Abstract

Caustics are responsible for a wide range of natural phenomena, from rainbows and mirages to sparkling seas. Here, we present caustics in space-time wavepackets, a class of pulsed beams featuring strong coupling between spatial and temporal frequencies. Space-time wavepackets have attracted much attention with their propagation-invariant intensity profiles that travel at tunable superluminal and subluminal group velocities. These intensity profiles, however, have been largely restricted to an X-shape or similar pattern. We show that space-time caustics combine the propagation invariance of space-time wavepackets with the flexible design of caustics, allowing for customizable intensity patterns in space-time wavepackets. Our method directly provides the phase distribution needed to realize user-designed caustic patterns in space-time wavepackets. We show that space-time caustics can feature in a broad range of intriguing optical phenomena, including backward traveling caustics formed from purely forward propagating waves, and nondiffracting beams that evolve with time. Our findings should open the doors to an even wider range of structured light with spatiotemporal coupling.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Caustics in light are commonly understood as spatial patterns created by singularities in ray optics. Caustics underlie a wide variety of everyday phenomena, from the formation of rainbows to bright lines at the bottom of swimming pools [1,2,3]. They have also inspired a diversity of shaped electromagnetic waves, including the Airy beam [4,5,6], Pearcy beam [7], higher-order optical catastrophes [8] and nondiffracting beams such as Bessel, Mathieu, Weber and accelerating beams [9,10,11,12,13,14,15,16,17]. It was recently demonstrated that arbitrary caustic patterns can be inscribed in the transverse plane of a nondiffracting beam [18]. These myriad forms of structured light have led to a wealth of applications including laser micromachining [19,20], optical micromanipulation [21], and light sheet microscopy [22].

The question arises as to what happens when caustics are formed from light with underlying space-time coupling. This is an especially pertinent question given rapidly growing interest in space-time wavepackets, a class of pulsed beams featuring correlations between spatial and temporal frequencies [23,24,25,26,27,28,29,30,31,32]. Space-time wavepackets are distinctive for their propagation-invariant intensity profiles, and their tunable group velocities that span the subluminal and superluminal ranges [33,34]. The study of such wavepackets goes back to historical predictions that include Brittingham’s focus wave mode [32], Mackinnon’s wavepacket [35] and X-waves [36,37,38]. Space-time wavepackets have been shown to be potentially useful in applications such as optical buffering [39], spectroscopy and microscopy [40]. To date, however, all instances of space-time wavepackets have been restricted in their intensity profiles (usually X-shaped or similar), and no explicit method of designing the intensity profile of a space-time wavepacket has been introduced.

Here, we show that caustics with space-time coupling result in space-time wavepackets whose intensity profiles are molded by the underlying caustic design. We present a formalism that allows users to design space-time wavepackets with customized intensity profiles, and also to obtain the phase pattern needed to realize the profile. The space-time caustic can then be experimentally realized by encoding the phase pattern into the spatial light modulator or phase plate used to generate the wavepacket. Using the method of stationary phase, we obtain closed form expressions for vectorial electromagnetic wavepackets that feature space-time caustic patterns. Notably, the intensity pattern and group velocity of these wavepackets are controlled via completely independent parameters: the intensity pattern is controlled via the phase delay of constituent plane waves, whereas the group velocity is tuned via the slope of the dispersion relation. This allows for customized combinations of caustic pattern and group velocity. We show intriguing optical phenomena, including backward traveling caustics formed from purely forward propagating waves, and nondiffracting beams that evolve with time. Our investigation reveals the enormous versatility in intensity patterns available for space-time wavepacket design, which could enable new modalities in applications for which intensity profile is important, such as light-sheet microscopy.

2. Results

A caustic is a focal line or surface touched by every ray in an ensemble of rays [3]. In electromagnetism, this ensemble of rays corresponds to a wavepacket, whose field profile is naturally affected by the properties of its constituent rays. In particular, a wavepacket tends to exhibit notable features in the vicinity of its underlying caustic pattern, such as higher intensities due to constructive interference of light along those points. Before we proceed to apply caustics to space-time wavepackets, we clarify that the term “space-time ray” has been used in the past for descriptions of pulse propagation along a ray trajectory (e.g., [41,42,43]). This should not be confused with the use of “space-time wavepacket” here and in many other works (e.g., [29,30,34]), where the term refers to a specific type of wavepacket that features strong correlations between spatial and temporal frequencies.

We begin by reviewing how typical space-time wavepackets are formed. A vectorial electric field in a linear, uniform medium can be described as

$${\textbf{E}({{\textbf r},t} )= \textrm{Re}\left\{ {\smallint d\phi d\theta {\boldsymbol \; }{{\hat{{\bf \epsilon }}}_{\theta ,\phi }}A({\theta ,\phi } ){e^{\textrm{i}{{\textbf k}_{\theta ,\phi }} \cdot {\boldsymbol r} - \textrm{i}{\omega_{\theta ,\phi }}t}}} \right\},}$$
where ${\textbf r} \equiv ({x,y,z} )$ is position and t is time. The corresponding magnetic field is readily obtained from the curl relations of Maxwell’s equations. The integrand in the expression for ${\textbf E}$ represents a plane wave associated with polar angle $\theta $ and azimuthal angle $\phi $, possessing angular frequency ${\omega _{\theta ,\phi }}$, wavevector ${{\textbf k}_{\theta ,\phi }}$, complex amplitude $A({\theta ,\phi } )$ and polarization unit vector ${\hat{{\bf \epsilon }}_{\theta ,\phi }}$. Maxwell’s equations require that ${\omega _{\theta ,\phi }} = |{{{\textbf k}_{\theta ,\phi }}} |c$, c being the speed of light, and ${{\textbf k}_{\theta ,\phi }} \cdot {\hat{{\bf \epsilon }}_{\theta ,\phi }} = 0$. Space-time coupling occurs when additional constraints in the dispersion relation beyond ${\omega _{\theta ,\phi }} = |{{{\textbf k}_{\theta ,\phi }}} |c$ are imposed, reducing the degrees of freedom available in the choice of ${{\textbf k}_{\theta ,\phi }}$ for a given ${\omega _{\theta ,\phi }}$. For instance, to create propagation-invariant wavepackets of group velocity ${v_g}$, one uses a linear spatiotemporal relationship ${\omega _{\theta ,\phi }} = {v_g}{k_{z;\theta ,\phi }} + {\omega _d}$ [33,34]. Simply setting $A({\theta ,\phi } )= 1$, as is often done, gives us the well-known X-shaped (or similar) profile that has come to be associated with many space-time wavepackets.

In this paper, we consider the case where $A({\theta ,\phi } )= {e^{\textrm{i}\varPhi ({\theta } )}}\delta (\phi )+ {e^{\textrm{i}\varPhi ({ - {\theta }} )}}\delta ({\phi - \pi } )$ in Eq. (1) ($\delta $ being the Dirac delta distribution), implying the wavepacket does not contain ${k_y} \ne 0$ components (i.e., is invariant in $y$), and a linear spatiotemporal relationship ${\omega _{\theta ,\phi }} = {v_g}{k_{z;\theta ,\phi }} + {\omega _d}$. Under these conditions, we simplify Eq. (1) to

$${\textbf{E}({{\textbf r},t} )= \textrm{Re}\left\{ {\smallint d{k_x}{\boldsymbol \; }{{\hat{{\bf \epsilon }}}_{{k_x}}}{e^{\textrm{i}{k_0}(\theta )({x\textrm{sin}\theta + z\textrm{cos}\theta - ct} )+ \textrm{i}\varPhi ({\textrm{sgn}({{k_x}} ){\theta }} )}}} \right\},\; \; \; ({\textrm{finite}\; {v_g}} ),}$$
where the variable $\theta = \theta ({{k_x}} )$ is a function of ${k_x}$ and $\textrm{sgn}({{k_x}} )$ is the sign of ${k_x}$. Specifically, ${k_x} \equiv {k_0}(\theta )\sin \theta $, ${k_0}(\theta )\equiv {k_d}{({1 - {\beta_g}\cos \theta } )^{ - 1}}$, ${k_d} \equiv {\omega _d}/c$ and ${\beta _g} \equiv {v_g}/c$. To evaluate Eq. (2), the integral over ${k_x}$ can be transformed into an integral over $\theta $. For the transverse magnetic (TM) mode, ${\hat{{\bf \epsilon }}_{{k_x}}} = \hat{{\textbf x}}\cos \theta - \textrm{sgn}({{k_x}} )\hat{{\textbf z}}\sin \theta $; For the transverse electric (TE) mode, ${\hat{{\bf \epsilon }}_{{k_x}}} = \hat{{\textbf y}}$. Note that the full range of $\; \theta $ is not needed to generate a caustic. In fact, one may note that all examples in the main text do not use the full range of $\theta $. To obtain an analytical formula for the caustic corresponding to phase delay $\varPhi $ (and a given range of $\theta $ that corresponds to the frequency bandwidth in question), we apply the method of stationary phase [44] on the integrand in Eq. (2):
$$\begin{aligned} {\frac{\partial }{{\partial \theta }}\chi (\theta )}&= 0,\\ {\frac{{{\partial ^2}}}{{\partial {\theta ^2}}}\chi (\theta )}&= 0,\\ \chi (\theta )\equiv {k_0}(\theta )({x\textrm{sin}\theta + z\textrm{cos}\theta - ct} )&+ \varPhi ({\textrm{sgn}({{k_x}} )\theta } ),\; \; \; ({\textrm{finite}\; {v_g}} ),\end{aligned}$$

We then evaluate Eq. (3) for x and z to obtain the following equation for a caustic (${x_c},{z_c}$) inscribed by phase $\varPhi (\theta )$ in the x-z plane at time $t$:

$$\begin{aligned} {x_c} &={{-} k_d^{ - 1}[{\varPhi^{\prime}(\theta)({\textrm{cos}\theta - {\beta_g}({1 + {{\sin }^2}\theta } )} )+ {\varPhi }^{\prime{\prime}}(\theta)({{\beta_g}\textrm{cos}\theta - 1} )\textrm{sin}\theta \; ],\; \;} }\\ {z_c} &= {k_d^{ - 1}[{\varPhi^{\prime}(\theta)({1 - 2\beta_\textrm{g}^2 + {\beta_g}cos\theta } )\textrm{sin}{\theta } + {\varPhi }^{\prime{\prime}}(\theta)({1 - {\beta_g}\textrm{cos}{\theta }} )({\textrm{cos}\theta - {\beta_g}} )} ]+ {v_g}t,\; \; ({\textrm{finite}\; {v_g}} )} \end{aligned}$$
where ${k_d} = {\omega _d}/c$, ${\beta _g} = {v_g}/c$, and a prime denotes a derivative with respect to $\theta $. For the purposes of plotting the caustic, the values of $\theta $ corresponding to $\phi = \pi $ should be given an artificial negative sign in Eq. (4). While this procedure bears similarities to the application of the method of stationary phase in Ref. [18] for caustics in the transverse plane of a nondiffracting wave, one key difference is the $\theta $-dependence of ${k_0}(\theta )$ in our case, which was absent from the scenarios considered in Ref. [18], but required for the description of a space-time caustic.

Equation (4) provides a way to determine the phase delay $\varPhi $ needed to realize a user-designed caustic pattern (${x_c},{z_c}$). While one may realize a wide range of caustic patterns, we note that Eq. (4) also imposes restrictions on the possible patterns. By solving Eq. (4) for $\varPhi ^{\prime}$ and $\varPhi ^{\prime{\prime}}$, and requiring that ${({\varPhi^{\prime}} )^{\prime}} = \varPhi ^{\prime{\prime}}$, we obtain (at a given $t$)

$${\sin \theta z_c^{\prime} = ({\cos \theta - {\beta_g}} )x_c^{\prime},}$$
which automatically yields ${z_c}(\theta )$ after ${x_c}(\theta )$ has been defined, or vice versa. Equation (5) is equivalent to the following constraint on the shape of caustic (at a given $t$):
$${\frac{{\textrm{d}{x_c}}}{{\textrm{d}{z_c}}} = \frac{{\sin \theta }}{{({\cos \theta - {\beta_g}} )}}\; .}$$

In other words, the slope in the resulting caustic is uniquely determined by the parameter $\theta $. A caustic pattern $({{x_c},{z_c}} )$ can be realized provided ${x_c}$ and ${z_c}$ are parametrized in $\theta $ such that Eq. (6) is satisfied, whereupon we obtain $\varPhi $ by solving

$${\varPhi ^{\prime} = \frac{{({{\beta_g} - \cos \theta } ){x_c} + ({{z_c} - {v_g}t} )\textrm{sin}\theta }}{{{{({{\beta_g}\textrm{cos}\theta - 1} )}^2}}}.}$$

Figure 1 shows the evolution of a space-time caustic traveling at subluminal group velocity ${v_g} = 0.02c$, c being the speed of light in free space. The intensity pattern is shown in Fig. 1(a), and the corresponding electric fields for the transverse magnetic (TM) and transverse electric (TE) modes shown in Figs. 1(b)-1(d). Throughout this paper, intensity is defined as the magnitude of the Poynting vector. Figures 1(e) and 1(f) show the dispersion relation. The linear relation between $\omega $ and ${k_z}$ underlies the tunable group velocity of propagation-invariant space-time wavepackets [33,34], and is also the reason behind the group velocity of the traveling caustic here. The shape of the caustic itself is determined by the phase profile $\varPhi $, plotted as a function of polar angle and wavelength in Figs. 1(g) and 1(h). In this case, we choose the phase delay $\varPhi ={-} 10{\sin ^2}({2\theta } )$, simply to give a star-like (as opposed to X-shape) space-time caustic as an example. Using this phase delay in Eq. (4) yields the predicted caustic (${x_c}$, ${z_c}$) at time t, which we plot in cyan lines in Fig. 1(a), showing the excellent agreement between theoretical prediction and the corresponding fully vectorial electromagnetic wavepacket. In this example, the wavelength range is 777 nm – 792 nm, well within bandwidths that have been experimentally achieved in shaping light with transmissive phase plates [45].

 figure: Fig. 1.

Fig. 1. Space-time caustics of light. (a) shows the intensity profile of a subluminal propagation-invariant space-time caustic of group velocity ${v_g} = 0.02c$, c being the speed of light in free space, at different instances in time. The caustic pattern predicted by Eq. (4) is overlain on the $t = 0$ instance in cyan lines, showing excellent agreement with the actual vectorial electromagnetic wavepacket. (b) and (c) show the electric fields (${E_y} = 0$ everywhere) of the transverse magnetic (TM) mode version corresponding to the intensity pattern in (a); and (d) shows the electric field (${E_{x,z}} = 0$ everywhere) of the transverse electric (TE) mode corresponding to the intensity pattern in (a). (e) and (f) show the dispersion relation and full frequency range of the space-time caustic. Notably the wavepacket is made up of only forward propagating (${k_z} > 0$) waves. Dashed black lines indicate the light cone. (g) and (h) show the phase delay of the constituent plane waves as a function of polar angle (negative polar angle corresponds to $\phi = \pi $) and wavelength respectively. Note that the range of $\; \omega $ in (e,f) exactly corresponds to the range of $\lambda $ in (h). The precise caustic pattern can be tuned through the phase profile, as we show in Fig. 2.

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Figure 1 reveals an important feature of space-time caustics: the intensity profile and group velocity are separately controlled via the phase profile and dispersion relation respectively. Indeed, the linear $\omega $-${k_z}$ dispersion relation in Figs. 1(e) and 1(f) is exactly what has been studied in previous works on space-time wavepackets, with the slope of this $\; \omega $-${k_z}$ curve being group velocity ${v_g}$ [33,34]. This is not surprising since the linear $\omega $-${k_z}$ dispersion relation underlies propagation-invariance in this and previous works. The intensity profile here, however, is dramatically different from the X-shaped profiles studied before, and is not affected by the dispersion relation. Instead, the intensity profile is independently determined though the choice of $\varPhi $. This highlights the possibility of realizing customized intensity profiles through additional phase engineering of space-time wavepackets, in a way that does not affect the propagation invariance or the group velocity.

To exemplify how different space-time caustics can be obtained for the exact same dispersion relation and frequency range used in Fig. 1, Fig. 2 shows the variety of intensity profiles achievable with different phase delays $\varPhi $. Figures 2(a), 2(c), and 2(e) correspond to the expressions $\varPhi ={-} 25\textrm{sin}({2\theta } )$ (illustrated in Fig. 2(b)), $\varPhi = 50\theta $ (Fig. 2(d)) and $\varPhi ={-} 4\theta - 5\textrm{sin}({4\theta } )$ (Fig. 2(f)) respectively. Just like Fig. 1 g, Figs. 2(b), 2(d), and 2(f) show that very different intensity profiles can be produced through a relatively gradual variation in phase over the entire bandwidth.

 figure: Fig. 2.

Fig. 2. Tailoring space-time caustics through the phase delay. (a), (c), (e) show intensity patterns based on the exact same dispersion relation and frequency range as in Fig. 1 (i.e. Figures 1(e) and 1(f) also apply to these intensity patterns), but with different phase delay profiles $\varPhi $ reflected in (b), (d), (f) respectively. The caustic pattern predicted by Eq. (4) is overlain on the $t = 0$ instances in cyan lines. These caustics propagate at the same group velocity ${v_g} = 0.02c$ as the Fig. 1 caustic but are markedly different from one another, showing that the caustic pattern can be controlled independently of the group velocity.

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Figure 3 shows the robustness of using caustics to design unique intensity profiles in space-time wavepackets. The examples of space-time caustics in Fig. 3 comprise a propagation-invariant caustic traveling at subluminal group velocity ${v_g} = 0.96c$ in Fig. 3(a); a caustic traveling at superluminal group velocity ${v_g} = 1.2c$ in Fig. 3(b); and a backward-traveling superluminal caustic whose group velocity ${v_g} ={-} 1.2c$ in Fig. 3(c). It is noteworthy that although the wavepacket is made up of only forward-propagating (i.e., ${k_z} > 0$) plane waves, the negative slope of the space-time relationship in Fig. 3(c)(ii) results in an intensity pattern that travels in the backwards direction. These examples show that space-time caustics are possible for group velocities beyond the very subluminal velocity studied in Figs. 1 and 2. The phase delay functions and wavelength ranges used in these examples are $\varPhi = 50\theta ^{\prime} - 12.5{\sin ^3}({2\theta^{\prime}} )$, from wavelength 500 nm to 717 nm for Fig. 3(a); $\varPhi = 20\theta ^{\prime} - 10{\sin ^3}({3\theta^{\prime}} )$, from wavelength 615 nm to 754 nm for Fig. 3(b); and $\varPhi = 75\theta ^{\prime} - 37.5{\sin ^3}({4\theta^{\prime}} )$, from wavelength 500 nm to 746 nm for Fig. 3(c). In the foregoing expressions, $\theta ^{\prime} = \arctan [{{\gamma^{ - 1}}\textrm{sin}\theta /({\cos \theta - \beta } )} ]$, $\beta = {v_g}/c$ for subluminal group velocities and $\beta = c/{v_g}$ for superluminal velocities, and $\gamma = {({1 - {\beta^2}} )^{ - 1/2}}$. The wavelength ranges used in Fig. 3 fall within the bandwidth of experimentally demonstrated ultra-broadband spatial light modulators [46].

 figure: Fig. 3.

Fig. 3. Superluminal, subluminal and backward-traveling space-time caustics. (a)(i) shows the intensity profiles for a subluminal (${v_g} = 0.96c$) space-time caustic at times 0 ps, 0.56 ps, 1.1 ps (in the order indicated by the white arrows). The earliest instance is overlain with cyan lines that correspond to the caustic predicted using Eq. (4), showing excellent agreement with the actual vectorial electromagnetic wavepacket. Here, the intensity was calculated from the fields of a transverse magnetic mode, although a transverse electric mode would also have given the same intensity pattern. (a)(ii) shows the linear relationship between angular frequency $\omega $ and longitudinal wavevector component ${k_z}$. The slope of this relationship gives the group velocity of the wavepacket. (a)(iii) shows the phase delay as a function of wavelength. The descriptions for the panels in (b) and (c) are exactly as in (a) except that they correspond to a superluminal forward-traveling wavepacket (${v_g} = 1.2c$) and a superluminal backward-traveling wavepacket (${v_g} ={-} 1.2c$) respectively. Note from (ii) that all wavepackets here – including the backward-traveling example in (c) – are made of purely forward propagating (${k_z} > 0$) waves. This shows that backward-traveling space-time wavepackets are also amenable to design using caustics.

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The space-time caustics we have considered so far travel at finite group velocities and comprise a range of longitudinal wavevector components ${k_z}$. In the limit where the wavepacket is formed from a single ${k_z}$, we obtain time-diffracting wavepackets, a subclass of space-time wavepackets that attracted interest due to their potential to realize needle-like or sheet-like nondiffracting wavepackets [25,26,27]. Made up of multi-frequency plane waves having the same (or having a narrow spread in) longitudinal spatial frequency ${k_z}$, the needle-like nondiffracting wavepacket features a long, narrow intensity hotspot along the longitudinal axis with high transverse confinement [26,27]. The sheet-like nondiffracting wavepacket refers to a two-dimensional version of the needle (with confinement in one transverse direction instead of both). In Fig. 4, we see that patterning the phase delay of the constituent plane waves allows for robust waveshaping beyond the design of needle-like or sheet-like wavepackets. Specifically, we achieve an evolving wavepacket featuring a single light sheet at an earlier time (Fig. 4(b)) and dual light sheet at a later time (Fig. 4(c)). Obtaining the caustic expression this time requires separate consideration from Eq. (4), since the slope of the space-time relationship (Fig. 4(d)) is infinite ($|{{v_g}} |\to \infty $). In this limiting case, which corresponds to the case of a temporally diffracting wavepacket (all components share the same ${k_z}$), we simplify Eq. (1) to

$${\textbf{E}({{\textbf r},t} )= \textrm{Re}\left\{ {\smallint d{k_x}{\boldsymbol \; }{{\hat{{\bf \epsilon }}}_{{k_x}}}{e^{\textrm{i}{k_z}({x\textrm{tan}\theta + z - ct\textrm{sec}\theta } )+ \textrm{i}\varPhi ({\theta } )}}} \right\},\; \; ({|{{v_g}} |\to \infty } ),}$$
where ${k_x} \equiv {k_z}\textrm{tan}\theta $ and $A({\theta ,\phi } )= {A_1}({\theta ,\phi } )\equiv {e^{\textrm{i}\varPhi ({\theta } )}}[{\delta (\phi )+ \delta ({\phi - \pi } )} ]$ in Eq. (1). We then apply Eq. (3), but with $\chi (\theta )\equiv {k_z}({x\textrm{tan}\theta + z - ct\textrm{sec}\theta } )+ \varPhi ({\theta } )$, to directly obtain
$$\begin{aligned}{x_c} &= {\textrm{sgn}({{k_x}} ){{({2{k_z}} )}^{ - 1}}[{({\cos ({2\mathrm{\theta }} )- 3} )\varPhi^{\prime} + \textrm{sin}2\theta \varPhi^{\prime{\prime}}} ],}\\ c{t_c} &= {{{({2{k_z}} )}^{ - 1}}[{ - 4\textrm{sin}\theta \varPhi^{\prime} + 2\textrm{cos}\theta \varPhi ^{\prime{\prime}}} ],\; \; ({|{{v_g}} |\to \infty } ),} \end{aligned}$$
where $\textrm{sgn}({{k_x}} )$ refers to the sign of ${k_x}$. In Fig. 4, the phase delay is given by $\varPhi ={-} 25{\sin ^{3/2}}({2\theta } )$, in wavelength range 545 nm – 770 nm. The prediction of Eq. (9) is overlain on the wavepacket intensity of Fig. 4(a) in cyan dashed lines, showing excellent agreement between the stationary phase approximation and the actual electromagnetic intensity pattern. Equation (9) provides a way to determine the phase delay $\varPhi $ needed to realize a customized user-designed caustic pattern (${x_c},{t_c}$). Equation (9) also provides constraints on the achievable caustic pattern. In particular, we find that a parametrization for ${x_c}$ and ${t_c}$ in $\theta $ must be found such that the following equation is satisfied:
$${\frac{{d{x_c}}}{{dc{t_c}}} ={\pm} \sin \theta ,}$$
whereupon we obtain $\varPhi $ by solving
$${\varPhi ^{\prime} = {k_z}\frac{{{x_c} + c{t_c}\sin \theta }}{{{{\sin }^2}\theta - 1}}.}$$

 figure: Fig. 4.

Fig. 4. Time-diffracting propagation-invariant caustics. (a) shows the intensity profile of a time-diffracting caustic as a function of x and t, with the caustic predicted by Eq. (4) overlain in dashed cyan lines. Note that the evolution of the intensity profile is in time, making the wavepacket propagation invariant in $z\; $ at any instant of t, as we see at different snapshots in time (b,c). (d) shows the dispersion relation, corresponding to a vertical line in the $\omega $-${k_z}$ plane. (f) shows the phase delay as a function of wavelength. Note that the range of $\; \omega $ in (d,e) exactly corresponds to the range of $\lambda $ in (f). This exemplifies how caustics can be used to design the evolution of time-diffracting propagation-invariant beams, resulting for instance in a single beam at an earlier time (b) splitting into dual beams at a later time (c).

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The examples in Figs. 14 were chosen with the bandwidth of realistic light sources in mind. Even with narrower bandwidths, space-time caustics can be realized, albeit with a truncated shape compared to what we have shown. To visualize the change in the caustic pattern, we show how different $\theta $ correspond to different parts of our presented caustics in Fig. 5. The precise caustic pattern can thus be controlled by varying the range of $\theta $ in the wavepacket. To complement the information presented on the wavepacket dispersion, we have also included plots of $\omega $, ${k_x}\; $ and ${k_z}$ as a function of $\theta $ in Fig. 5.

 figure: Fig. 5.

Fig. 5. Controlling caustic patterns via angular range. The precise caustic pattern can be controlled by varying the range of $\theta $ included in the wavepacket. (a-d) show the caustic patterns of the examples in Figs. 1, (2a,b), (2c,d), (2e,f), respectively. The blue curves correspond to ${k_x} > 0$ and the red curves to ${k_x} < 0$. Their shared plots of ${k_x}$, ${k_z}$ and $\omega /c$ are shown in (e) for reference. Similarly, (f,g), (h,i), (j,k) and (l,m) belong to the examples in Figs. 3(a), 3(b), 3(c) and 4 respectively. Circular markers along the caustic patterns denote where the pattern would terminate if $\theta $ were to be terminated at that value. In (a,b,c,d), the markers, in order of progression indicated by the arrows, indicate $\theta = \{{0,0.53,1.05,1.57} \}$ rad. In (f), $\theta = \{{0,0.065,0.13,0.19} \}$ rad. In (h), $\theta = \{{0,0.083,0.17,0.25} \}$ rad. In (j), $\theta = \{{0,0.39,0.78,1.17} \}$ rad. In (l), $\theta = \{{7.9 \times {{10}^{ - 4}},0.26,0.52,0.79} \}$ rad.

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3. Discussion

Space-time caustics can be experimentally realized by modifying the phase pattern in the spatial light modulator (SLM) or phase plate in existing setups for creating space-time wavepackets, according to the prescription of $\varPhi $ [34,40,45]. Conventional space-time wavepackets can be realized through two essential steps [34]: 1) using a diffractive grating to map each temporal frequency of the multi-frequency input to a unique spatial position (along one spatial dimension, say $x$); and 2) using an SLM or phase plate to assign the desired spatial frequency to each of these temporal frequencies, thereby enforcing the required space-time coupling. It should be feasible to realize space-time caustics via a similar procedure, where the main modification lies in the required pattern for the SLM or phase plate. This pattern (for space-time caustics) would differ from that for conventional space-time wavepackets in featuring an additional phase modulation – determined by the frequency-dependent phase delay $\varPhi $ – along the spatial dimension to which the temporal frequency has been mapped (x in this example). Further development in waveshaping – for instance, freeform refracting surfaces [47] – will lead to still more flexibility in realizing space-time caustics.

A convenient fact to note in the design of space-time caustics is that space-time caustics that travel at different group velocities are related via the Lorentz boost, as is the case for the conventional space-time wavepacket [33]. Specifically, all subluminal space-time caustics may be obtained as the Lorentz boost of a monochromatic (i.e. spatially diffracting) caustic; and all superluminal space-time caustics may be obtained as the Lorentz boost of a temporally diffracting caustic, namely, one that comprises a single ${k_z}$, such as the example in Fig. 4.

To see this, consider first the monochromatic caustic, where all components share the same temporal angular frequency $\omega ^{\prime}$. By definition, the group velocity is zero in this case (since the dispersion relation in $\omega $ vs ${k_z}$ has zero slope). The phase of the constituent plane waves (a.k.a., the integrand in Eq. (2)), is given by $\chi ^{\prime}({\theta^{\prime}} )\equiv ({\omega^{\prime}/c} )({x^{\prime}\textrm{sin}\theta^{\prime} + z^{\prime}\textrm{cos}\theta^{\prime} - ct^{\prime}} )+ \varPhi ^{\prime}({\textrm{sgn}({k_{x}^{\prime}} )\mathrm{\theta }{^{\prime}}} )$, where primed variables denote the frame of reference traveling at speed ${v_g}$ in the + z direction with respect to the observation frame. Noting the Lorentz transform relations $z^{\prime} = \gamma ({z - {v_g}t} )$, $t^{\prime} = \gamma ({t - {v_g}z/{c^2}} )$, ${\beta _g} \equiv {v_g}/c$, $\gamma \equiv {({1 - \beta_g^2} )^{ - 1/2}}$, and the Lorentz invariance of $\chi $, we find that $\chi (\theta )= \chi ^{\prime}({\theta^{\prime}} )= {k_0}(\theta )({x\textrm{sin}\theta + z\textrm{cos}\theta - ct} )+ \varPhi ({\textrm{sgn}({{k_x}} )\mathrm{\theta }} )$, where

$$\begin{aligned} \cos \theta &= {\frac{{\cos \theta ^{\prime} + {\beta _g}}}{{1 + {\beta _g}\cos \theta ^{\prime}}},}\\ \sin \theta &= {\frac{{\sin \theta ^{\prime}}}{{\gamma ({1 + {\beta_g}\cos \theta^{\prime}} )}},}\\ \omega &= {\omega ^{\prime} + {v_g}{k_z},\; \; ({{v_g} < c} ),} \end{aligned}$$
and $\varPhi (\theta )= \varPhi ^{\prime}({\theta^{\prime}} )\; $. Note that the last line of Eq. (12) is simply the space-time correlation for a space-time wavepacket or caustic traveling at group velocity ${v_g}$, with ${\omega _\textrm{d}} = \omega ^{\prime}$, where $\omega ^{\prime}$ was a constant in the rest frame. This shows that space-time caustics traveling at non-zero group velocities can be obtained through the Lorentz transform of a monochromatic caustic. However, since the Lorentz transform only applies to subluminal velocities, only subluminal space-time caustics can be achieved this way.

To obtain superluminal space-time caustics, we can begin with a temporally diffracting caustic, which is exactly what is presented in Eq. (8) above. Treating Eq. (8) as the solution in the rest frame (primed variables), which is traveling at velocity ${c^2}/{v_g}$ in the + z direction with respect to the observation frame (unprimed variables), we may apply the Lorentz transform to obtain

$$\begin{aligned} \cos \theta &= {\frac{{\cos \theta ^{\prime} + \beta _g^{ - 1}}}{{1 + \beta _g^{ - 1}\cos \theta ^{\prime}}},}\\ \sin \theta &= {\frac{{\sin \theta ^{\prime}}}{{{\gamma _s}({1 + \beta_g^{ - 1}\cos \theta^{\prime}} )}},}\\ \omega &={{-} \frac{{k_z^{\prime}{v_g}}}{{{\gamma _s}}} + {v_g}{k_z},\; \; ({{v_g} > c} ),} \end{aligned}$$
with $\varPhi (\theta )= \varPhi ^{\prime}({\theta^{\prime}} )\; $ and ${\gamma _s} \equiv {({1 - \beta_g^{ - 2}} )^{ - 1/2}}$. Note that the last line of Eq. (13) is simply the space-time correlation for a space-time wavepacket or caustic traveling at group velocity ${v_g}$, with ${\omega _{\textrm{d}}} ={-} k_z^{\prime}{v_g}/{\gamma _s}$, where $k_z^{\prime}$ was a constant in the rest frame. This shows that superluminal space-time caustics traveling at non-zero group velocities can be obtained through the Lorentz transform of a temporally diffracting wavepacket.

In conclusion, we show that propagation-invariant space-time wavepackets can be shaped by introducing wavelength-dependent phase delays in space-time wavepackets, resulting in intensity profiles molded by the underlying caustic design. Using the method of stationary phase, we obtained closed form expressions for the design of these space-time caustics that are in excellent agreement with the resulting exact, vectorial electromagnetic wavepacket. This enables the design of customized space-time wavepackets beyond the typical X-shaped intensity patterns. Notably, the intensity pattern and group velocity are controlled via completely independent parameters – the intensity pattern via the phase delay of constituent plane waves, and the group velocity via the slope of the dispersion relation – allowing for customizable combinations of caustic pattern and group velocity. We show intriguing space-time caustic phenomena, including backward traveling caustics formed from purely forward propagating waves, and nondiffracting beams that evolve with time. Our study extends the concept of optical caustics to the space-time domain, and reveals the promise of caustics in designing new kinds of space-time wavepackets.

Funding

Agency for Science, Technology and Research (A1984c0043); Nanyang Technological University.

Acknowledgments

This work was supported by the Agency for Science, Technology and Research (A*STAR) Science & Engineering Research Council (Grant No. A1984c0043), and the Nanyang Assistant Professorship Start-up grant.

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. M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A: Math. Gen. 8(4), 566–584 (1975). [CrossRef]  

2. M. Berry, “Making waves in physics,” Nature 403(6765), 21 (2000). [CrossRef]  

3. M. V. Berry, “Nature’s optics and our understanding of light,” Contemp. Phys. 56, 1–15 (2014). [CrossRef]  

4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]  

5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]  

6. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]  

7. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012). [CrossRef]  

8. A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19(5), 053004 (2017). [CrossRef]  

9. J. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651 (1987). [CrossRef]  

10. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef]  

11. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef]  

12. J. Turunen and A. T. Friberg, “Chapter 1 - Propagation-Invariant Optical Fields,” Prog. Opt. 54, 1–88 (2010). [CrossRef]  

13. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. 108(16), 163901 (2012). [CrossRef]  

14. L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19(17), 16455–16465 (2011). [CrossRef]  

15. P. Rose, M. Boguslawski, and M. C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14(3), 033018 (2012). [CrossRef]  

16. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012). [CrossRef]  

17. I. Julián-Macías, C. Rickenstorff-Parrao, O. D. J. Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, P. Ortega-Vidals, G. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustics associated with Mathieu beams,” J. Opt. Soc. Am. A 35(2), 267–274 (2018). [CrossRef]  

18. A. Zannotti, C. Denz, M. A. Alonso, and M. R. Dennis, “Shaping caustics into propagation-invariant light,” Nat. Commun. 11(1), 3597 (2020). [CrossRef]  

19. M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012). [CrossRef]  

20. F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams: Invited paper for the section: Hot topics in Ultrafast Lasers,” Opt. Laser Technol. 80, 125–137 (2016). [CrossRef]  

21. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]  

22. J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018). [CrossRef]  

23. S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004). [CrossRef]  

24. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69(3), 036612 (2004). [CrossRef]  

25. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24(25), 28659–28668 (2016). [CrossRef]  

26. K. J. Parker and M. A. Alonso, “Longitudinal iso-phase condition and needle pulses,” Opt. Express 24(25), 28669–28677 (2016). [CrossRef]  

27. L. J. Wong and I. Kaminer, “Abruptly Focusing and Defocusing Needles of Light and Closed-Form Electromagnetic Wavepackets,” ACS Photonics 4(5), 1131–1137 (2017). [CrossRef]  

28. M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017). [CrossRef]  

29. H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-Pulse-Front Space-Time Wave Packets,” ACS Photonics 6(2), 475–481 (2019). [CrossRef]  

30. H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets having arbitrary group velocities in free space,” Nat. Commun. 10(1), 929 (2019). [CrossRef]  

31. M. A. Porras and C. Conti, “Couplings between the temporal and orbital angular momentum degrees of freedom in ultrafast optical vortices,” Phys. Rev. A 101(6), 063803 (2020). [CrossRef]  

32. J. N. Brittingham, “Focus waves modes in homogeneous Maxwell's equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983). [CrossRef]  

33. L. J. Wong and I. Kaminer, “Ultrashort Tilted-Pulse-Front Pulses and Nonparaxial Tilted-Phase-Front Beams,” ACS Photonics 4(9), 2257–2264 (2017). [CrossRef]  

34. M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time wave packets in free space: Theory and experiments,” Phys. Rev. A 99(2), 023856 (2019). [CrossRef]  

35. L. Mackinnon, “A nondispersive de Broglie wave packet,” Found Phys 8(3-4), 157–176 (1978). [CrossRef]  

36. J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(1), 19–31 (1992). [CrossRef]  

37. J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(3), 441–446 (1992). [CrossRef]  

38. P. Saari and K. Reivelt, “Evidence of X-Shaped Propagation-Invariant Localized Light Waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997). [CrossRef]  

39. M. Yessenov, B. Bhaduri, P. J. Delfyett, and A. F. Abouraddy, “Free-space optical delay line using space-time wave packets,” Nat. Commun. 11(1), 5782 (2020). [CrossRef]  

40. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space–time light sheets,” Nat. Photonics 11(11), 733–740 (2017). [CrossRef]  

41. V. M. Babich and V. V. Ulin, “Complex space-time ray method and quasiphotons,” J. Math. Sci. 24(3), 269–273 (1984). [CrossRef]  

42. L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18(2), 242–253 (1970). [CrossRef]  

43. R. M. Lewis, “Asymptotic theory of wave-propagation,” Arch. Rational Mech. Anal. 20(3), 191–250 (1965). [CrossRef]  

44. M. V. Berry and C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257–346 (1980). [CrossRef]  

45. H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018). [CrossRef]  

46. D.-M. Spangenberg, A. Dudley, P. H. Neethling, E. G. Rohwer, and A. Forbes, “White light wavefront control with a spatial light modulator,” Opt. Express 22(11), 13870–13879 (2014). [CrossRef]  

47. J. DelOlmo-Márquez, G. Castillo-Santiago, M. Avendaño-Alejo, I. Moreno, E. Román-Hernández, and M. C. López-Bautista, Opt. Express 29(15), 23300–23314 (2021).

References

  • View by:

  1. M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A: Math. Gen. 8(4), 566–584 (1975).
    [Crossref]
  2. M. Berry, “Making waves in physics,” Nature 403(6765), 21 (2000).
    [Crossref]
  3. M. V. Berry, “Nature’s optics and our understanding of light,” Contemp. Phys. 56, 1–15 (2014).
    [Crossref]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [Crossref]
  5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref]
  6. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [Crossref]
  7. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
    [Crossref]
  8. A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19(5), 053004 (2017).
    [Crossref]
  9. J. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651 (1987).
    [Crossref]
  10. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [Crossref]
  11. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004).
    [Crossref]
  12. J. Turunen and A. T. Friberg, “Chapter 1 - Propagation-Invariant Optical Fields,” Prog. Opt. 54, 1–88 (2010).
    [Crossref]
  13. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
    [Crossref]
  14. L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19(17), 16455–16465 (2011).
    [Crossref]
  15. P. Rose, M. Boguslawski, and M. C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14(3), 033018 (2012).
    [Crossref]
  16. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
    [Crossref]
  17. I. Julián-Macías, C. Rickenstorff-Parrao, O. D. J. Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, P. Ortega-Vidals, G. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustics associated with Mathieu beams,” J. Opt. Soc. Am. A 35(2), 267–274 (2018).
    [Crossref]
  18. A. Zannotti, C. Denz, M. A. Alonso, and M. R. Dennis, “Shaping caustics into propagation-invariant light,” Nat. Commun. 11(1), 3597 (2020).
    [Crossref]
  19. M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
    [Crossref]
  20. F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams: Invited paper for the section: Hot topics in Ultrafast Lasers,” Opt. Laser Technol. 80, 125–137 (2016).
    [Crossref]
  21. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
    [Crossref]
  22. J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
    [Crossref]
  23. S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12(5), 935–940 (2004).
    [Crossref]
  24. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69(3), 036612 (2004).
    [Crossref]
  25. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24(25), 28659–28668 (2016).
    [Crossref]
  26. K. J. Parker and M. A. Alonso, “Longitudinal iso-phase condition and needle pulses,” Opt. Express 24(25), 28669–28677 (2016).
    [Crossref]
  27. L. J. Wong and I. Kaminer, “Abruptly Focusing and Defocusing Needles of Light and Closed-Form Electromagnetic Wavepackets,” ACS Photonics 4(5), 1131–1137 (2017).
    [Crossref]
  28. M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017).
    [Crossref]
  29. H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-Pulse-Front Space-Time Wave Packets,” ACS Photonics 6(2), 475–481 (2019).
    [Crossref]
  30. H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets having arbitrary group velocities in free space,” Nat. Commun. 10(1), 929 (2019).
    [Crossref]
  31. M. A. Porras and C. Conti, “Couplings between the temporal and orbital angular momentum degrees of freedom in ultrafast optical vortices,” Phys. Rev. A 101(6), 063803 (2020).
    [Crossref]
  32. J. N. Brittingham, “Focus waves modes in homogeneous Maxwell's equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
    [Crossref]
  33. L. J. Wong and I. Kaminer, “Ultrashort Tilted-Pulse-Front Pulses and Nonparaxial Tilted-Phase-Front Beams,” ACS Photonics 4(9), 2257–2264 (2017).
    [Crossref]
  34. M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time wave packets in free space: Theory and experiments,” Phys. Rev. A 99(2), 023856 (2019).
    [Crossref]
  35. L. Mackinnon, “A nondispersive de Broglie wave packet,” Found Phys 8(3-4), 157–176 (1978).
    [Crossref]
  36. J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(1), 19–31 (1992).
    [Crossref]
  37. J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(3), 441–446 (1992).
    [Crossref]
  38. P. Saari and K. Reivelt, “Evidence of X-Shaped Propagation-Invariant Localized Light Waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
    [Crossref]
  39. M. Yessenov, B. Bhaduri, P. J. Delfyett, and A. F. Abouraddy, “Free-space optical delay line using space-time wave packets,” Nat. Commun. 11(1), 5782 (2020).
    [Crossref]
  40. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space–time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
    [Crossref]
  41. V. M. Babich and V. V. Ulin, “Complex space-time ray method and quasiphotons,” J. Math. Sci. 24(3), 269–273 (1984).
    [Crossref]
  42. L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18(2), 242–253 (1970).
    [Crossref]
  43. R. M. Lewis, “Asymptotic theory of wave-propagation,” Arch. Rational Mech. Anal. 20(3), 191–250 (1965).
    [Crossref]
  44. M. V. Berry and C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257–346 (1980).
    [Crossref]
  45. H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
    [Crossref]
  46. D.-M. Spangenberg, A. Dudley, P. H. Neethling, E. G. Rohwer, and A. Forbes, “White light wavefront control with a spatial light modulator,” Opt. Express 22(11), 13870–13879 (2014).
    [Crossref]
  47. J. DelOlmo-Márquez, G. Castillo-Santiago, M. Avendaño-Alejo, I. Moreno, E. Román-Hernández, and M. C. López-Bautista, Opt. Express 29(15), 23300–23314 (2021).

2020 (3)

A. Zannotti, C. Denz, M. A. Alonso, and M. R. Dennis, “Shaping caustics into propagation-invariant light,” Nat. Commun. 11(1), 3597 (2020).
[Crossref]

M. A. Porras and C. Conti, “Couplings between the temporal and orbital angular momentum degrees of freedom in ultrafast optical vortices,” Phys. Rev. A 101(6), 063803 (2020).
[Crossref]

M. Yessenov, B. Bhaduri, P. J. Delfyett, and A. F. Abouraddy, “Free-space optical delay line using space-time wave packets,” Nat. Commun. 11(1), 5782 (2020).
[Crossref]

2019 (3)

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time wave packets in free space: Theory and experiments,” Phys. Rev. A 99(2), 023856 (2019).
[Crossref]

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-Pulse-Front Space-Time Wave Packets,” ACS Photonics 6(2), 475–481 (2019).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets having arbitrary group velocities in free space,” Nat. Commun. 10(1), 929 (2019).
[Crossref]

2018 (3)

2017 (5)

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space–time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

L. J. Wong and I. Kaminer, “Ultrashort Tilted-Pulse-Front Pulses and Nonparaxial Tilted-Phase-Front Beams,” ACS Photonics 4(9), 2257–2264 (2017).
[Crossref]

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19(5), 053004 (2017).
[Crossref]

L. J. Wong and I. Kaminer, “Abruptly Focusing and Defocusing Needles of Light and Closed-Form Electromagnetic Wavepackets,” ACS Photonics 4(5), 1131–1137 (2017).
[Crossref]

M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017).
[Crossref]

2016 (3)

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24(25), 28659–28668 (2016).
[Crossref]

K. J. Parker and M. A. Alonso, “Longitudinal iso-phase condition and needle pulses,” Opt. Express 24(25), 28669–28677 (2016).
[Crossref]

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams: Invited paper for the section: Hot topics in Ultrafast Lasers,” Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

2014 (2)

2012 (5)

P. Rose, M. Boguslawski, and M. C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14(3), 033018 (2012).
[Crossref]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
[Crossref]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[Crossref]

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

2011 (1)

2010 (1)

J. Turunen and A. T. Friberg, “Chapter 1 - Propagation-Invariant Optical Fields,” Prog. Opt. 54, 1–88 (2010).
[Crossref]

2008 (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

2007 (2)

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref]

2004 (3)

2000 (1)

M. Berry, “Making waves in physics,” Nature 403(6765), 21 (2000).
[Crossref]

1997 (1)

P. Saari and K. Reivelt, “Evidence of X-Shaped Propagation-Invariant Localized Light Waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

1992 (2)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(1), 19–31 (1992).
[Crossref]

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(3), 441–446 (1992).
[Crossref]

1987 (2)

J. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

1984 (1)

V. M. Babich and V. V. Ulin, “Complex space-time ray method and quasiphotons,” J. Math. Sci. 24(3), 269–273 (1984).
[Crossref]

1983 (1)

J. N. Brittingham, “Focus waves modes in homogeneous Maxwell's equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

1980 (1)

M. V. Berry and C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

1978 (1)

L. Mackinnon, “A nondispersive de Broglie wave packet,” Found Phys 8(3-4), 157–176 (1978).
[Crossref]

1975 (1)

M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A: Math. Gen. 8(4), 566–584 (1975).
[Crossref]

1970 (1)

L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18(2), 242–253 (1970).
[Crossref]

1965 (1)

R. M. Lewis, “Asymptotic theory of wave-propagation,” Arch. Rational Mech. Anal. 20(3), 191–250 (1965).
[Crossref]

Abouraddy, A. F.

M. Yessenov, B. Bhaduri, P. J. Delfyett, and A. F. Abouraddy, “Free-space optical delay line using space-time wave packets,” Nat. Commun. 11(1), 5782 (2020).
[Crossref]

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time wave packets in free space: Theory and experiments,” Phys. Rev. A 99(2), 023856 (2019).
[Crossref]

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-Pulse-Front Space-Time Wave Packets,” ACS Photonics 6(2), 475–481 (2019).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets having arbitrary group velocities in free space,” Nat. Commun. 10(1), 929 (2019).
[Crossref]

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space–time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24(25), 28659–28668 (2016).
[Crossref]

Aggarwal, S.

J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
[Crossref]

Alonso, M. A.

A. Zannotti, C. Denz, M. A. Alonso, and M. R. Dennis, “Shaping caustics into propagation-invariant light,” Nat. Commun. 11(1), 3597 (2020).
[Crossref]

K. J. Parker and M. A. Alonso, “Longitudinal iso-phase condition and needle pulses,” Opt. Express 24(25), 28669–28677 (2016).
[Crossref]

Arnold, C.

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

Avendaño-Alejo, M.

J. DelOlmo-Márquez, G. Castillo-Santiago, M. Avendaño-Alejo, I. Moreno, E. Román-Hernández, and M. C. López-Bautista, Opt. Express 29(15), 23300–23314 (2021).

Babich, V. M.

V. M. Babich and V. V. Ulin, “Complex space-time ray method and quasiphotons,” J. Math. Sci. 24(3), 269–273 (1984).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Bekenstein, R.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[Crossref]

Berry, M.

M. Berry, “Making waves in physics,” Nature 403(6765), 21 (2000).
[Crossref]

Berry, M. V.

M. V. Berry, “Nature’s optics and our understanding of light,” Contemp. Phys. 56, 1–15 (2014).
[Crossref]

M. V. Berry and C. Upstill, “Catastrophe Optics: Morphologies of Caustics and Their Diffraction Patterns,” Prog. Opt. 18, 257–346 (1980).
[Crossref]

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979).
[Crossref]

M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A: Math. Gen. 8(4), 566–584 (1975).
[Crossref]

Bhaduri, B.

M. Yessenov, B. Bhaduri, P. J. Delfyett, and A. F. Abouraddy, “Free-space optical delay line using space-time wave packets,” Nat. Commun. 11(1), 5782 (2020).
[Crossref]

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time wave packets in free space: Theory and experiments,” Phys. Rev. A 99(2), 023856 (2019).
[Crossref]

Boguslawski, M.

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19(5), 053004 (2017).
[Crossref]

P. Rose, M. Boguslawski, and M. C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14(3), 033018 (2012).
[Crossref]

Brittingham, J. N.

J. N. Brittingham, “Focus waves modes in homogeneous Maxwell's equations: Transverse electric mode,” J. Appl. Phys. 54(3), 1179–1189 (1983).
[Crossref]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Cabrera-Rosas, O. D. J.

Cannan, D.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Castillo-Santiago, G.

J. DelOlmo-Márquez, G. Castillo-Santiago, M. Avendaño-Alejo, I. Moreno, E. Román-Hernández, and M. C. López-Bautista, Opt. Express 29(15), 23300–23314 (2021).

Chávez-Cerda, S.

Chen, Z.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Christodoulides, D. N.

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-Pulse-Front Space-Time Wave Packets,” ACS Photonics 6(2), 475–481 (2019).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Conti, C.

M. A. Porras and C. Conti, “Couplings between the temporal and orbital angular momentum degrees of freedom in ultrafast optical vortices,” Phys. Rev. A 101(6), 063803 (2020).
[Crossref]

Couairon, A.

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams: Invited paper for the section: Hot topics in Ultrafast Lasers,” Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

Courvoisier, F.

F. Courvoisier, R. Stoian, and A. Couairon, “Ultrafast laser micro- and nano-processing with nondiffracting and curved beams: Invited paper for the section: Hot topics in Ultrafast Lasers,” Opt. Laser Technol. 80, 125–137 (2016).
[Crossref]

L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19(17), 16455–16465 (2011).
[Crossref]

Delfyett, P. J.

M. Yessenov, B. Bhaduri, P. J. Delfyett, and A. F. Abouraddy, “Free-space optical delay line using space-time wave packets,” Nat. Commun. 11(1), 5782 (2020).
[Crossref]

DelOlmo-Márquez, J.

J. DelOlmo-Márquez, G. Castillo-Santiago, M. Avendaño-Alejo, I. Moreno, E. Román-Hernández, and M. C. López-Bautista, Opt. Express 29(15), 23300–23314 (2021).

Dennis, M. R.

A. Zannotti, C. Denz, M. A. Alonso, and M. R. Dennis, “Shaping caustics into propagation-invariant light,” Nat. Commun. 11(1), 3597 (2020).
[Crossref]

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
[Crossref]

Denz, C.

A. Zannotti, C. Denz, M. A. Alonso, and M. R. Dennis, “Shaping caustics into propagation-invariant light,” Nat. Commun. 11(1), 3597 (2020).
[Crossref]

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19(5), 053004 (2017).
[Crossref]

Denz, M. C.

P. Rose, M. Boguslawski, and M. C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14(3), 033018 (2012).
[Crossref]

Dholakia, K.

J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
[Crossref]

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Diebel, F.

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19(5), 053004 (2017).
[Crossref]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref]

Dudley, A.

Dudley, J. M.

Duocastella, M.

M. Duocastella and C. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Durnin, J. J.

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Espíndola-Ramos, E.

Fairchild, S. R.

Felsen, L.

L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18(2), 242–253 (1970).
[Crossref]

Ferrier, D. E. K.

J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
[Crossref]

Forbes, A.

Friberg, A. T.

J. Turunen and A. T. Friberg, “Chapter 1 - Propagation-Invariant Optical Fields,” Prog. Opt. 54, 1–88 (2010).
[Crossref]

Froehly, L.

Furfaro, L.

Giust, R.

Greenleaf, J. F.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(1), 19–31 (1992).
[Crossref]

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(3), 441–446 (1992).
[Crossref]

Gunn-Moore, F. J.

J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
[Crossref]

Gutiérrez-Vega, J. C.

Hu, Y.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Jacquot, M.

Juárez-Reyes, S. A.

Julián-Macías, I.

Kaminer, I.

L. J. Wong and I. Kaminer, “Ultrashort Tilted-Pulse-Front Pulses and Nonparaxial Tilted-Phase-Front Beams,” ACS Photonics 4(9), 2257–2264 (2017).
[Crossref]

L. J. Wong and I. Kaminer, “Abruptly Focusing and Defocusing Needles of Light and Closed-Form Electromagnetic Wavepackets,” ACS Photonics 4(5), 1131–1137 (2017).
[Crossref]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[Crossref]

Kondakci, H. E.

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-Pulse-Front Space-Time Wave Packets,” ACS Photonics 6(2), 475–481 (2019).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets having arbitrary group velocities in free space,” Nat. Commun. 10(1), 929 (2019).
[Crossref]

M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Classification of propagation-invariant space-time wave packets in free space: Theory and experiments,” Phys. Rev. A 99(2), 023856 (2019).
[Crossref]

H. E. Kondakci, M. Yessenov, M. Meem, D. Reyes, D. Thul, S. R. Fairchild, M. Richardson, R. Menon, and A. F. Abouraddy, “Synthesizing broadband propagation-invariant space-time wave packets using transmissive phase plates,” Opt. Express 26(10), 13628–13638 (2018).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space–time light sheets,” Nat. Photonics 11(11), 733–740 (2017).
[Crossref]

H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24(25), 28659–28668 (2016).
[Crossref]

Lacourt, P. A.

Lewis, R. M.

R. M. Lewis, “Asymptotic theory of wave-propagation,” Arch. Rational Mech. Anal. 20(3), 191–250 (1965).
[Crossref]

Li, T.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Lindberg, J.

Longhi, S.

López-Bautista, M. C.

J. DelOlmo-Márquez, G. Castillo-Santiago, M. Avendaño-Alejo, I. Moreno, E. Román-Hernández, and M. C. López-Bautista, Opt. Express 29(15), 23300–23314 (2021).

Lu, J.-Y.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves-exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(1), 19–31 (1992).
[Crossref]

J.-Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 39(3), 441–446 (1992).
[Crossref]

Mackinnon, L.

L. Mackinnon, “A nondispersive de Broglie wave packet,” Found Phys 8(3-4), 157–176 (1978).
[Crossref]

Mathis, A.

Mazilu, M.

J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
[Crossref]

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

McCluskey, K.

J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
[Crossref]

Meem, M.

Menon, R.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref]

Morandotti, R.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber Accelerating Beams,” Phys. Rev. Lett. 109(19), 193901 (2012).
[Crossref]

Moreno, I.

J. DelOlmo-Márquez, G. Castillo-Santiago, M. Avendaño-Alejo, I. Moreno, E. Román-Hernández, and M. C. López-Bautista, Opt. Express 29(15), 23300–23314 (2021).

Mourka, A.

Neethling, P. H.

Nemirovsky, J.

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting Accelerating Wave Packets of Maxwell’s Equations,” Phys. Rev. Lett. 108(16), 163901 (2012).
[Crossref]

Nye, N. S.

H. E. Kondakci, N. S. Nye, D. N. Christodoulides, and A. F. Abouraddy, “Tilted-Pulse-Front Space-Time Wave Packets,” ACS Photonics 6(2), 475–481 (2019).
[Crossref]

Nylk, J.

J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
[Crossref]

Ortega-Vidals, P.

Parker, K. J.

Porras, M. A.

M. A. Porras and C. Conti, “Couplings between the temporal and orbital angular momentum degrees of freedom in ultrafast optical vortices,” Phys. Rev. A 101(6), 063803 (2020).
[Crossref]

M. A. Porras, “Gaussian beams diffracting in time,” Opt. Lett. 42(22), 4679–4682 (2017).
[Crossref]

Preciado, M. A.

J. Nylk, K. McCluskey, M. A. Preciado, M. Mazilu, Z. Yang, F. J. Gunn-Moore, S. Aggarwal, J. A. Tello, D. E. K. Ferrier, and K. Dholakia, “Light-sheet microscopy with attenuation-compensated propagation-invariant beams,” Sci. Adv. 4(4), eaar4817 (2018).
[Crossref]

Reivelt, K.

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69(3), 036612 (2004).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-Shaped Propagation-Invariant Localized Light Waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997).
[Crossref]

Reyes, D.

Richardson, M.

Rickenstorff-Parrao, C.

Ring, J. D.

Rohwer, E. G.

Román-Hernández, E.

J. DelOlmo-Márquez, G. Castillo-Santiago, M. Avendaño-Alejo, I. Moreno, E. Román-Hernández, and M. C. López-Bautista, Opt. Express 29(15), 23300–23314 (2021).

Rose, P.

P. Rose, M. Boguslawski, and M. C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14(3), 033018 (2012).
[Crossref]

Saari, P.

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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.

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Figures (5)

Fig. 1.
Fig. 1. Space-time caustics of light. (a) shows the intensity profile of a subluminal propagation-invariant space-time caustic of group velocity ${v_g} = 0.02c$, c being the speed of light in free space, at different instances in time. The caustic pattern predicted by Eq. (4) is overlain on the $t = 0$ instance in cyan lines, showing excellent agreement with the actual vectorial electromagnetic wavepacket. (b) and (c) show the electric fields (${E_y} = 0$ everywhere) of the transverse magnetic (TM) mode version corresponding to the intensity pattern in (a); and (d) shows the electric field (${E_{x,z}} = 0$ everywhere) of the transverse electric (TE) mode corresponding to the intensity pattern in (a). (e) and (f) show the dispersion relation and full frequency range of the space-time caustic. Notably the wavepacket is made up of only forward propagating (${k_z} > 0$) waves. Dashed black lines indicate the light cone. (g) and (h) show the phase delay of the constituent plane waves as a function of polar angle (negative polar angle corresponds to $\phi = \pi $) and wavelength respectively. Note that the range of $\; \omega $ in (e,f) exactly corresponds to the range of $\lambda $ in (h). The precise caustic pattern can be tuned through the phase profile, as we show in Fig. 2.
Fig. 2.
Fig. 2. Tailoring space-time caustics through the phase delay. (a), (c), (e) show intensity patterns based on the exact same dispersion relation and frequency range as in Fig. 1 (i.e. Figures 1(e) and 1(f) also apply to these intensity patterns), but with different phase delay profiles $\varPhi $ reflected in (b), (d), (f) respectively. The caustic pattern predicted by Eq. (4) is overlain on the $t = 0$ instances in cyan lines. These caustics propagate at the same group velocity ${v_g} = 0.02c$ as the Fig. 1 caustic but are markedly different from one another, showing that the caustic pattern can be controlled independently of the group velocity.
Fig. 3.
Fig. 3. Superluminal, subluminal and backward-traveling space-time caustics. (a)(i) shows the intensity profiles for a subluminal (${v_g} = 0.96c$) space-time caustic at times 0 ps, 0.56 ps, 1.1 ps (in the order indicated by the white arrows). The earliest instance is overlain with cyan lines that correspond to the caustic predicted using Eq. (4), showing excellent agreement with the actual vectorial electromagnetic wavepacket. Here, the intensity was calculated from the fields of a transverse magnetic mode, although a transverse electric mode would also have given the same intensity pattern. (a)(ii) shows the linear relationship between angular frequency $\omega $ and longitudinal wavevector component ${k_z}$. The slope of this relationship gives the group velocity of the wavepacket. (a)(iii) shows the phase delay as a function of wavelength. The descriptions for the panels in (b) and (c) are exactly as in (a) except that they correspond to a superluminal forward-traveling wavepacket (${v_g} = 1.2c$) and a superluminal backward-traveling wavepacket (${v_g} ={-} 1.2c$) respectively. Note from (ii) that all wavepackets here – including the backward-traveling example in (c) – are made of purely forward propagating (${k_z} > 0$) waves. This shows that backward-traveling space-time wavepackets are also amenable to design using caustics.
Fig. 4.
Fig. 4. Time-diffracting propagation-invariant caustics. (a) shows the intensity profile of a time-diffracting caustic as a function of x and t, with the caustic predicted by Eq. (4) overlain in dashed cyan lines. Note that the evolution of the intensity profile is in time, making the wavepacket propagation invariant in $z\; $ at any instant of t, as we see at different snapshots in time (b,c). (d) shows the dispersion relation, corresponding to a vertical line in the $\omega $-${k_z}$ plane. (f) shows the phase delay as a function of wavelength. Note that the range of $\; \omega $ in (d,e) exactly corresponds to the range of $\lambda $ in (f). This exemplifies how caustics can be used to design the evolution of time-diffracting propagation-invariant beams, resulting for instance in a single beam at an earlier time (b) splitting into dual beams at a later time (c).
Fig. 5.
Fig. 5. Controlling caustic patterns via angular range. The precise caustic pattern can be controlled by varying the range of $\theta $ included in the wavepacket. (a-d) show the caustic patterns of the examples in Figs. 1, (2a,b), (2c,d), (2e,f), respectively. The blue curves correspond to ${k_x} > 0$ and the red curves to ${k_x} < 0$. Their shared plots of ${k_x}$, ${k_z}$ and $\omega /c$ are shown in (e) for reference. Similarly, (f,g), (h,i), (j,k) and (l,m) belong to the examples in Figs. 3(a), 3(b), 3(c) and 4 respectively. Circular markers along the caustic patterns denote where the pattern would terminate if $\theta $ were to be terminated at that value. In (a,b,c,d), the markers, in order of progression indicated by the arrows, indicate $\theta = \{{0,0.53,1.05,1.57} \}$ rad. In (f), $\theta = \{{0,0.065,0.13,0.19} \}$ rad. In (h), $\theta = \{{0,0.083,0.17,0.25} \}$ rad. In (j), $\theta = \{{0,0.39,0.78,1.17} \}$ rad. In (l), $\theta = \{{7.9 \times {{10}^{ - 4}},0.26,0.52,0.79} \}$ rad.

Equations (13)

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E ( r , t ) = Re { d ϕ d θ ϵ ^ θ , ϕ A ( θ , ϕ ) e i k θ , ϕ r i ω θ , ϕ t } ,
E ( r , t ) = Re { d k x ϵ ^ k x e i k 0 ( θ ) ( x sin θ + z cos θ c t ) + i Φ ( sgn ( k x ) θ ) } , ( finite v g ) ,
θ χ ( θ ) = 0 , 2 θ 2 χ ( θ ) = 0 , χ ( θ ) k 0 ( θ ) ( x sin θ + z cos θ c t ) + Φ ( sgn ( k x ) θ ) , ( finite v g ) ,
x c = k d 1 [ Φ ( θ ) ( cos θ β g ( 1 + sin 2 θ ) ) + Φ ( θ ) ( β g cos θ 1 ) sin θ ] , z c = k d 1 [ Φ ( θ ) ( 1 2 β g 2 + β g c o s θ ) sin θ + Φ ( θ ) ( 1 β g cos θ ) ( cos θ β g ) ] + v g t , ( finite v g )
sin θ z c = ( cos θ β g ) x c ,
d x c d z c = sin θ ( cos θ β g ) .
Φ = ( β g cos θ ) x c + ( z c v g t ) sin θ ( β g cos θ 1 ) 2 .
E ( r , t ) = Re { d k x ϵ ^ k x e i k z ( x tan θ + z c t sec θ ) + i Φ ( θ ) } , ( | v g | ) ,
x c = sgn ( k x ) ( 2 k z ) 1 [ ( cos ( 2 θ ) 3 ) Φ + sin 2 θ Φ ] , c t c = ( 2 k z ) 1 [ 4 sin θ Φ + 2 cos θ Φ ] , ( | v g | ) ,
d x c d c t c = ± sin θ ,
Φ = k z x c + c t c sin θ sin 2 θ 1 .
cos θ = cos θ + β g 1 + β g cos θ , sin θ = sin θ γ ( 1 + β g cos θ ) , ω = ω + v g k z , ( v g < c ) ,
cos θ = cos θ + β g 1 1 + β g 1 cos θ , sin θ = sin θ γ s ( 1 + β g 1 cos θ ) , ω = k z v g γ s + v g k z , ( v g > c ) ,

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