary mode occurs when $\gamma \equiv 1$. The solution for the parameter $q$ and general $\gamma$ is
From this equation, it is clear that the normalized beam width of the propagating stationary mode is constant, $w(\zeta ;1)=1$. When $\gamma \neq 1$ the initial value of the width is unitary and will increase or decrease presenting periodical breathing depending on the value of $\gamma$ [21,33,34]. Observe that for $\gamma \neq 1$ the period of the breathing is half of that in the $q$ parameter and the whole period is scaled to $\Lambda _{\gamma } = \Lambda /\gamma$.
3. Ray-wave duality revisited
It is well known that the propagation of rays in a medium with refractive index $n=n(\textbf {r})$ is governed by [35]
where r is the position vector of a given point on the ray, $ds$ is the distance element along the ray and $n$ is the refractive index of the medium in which the ray is propagating. In agreement with the wave study of the previous section, we assume paraxial rays $ds\simeq dz$ and using the refractive index in Eq. (1) to obtain that the equation for the rays in lenslike medium that after normalizing the longitudinal coordinate becomes [20,31,32] whose general solutions $\textbf {r}(x(\zeta ),y(\zeta )$) are $x(z) = A\cos (\frac {\gamma }{2}\zeta ) + B\sin (\frac {\gamma }{2}\zeta )$ and $y(z) = C\cos (\frac {\gamma }{2}\zeta ) + D\sin (\frac {\gamma }{2}\zeta )$, where $\textit {A,B,C,D}$ are constants determined by the initial conditions. It is clear that any ray starting at $\zeta =0$ launched at some point off-axis, say at $\textbf {r}_0 = (0,A)$, will follow a sinusoidal trajectory and complete a full period at $\Lambda _\gamma =\Lambda /\gamma$ as it happens in the wave-optics case discussed in the previous section. Since we are dealing with rays, the effect of having different values for $\gamma$ just modifies its period of the oscillations.A known and remarkable feature of the stationary wave modes in lenslike media is that they follow the trajectory determined by the ray equation when they enter normal to the medium at any position. Any point within the beam cross section will follow the trajectory determined by the ray equation [20]. Moreover, if at the entrance the beam is off axis and slightly tilted perpendicular to the offset, the modes will follow helical trajectories with the same period $\Lambda _\gamma$. The trajectories can be either elliptic or circular helices [24,32].
As an example, in Fig. 1 it is shown the propagation of two stationary Gaussian beam modes with $\gamma =1$ entering a lenslike medium at $\eta (\zeta =0)=\pm 7$ following the sinusoidal ray trajectories determined by (9). From this figure, we observe that since the intersection of the trajectories is independent of the offset, it can be regarded as a first focus of the lens-like medium occurring at $\zeta = \Lambda /4$. The QGRIN medium acts as a Fourier lens with focal distance $\Lambda /4$, in other words at this plane will appear the analogous to the diffraction pattern at infinity of any input field distribution.
Fig. 1. Propagation of two Gaussian modes in lenslike medium. Observe the interference of the modes when they encounter/collide.
By continuing propagation we observe that an input image will be inverted at $\zeta =\Lambda /2$ and recovered after propagating the whole period $\zeta =\Lambda$. The bundle of rays forming the beams follow the sinusoidal trajectories but then simultaneously behaving as waves interfering at the focal planes at $\zeta = \Lambda /4$ and $\zeta = 3\Lambda /4$. This effect can occur even for the higher order stationary modes, either Hermite-Gauss beams, Laguerre-Gauss beams or Ince-Gauss beams.
4. Higher order Gaussian modes in lenslike medium
Looking for high order stationary solutions of the form $u(\xi,\eta,\zeta )=v(\xi,\eta )\exp (i\beta \zeta /4)$ the wave equation becomes independent of the propagation coordinate, namely
It is easy to see that this equation is separable, $v(\xi,\eta )=U(\xi )V(\eta )$. Recalling that the condition for stationarity states that $\gamma =1$, we get
These equations have exact solutions if we make $C=2m+1$ and $\beta -C=2n+1$, the solutions are $U(\xi )=H_m(\xi )e^{-\frac {\xi ^2}{2}}$ and $V(\eta )=H_n(\eta )e^{-\frac {\eta ^2}{2}}$, respectively. Notice that the parameter $\gamma$ can be defined analogously as it is done for spatial solitons, as the ratio of two lengths, $\gamma = L_D/L_{GR}$, where $L_D$ the Rayleigh distance and $L_{GR}=1/2g$. Eliminating the separation constant $C$ we obtain the GRIN wavenumber $\beta _{m,n}=2(m+n+1)$. Collecting all the results, the stationary solution is
A similar calculation can be done for the Laguerre-Gauss beams in cylindrical coordinates associated to the conformal mapping $\xi +i\eta =\rho \exp (i\varphi )$ [36]
Another family of modes in GRIN media are the Ince-Gauss beams that are defined transversely in elliptic coordinates. The even modes are described by
In the previous section we mentioned that the propagation trajectory of all the higher order Gaussian modes will coincide with the path determined by the ray equation when they enter the lenslike medium medium off-axis. However, this is only true for the particular case in which $\gamma =1$, this is, for the stationary modes. When $\gamma \neq 1$ the propagation will present breathing on propagation governed by Eq. (7) and modifying the period of their oscillations as we will see below.
5. Propagation of Mo Zi shadows in a lenslike medium: self-healing of high order Gaussian modes
Chinese philosopher Mo Zi did several experiments on optics more than two thousand years ago. In one of his studies he observed that if an illuminated object cast two shadows it is due to the existence of two sources of light coinciding at the illuminated object [28]. In this section we use the phenomenon of self-healing to show that a structured beam is composed by the interference of a number of traveling waves. When a structured beam is partially obstructed by an opaque object, this will cast a number of shadows corresponding to the same number of interfering wavefronts. For the case of Laguerre-Gauss beams it is only two wavefronts but for the Hermite-Gauss beams four wavefronts are needed [9,11]. Similarly, 1D Airy beams require two wavefronts and four are needed for 2D beams [4].
In Fig. 2 it is shown the propagation of intensity on the plane $x-z$ of a stationary Laguerre-Gauss beam mode ($\gamma =1$, $n=12$ and $m=0$) through the lenslike medium with an obstruction placed off-axis. The yellow arrows correspond to the normals of each of the conical wavefronts that compose the Laguerre-Gauss beam, as observed by Mo Zi, two shadows, two wavefronts coinciding and interfering at the position of the obstruction [11]. Behind the obstruction, the shadows due to these composing conical waves are very well defined and follow sinusoidal trajectories. They meet at half the period $\zeta =\Lambda /2$ but at the opposite side of the axis to reconstruct the inverted pattern. They follow the sinusoidal path to meet again at the whole period reconstructing the pattern of the beam at the entrance of the lenslike medium. If the obstruction is placed on-axis at the input end of the media, the shadows will still follow sinusoidal trajectories and meet at the same distance, consistent with the off-axis case.
Fig. 2. Propagation of a Laguerre-Gauss beam $LG_{12,0}(\rho )$ along the plane $x-z$. Observe the shadows formed at onset of proagation created by each of the component traveling semi-conical waves. These shadows follow the characteristic sinusoidal propagation of rays in QGRIN media.
In Fig. 3 we can see snapshots at key positions of a complete period propagating in a lenslike medium. The beam is set on axis. When the beam propagates reaching the plane at $\Lambda /4$ we observe that the Laguerre-Gauss beam has been practically self-healed. The physical reason can be understood recalling the the lenslike medium behaves like a Fourier lens and a wavefield entering normal to the medium will focus at $\Lambda /4$. At this plane the beam presents its best self-healing pattern as it corresponds to what would be at infinity if it was propagating in free space. Propagating to $\zeta =\Lambda /2$ we recover the initial pattern but inverted as it occurs in a $4f$ lens system. The beam self-heals again at $\zeta = 3\Lambda /4$ to finally recover its initial input pattern after having propagated the whole period.
Fig. 3. Snapshots of the propagation of an obstructed on-axis Laguerre-Gauss beam. $LG_{12,0}(\rho )$. The propagation is presented in a zig-zag distribution. At the focal planes $\zeta = \Lambda /4$ and $\zeta = 3\Lambda /4$, bottom images, the beam presents the maximum self-healing.
We have chose to use obstructions off-axis with different geometries to brake the symmetry of the patterns and make clear that self-healing is neither due to Babinet’s principle nor is related to diffraction responsible for the poison spot. Since at one fourth of the whole period within the quadratic GRIN medium is equivalent to observe in free space the field pattern at infinity, it is the best place to observe self-healing. The size of the obstruction is relevant regarding to how much of the beam and its composing waves are obstructed. For instance, for Laguerre-Gauss modes if the beam is blocked such that only the external ring is allowed to pass, its incoming conica wavefront, after passing the axis and becoming outgoing wave, will interfere with the former reconstructing at infinity the central part of the Laguerre-Gauss beam (see Visualization 1). An extreme case occurs if half beam is blocked, the resulting pattern will be substantially altered and self-healing can not be considered as such (see Visualization 2 and Visualization 3). However, if one considers that self-healing is local, then self-healing occurs in 1D along the axis parallel to the obstruction for any of the Gaussian modes [9].
6. Experimental observation of self-healing of Gaussian modes in lenslike media
In this section we show the experimental demonstration of the predicted phenomena in the previous sections. Figure 4(a) illustrates the experimental setup for observing the self-healing process. The $632.8$ $nm$ He-Ne laser is divided into two beams by the beam splitter after passing through a beam expander. Composed of L1 and L2, the beam expander enlarges the linear polarized beam by 10 times. The phase-type spatial light modulator modulates the reflected one into high order Gaussian modes, where the phase mask is generated by the encoding method provided by Bolduc [38,39]. To code the informations of target beam $\Psi \left ( {{\omega _x},{\tau _y}} \right ) = A\left ( {{\omega _x},{\tau _y}} \right )\exp \left [ {i\phi \left ( {{\omega _x},{\tau _y}} \right )} \right ]$ (where ${{\omega _x}}$ and ${{\tau _y}}$ are the pixel coordinates) into a SLM, the expression of the phase-only hologram is given by:
Fig. 4. (a) Experimental setup. L, lens; M, mirror; SLM, spatial light modulator; AP, aperture; OL, objective lens; TL, tube lens; CCD, charge coupled device. (b)(c) The phase masks for generating obstructed Laguerre-Gaussian beam and titled Hermite-Gaussian beam, respectively.
Without loss of generality, let us first consider the Laguerre-Gaussian beam with $n=12$ and $m=0$, generated by the phase mask in Fig. 4(b). It was determined that the stationary Laguerre-Gaussian mode, $\gamma = 1$, can be obtained by setting its width ${w _0} = 10.2$ $\mu m$. The Laguerre-Gaussian beam launched into the center of the GRIN media, after being obstructed by an opaque circular obstacle with $r=15$ $\mu m$. The information of the obstacle was encoded in the phase mask. Figure 5(a) captured by the CCD presents the intensity distributions of Laguerre-Gaussian at different propagation distances. As shown in Fig. 5(a), at ${z_1} = 0$ the beam is partially blocked by the circular obstacle. By propagating within the medium to the distance ${z_2}=0.25\Lambda$ the central part of the Laguerre-Gaussian beam is practically reconstructed, as expected and in agreement with the below accompanying simulations, Fig. 5(b). When the beam propagates to the plane at ${z_3}=0.5\Lambda$, the pattern is symmetrically reversed compared to the initial plane. Further propagation to ${z_4}=0.75\Lambda$ self-healing is present again. After having propagated a full period the initial pattern is recovered. Notice that the snapshots in the first row, Fig. 5(a), are perfectly consistent with those obtained in the simulations shown at the same positions in Fig. 5(b).
Fig. 5. The experimental (a) and theoretically (b) intensity distribution of the Laguerre-Gauss beam at different propagation distances. The beam is launched into the center of the parabolic potential. Parameters are chosen as $n=12$ $m=0$ and ${w _0} = 10.2$ $\mu m$. Scale bars: $50$ $\mu m$.
Next, we investigated the self-healing of a high order Hermite-Gaussian beam, verifying that after certain distances, the intensity distribution of the recovered beam is the substantially same as the unblocked beams. In an attempt to perturb the symmetry of the self-healing of the beam we investigated the simultaneous effect of multiple obstacles with different geometries. We generated a Hermite-Gaussian beam with three obstacles of different shapes encoded in the spatial light modulator, these are a triangle, a circle and a square. Again, the beam width of the stationary HG beam must be ${w _0} = 10.2$ $\mu m$, and we have chosen a square HG beam with $m=8$ and $n=8$. The experimental and simulated evolution of the Hermite-Gaussian beam launched on axis is shown in Figs. 6(a) and 6(b), respectively. Notice that the self-healing process occurs at the expected focal planes at ${z_2} = 0.25\Lambda$ and ${z_4} = 0.75\Lambda$. There is a clear high degree of agreement between experiment and simulation.
Fig. 6. The experimental (a), (c) and theoretically (b), (d) intensity distribution of the Hermite-Gauss beam at different propagation distances: ${z_m} = m\Lambda /4$ with $m = 1,2,3,4$. The Hermite-beam with $m=8$ and $n=8$ is launched into the lenslike media vertically (a) (b) and slantingly (c) (d). The parameters for (c), (d) trajectories are chosen as ${x_0} = 100$ $\mu m$, ${y_0} = 0$, ${k_\xi } = 0$ and ${k_\eta } = 4.95$. Scale bars: $50$ $\mu m$.
Finally we investigated a structured light beam with an initial transverse linear momentum to demonstrate that it can accelerate along a trajectory governed by a general equation of an elliptical helix in the lenslike medium orbiting the propagation axis [24]. For this purpose we used a partially obstructed Hermite-Gaussian with a beam width to induce a breading behavior, this is, the beam has been subjected to propagate breathing along an helical trajectory. Typical set of parameters was chosen as: ${x_0} = 100$ $\mu m$, ${y_0} = 0$, ${k_\xi } = 0$ and ${k_\eta } = 4.95$. The initial light beam was generated using the phase mask shown in Fig. 4(c). In this case, it was observed that the propagation trajectory of the Hermite-Gaussian beam was an elliptical helix, whose projected ellipse has a semi-major axis is $100$ $\mu m$ and the semi-minor axis $50$ $\mu m$. Here the Gaussian beam width was set as ${w _0} = 7.78$ $\mu m$, that is smaller than that for stationary modes. As shown in rows Figs. 6(c) and 6(d), self-healing process still occurs at the expected propagation distances when the beam propagates along the helix and clearly exhibits a breathing behavior. In the planes $0.25\Lambda _{\gamma }$ and $0.75\Lambda _{\gamma }$, the Hermite-Gaussian beam expanded showing clearly self-healing, very similar to the intensity pattern of an unperturbed beam. Again, our experimental results of the Hermite-Gauss beam agree with the theoretical predictions. Observe that the pattern does not rotate maintaining its initial orientation while it is undergoing self-healing, despite obstacles being reversed when propagated to $0.5\Lambda _{\gamma }$. There are optimal self-healing propagation distances, which are $0.25\Lambda _{\gamma }$ and $0.75\Lambda _{\gamma }$ respectively. Because the lenslike medium behaves like a Fourier lens, it is equivalent to the pattern at an infinite distance.
7. Conclusion
In conclusion, we demonstrated for the first time that the self-healing of structured beams occurs in lenslike media and it is due to the superposition of the composing traveling waves that follow the trajectories determined by ray optics. We investigated the evolution of higher order Gaussian modes, Laguerre-Gaussian beams and Hermite-Gaussian beams, partially obstructed at onset of propagation in the lenslike medium with obstacles of different shapes. Self-healing was clearly observed at the predicted planes that correspond to the focal planes within the lenslike medium. We have demonstrated that the opaque object will cast shadows consistent with the fundamental traveling waves that constitute the structured beam. Moreover, these shadows of the wave fields followed sinusoidal trajectories governed by the ray equation in lenslike media. At half the period, the shadows re-encounter to form the inverted image of the input obstructed beam. This phenomenon has never been reported before in the literature. Moreover, the self-healing process was also observed when the propagation dynamics of light beams included orbital motion and breathing. We expect that the Ince-Gaussian beam will also exhibit similar propagation characteristics as those reported here for Hermite-Gauss beams and Laguerre-Gauss beams.
Funding
National Natural Science Foundation of China (11974146, 12374306, 62175091); Guangzhou Science and Technology Program (202201020061).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. 32(16), 2447–2449 (2007). [CrossRef]
2. G. A. Siviloglou, J. Broky, A. Dogariu, et al., “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]
3. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef]
4. J. Rogel-Salazar, H. A. Jiménez-Romero, and S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A: At., Mol., Opt. Phys. 89(2), 023807 (2014). [CrossRef]
5. J. Broky, G. A. Siviloglou, A. Dogariu, et al., “Self-healing properties of optical airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef]
6. G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-gauss beam,” Phys. Rev. A: At., Mol., Opt. Phys. 89(4), 043807 (2014). [CrossRef]
7. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998). [CrossRef]
8. D. Aguirre-Olivas, G. Mellado-Villaseñor, V. Arrizón, et al., “Self-healing of hermite-gauss and ince-gauss beams,” Proc. SPIE 9581, 958105 (2015). [CrossRef]
9. J. A. Ugalde-Ontiveros, A. Jaimes-Nájera, S. Luo, et al., “What are the traveling waves composing the Hermite-gauss beams that make them structured wavefields?” Opt. Express 29(18), 29068–29081 (2021). [CrossRef]
10. J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, et al., “Laguerre-Gauss beams versus Bessel beams showdown: peer comparison,” Opt. Lett. 40(16), 3739–3742 (2015). [CrossRef]
11. J. Mendoza-Hernández, M. L. Arroyo-Carrasco, M. D. Iturbe-Castillo, et al., “Structured light beams constituted of incoming and outgoing waves,” Phys. Rev. A: At., Mol., Opt. Phys. 100(5), 053847 (2019). [CrossRef]
12. J. Pinnell, V. Rodríguez-Fajardo, A. Forbes, et al., “Revealing the modal content of obstructed beams,” Phys. Rev. A: At., Mol., Opt. Phys. 102(3), 033524 (2020). [CrossRef]
13. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
14. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8(4), 302–306 (2014). [CrossRef]
15. M. McLaren, T. Mhlanga, M. J. Padgett, et al., “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5(1), 3248 (2014). [CrossRef]
16. A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured light,” Nat. Photonics 15(4), 253–262 (2021). [CrossRef]
17. Y. Zhang, M. R. Belić, L. Zhang, et al., “Periodic inversion and phase transition of finite energy airy beams in a medium with parabolic potential,” Opt. Express 23(8), 10467–10480 (2015). [CrossRef]
18. L. Zhang, H. Li, Z. Liu, et al., “Controlling cosine-gaussian beams in linear media with quadratic external potential,” Opt. Express 29(4), 5128–5140 (2021). [CrossRef]
19. E. A. J. Marcatili, “Modes in a sequence of thick astigmatic lens-like focusers,” Bell Syst. Tech. J. 43(6), 2887–2904 (1964). [CrossRef]
20. P. Tien, J. Gordon, and J. Whinnery, “Focusing of a light beam of Gaussian field distribution in continuous and periodic lens-like media,” Proc. IEEE 53(2), 129–136 (1965). [CrossRef]
21. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4(12), 1562–1569 (1965). [CrossRef]
22. Y. Zong, S. Xia, L. Tang, et al., “Observation of localized flat-band states in kagome photonic lattices,” Opt. Express 24(8), 8877–8885 (2016). [CrossRef]
23. J. Zeng, Y. Hu, X. Zhang, et al., “Localization-to-delocalization transition of light in frequency-tuned photonic moiré lattices,” Opt. Express 29(16), 25388–25398 (2021). [CrossRef]
24. J. Jia, H. Lin, Y. Liao, et al., “Pendulum-type light beams,” Optica 10(1), 90–96 (2023). [CrossRef]
25. X. Chu, G. Zhou, and R. Chen, “Analytical study of the self-healing property of airy beams,” Phys. Rev. A: At., Mol., Opt. Phys. 85(1), 013815 (2012). [CrossRef]
26. P. Vaveliuk, O. Martínez-Matos, Y.-X. Ren, et al., “Dual behavior of caustic optical beams facing obstacles,” Phys. Rev. A: At., Mol., Opt. Phys. 95(6), 063838 (2017). [CrossRef]
27. M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, et al., “Self-healing property of a caustic optical beam,” Appl. Opt. 46(34), 8284–8290 (2007). [CrossRef]
28. L.-A. Wu, G. L. Long, Q. Gong, et al., “Optics in ancient China,” AAPPS Bull. 25, 6–10 (2015).
29. S. Chávez-Cerda, “A new approach to bessel beams,” J. Mod. Opt. 46(6), 923–930 (1999). [CrossRef]
30. S. Cunzhi, J. Pu, and S. Chávez-Cerda, “Elegant Cartesian Laguerre-Hermite-Gaussian laser cavity modes,” Opt. Lett. 40(6), 1105–1108 (2015). [CrossRef]
31. A. Yariv and P. Yeh, Photonics, Electrical and Computer Engineering (Oxford University, 2007).
32. V. Lakshminarayanan and A. G. K. Thyagarajan, Lagrangian Optics (Kluwer Academic Publishers, 2002).
33. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12(10), 2434–2441 (1973). [CrossRef]
34. J. C. Gutiérrez-Vega and M. A. Bandres, “Ince–Gaussian beams in a quadratic-index medium,” J. Opt. Soc. Am. A 22(2), 306–309 (2005). [CrossRef]
35. M. Born, E. Wolf, A. B. Bhatia, et al., Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
36. M. Newstein and B. Rudman, “Laguerre-gaussian periodically focusing beams in a quadratic index medium,” IEEE J. Quantum Electron. 23(5), 481–482 (1987). [CrossRef]
37. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–Gaussian beams,” Opt. Lett. 29(2), 144–146 (2004). [CrossRef]
38. H. Lin, S. Fu, H. Yin, et al., “Intrinsic vortex–antivortex interaction of light,” Laser Photonics Rev. 16(8), 2100648 (2022). [CrossRef]
39. E. Bolduc, N. Bent, E. Santamato, et al., “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38(18), 3546–3549 (2013). [CrossRef]
