# Observation of optical de Broglie-Mackinnon wave packets

Layton A. Hall1 and Ayman F. Abouraddy1, Corresponding authors:
<EMAIL_ADDRESS>1CREOL, The College of Optics & Photonics, University of
Central Florida, Orlando, Florida 32816, USA

###### Abstract

de Broglie wave packets accompanying moving particles are dispersive and lack
an intrinsic length scale dictated solely by the particle mass and velocity.
Mackinnon proposed almost 45 years ago a localized non-dispersive wave packet
constructed out of dispersive de Broglie phase waves via a Copernican
inversion of the roles of particle and observer, whereupon an intrinsic length
scale emerges by accounting for every possible observer – rather than by
introducing an ad hoc uncertainty in the particle velocity. The de Broglie-
Mackinnon (dBM) wave packet has nevertheless remained to date a theoretical
entity. Here, we report the observation of optical dBM wave packets using
paraxial space-time-coupled pulsed laser fields in presence of anomalous
group-velocity dispersion. Crucially, the bandwidth of dBM wave packets has an
upper limit that is compatible with the wave-packet group velocity and
equivalent mass. In contrast to previously observed linear propagation-
invariant wave packets whose spatio-temporal profiles at any axial plane are
X-shaped, those for dBM wave packets are uniquely O-shaped (circularly
symmetric with respect to space and time). By sculpting their spatio-temporal
spectral structure, we produce dispersion-free dBM wave packets in the
dispersive medium, observe their circularly symmetric spatio-temporal
profiles, and tune the field parameters corresponding to particle mass and
velocity that uniquely determine the wave-packet length scale.

It is well-known that there are no dispersion-free wave-packet solutions to
the $(1+1)$D potential-free Schrödinger equation – with the sole exception of
the Airy wave packet identified by Berry and Balasz in 1979 Berry and Balazs
(1979), which does not travel at a fixed group velocity, but rather
accelerates despite the absence of an external force Greenberger (1980). The
Airy wave packet has impacted all areas of wave physics (e.g., optics
Siviloglou and Christodoulides (2007), acoustics Zhang  _et al._ (2014), water
waves Fu  _et al._ (2015), electron beams Voloch-Bloch  _et al._ (2013), and
as a model for Dirac particles Kaminer  _et al._ (2015)). Less-known is that
in the year preceding the discovery of the Airy wave packet, Mackinnon
identified a non-dispersive $(1+1)$D wave packet that travels at a constant
group velocity Mackinnon (1978), but is constructed out of dispersive de
Broglie ‘phase waves’ that accompany the motion of a massive particle and are
solutions to the Klein-Gordon equation. de Broglie had originally demonstrated
that the group velocity $\widetilde{v}$ of a wave packet constructed of phase-
waves is equal to the particle velocity $v$ de Broglie (1925). However,
localized de Broglie wave packets are dispersive, as are Schrödinger wave
packets Cohen-Tannoudji  _et al._ (1991). Moreover, because de Broglie wave
packets necessitate introducing an ad hoc uncertainty in the particle velocity
de Broglie (1925), and there is no upper limit on the exploitable bandwidth,
such wave packets lack an intrinsic length scale (i.e., a scale uniquely
determined by the particle mass and velocity).

Through a Copernican inversion of the roles of particle and observer,
Mackinnon constructed out of dispersive de Broglie phase waves a non-
dispersive wave packet Mackinnon (1978) – which we refer to henceforth as the
de Broglie-Mackinnon (dBM) wave packet. Instead of introducing an ad hoc
uncertainty into the particle velocity from the perspective of a privileged
reference frame, Mackinnon suggested accounting for all possible observers,
who cooperatively report observations made in their reference frames to a
single agreed-upon frame in which Lorentz contraction and time dilation are
corrected for Mackinnon (1978). Besides retaining the salutary features of
conventional de Broglie wave packets, Mackinnon’s construction unveiled an
intrinsic length scale for the dBM wave packet determined solely by the
particle mass and velocity. However, despite the clear algorithmic process for
constructing the dBM wave packet, it is not a solution to the Klein-Gordon
equation Mackinnon (1978), and is instead constructed only epistemologically
in the selected reference frame. As such, dBM wave packets have yet to be
realized in any physical wave.

Nevertheless, it has been recognized that the $(1+1)$D dBM wave packet can be
mapped to physical solutions of the optical wave equation by first enlarging
the field dimensionality to $(2+1)$D, which allows introducing angular
dispersion Torres  _et al._ (2010); Fülöp and Hebling (2010). This procedure
enables realizing the dBM dispersion relationship for propagation along the
optical axis in the initial reduced-dimensionality $(1+1)$D space Saari and
Reivelt (2004). However, observing optical dBM wave packets in free space
faces insurmountable practical difficulties Zamboni-Rached and Recami (2008);
Yessenov  _et al._ (2022). Specifically, such wave packets are produced by
relativistic optical dipoles and are observed by stationary, coherent field
detectors that nevertheless fully encircle the moving dipole. We investigate
here a different strategy that makes use of the unique characteristics of
optical-wave propagation in the anomalous group-velocity dispersion (GVD)
regime Saleh and Teich (2007) to produce paraxial dBM wave packets. In this
conception, an optical dBM wave packet is a particular realization of so-
called ‘space-time’ (ST) wave packets Kondakci and Abouraddy (2016); Parker
and Alonso (2016); Wong and Kaminer (2017); Yessenov  _et al._ (2019a, 2022)
in dispersive media Porras  _et al._ (2003a, b); Longhi (2004a); Porras and Di
Trapani (2004); Malaguti  _et al._ (2008); Malaguti and Trillo (2009); Hall
and Abouraddy (2022). In general, ST wave packets are pulsed optical beams
whose unique characteristics (e.g., tunable group velocity Kondakci and
Abouraddy (2019) and anomalous refraction Bhaduri  _et al._ (2020)) stem from
their spatio-temporal spectral structure rather than their particular spatial
or temporal profiles. Recent advances in the synthesis of ST wave packets make
them a convenient platform for producing a wide variety of structured pulsed
fields Yessenov  _et al._ (2022), including dBM wave packets.

Here, we provide unambiguous observations of optical dBM wave packets in
presence of anomalous GVD. Starting with generic femtosecond pulses, we make
use of a universal angular-dispersion synthesizer Hall and Abouraddy (2021) to
construct spatio-temporally structured optical fields in which the spatial and
temporal degrees-of-freedom are no longer separable. Critically, the
association between the propagation angle and wavelength is two-to-one rather
than one-to-one as in conventional tilted pulse fronts Torres  _et al._
(2010). This feature allows folding the spatio-temporal spectrum back on
itself, thereby guaranteeing the paraxiality of the synthesized dBM wave
packets. Consequently, these wave packets retain in the medium all the
characteristic features of their free-space counterparts while circumventing
the above-mentioned difficulties. Such space-time-coupled wave packets are
dispersive in free space, but become propagation-invariant once coupled to a
medium in the anomalous-GVD regime, where they travel at a tunable group
velocity $\widetilde{v}$. Although all previously observed linear,
propagation-invariant wave packets have at a fixed axial plane been either
X-shaped Saari and Reivelt (1997); Turunen and Friberg (2010); Hernández-
Figueroa  _et al._ (2014); Yessenov  _et al._ (2022) or separable Chong  _et
al._ (2010) with respect to the transverse coordinate and time, the spatio-
temporal profiles of dBM wave packets are – in contrast – circularly symmetric
(O-shaped). In addition to verifying this long-theorized O-shaped spatio-
temporal structure Longhi (2004a); Porras and Di Trapani (2004); Malaguti  _et
al._ (2008), we confirm the impact of the two identifying parameters
(equivalent to particle mass and velocity) on the bandwidth and length scale
of the non-dispersive dBM wave packets. Propagation invariance in the
dispersive medium constrains the maximum bandwidth (minimum wave-packet
length) according to these selected parameters. Finally, in contrast to Airy
wave packets that are the unique non-dispersive solution to the Schrödinger
equation, the axial profile of dBM wave packets can be varied almost
arbitrarily, which we confirm by modulating their spatio-temporal spectral
phase distribution. These results may pave the way to optical tests of the
solutions of the Klein-Gordon equation for massive particles.

## Results

### Theory of de Broglie wave packets

de Broglie posited two distinct entities accompanying massive particles: an
internal ‘clock’ and an external ‘phase wave’ MacKinnon (1976); Espinosa
(1982). For a particle of rest mass $m_{\mathrm{o}}$ whose energy is expressed
as $E_{\mathrm{o}}\\!=\\!m_{\mathrm{o}}c^{2}\\!=\\!\hbar\omega_{\mathrm{o}}$,
the internal clock and the infinite-wavelength phase wave coincide at the same
de Broglie frequency $\omega_{\mathrm{o}}$ in the particle’s rest frame [Fig.
1(a)]; here $c$ is the speed of light in vacuum, and $\hbar$ is the modified
Planck constant. When the particle moves at a velocity $v$, the frequencies
observed in the rest frame diverge: the internal frequency drops to
$\omega\\!=\\!\omega_{\mathrm{o}}\sqrt{1-\beta_{v}^{2}}$ whereas the phase-
wave frequency increases to
$\omega\\!=\\!\omega_{\mathrm{o}}\big{/}\sqrt{1-\beta_{v}^{2}}$ and takes on a
finite wavelength $\lambda$, where $\beta_{v}\\!=\\!\tfrac{v}{c}$ [Fig. 1(b)].
The wave number $k\\!=\\!\tfrac{2\pi}{\lambda}$ for the phase wave is
determined by the de Broglie dispersion relationship
$\omega^{2}\\!=\\!\omega_{\mathrm{o}}^{2}+c^{2}k^{2}$ [Fig. 1(c)], so that it
is a solution to the Klein-Gordon equation. Because de Broglie phase waves are
extended, a particle with a well-defined velocity cannot be localized.
Instead, spatially localizing the particle requires introducing an ad hoc
uncertainty in the particle velocity (a spread from $v$ to $v+\Delta v$) to
induce a bandwidth $\Delta\omega$ (from $\omega_{\mathrm{c}}$ to
$\omega_{\mathrm{c}}+\Delta\omega$), or $\Delta k$ (from $k_{\mathrm{c}}$ to
$k_{\mathrm{c}}+\Delta k$) de Broglie (1925); Mackinnon (1978), thus resulting
in a finite-width wave packet that is also a solution to the Klein-Gordon
equation [Fig. 1(c)]. The wave-packet group velocity
$\widetilde{v}\\!=\\!1\big{/}\tfrac{dk}{d\omega}\big{|}_{\omega_{\mathrm{c}}}\\!=\\!v$
is equal to the particle velocity, whereas its phase velocity is
$v_{\mathrm{ph}}\\!=\\!\tfrac{\omega}{k}\\!=\\!\tfrac{c^{2}}{v}$
($v_{\mathrm{ph}}\widetilde{v}\\!=\\!c^{2}$; see Methods). However, de Broglie
wave packets are dispersive $\tfrac{d\widetilde{v}}{d\omega}\\!\neq\\!0$.
Moreover, because there is no upper limit on the exploitable bandwidth [Fig.
1(c)], de Broglie wave packets lack an intrinsic length scale; that is, there
is no minimum wave-packet length that is uniquely identified by the particle
parameters (mass $m_{\mathrm{o}}$ and velocity $v$).

### Non-dispersive de Broglie-Mackinnon (dBM) wave packets

Mackinnon proposed an altogether different conception for constructing
localized non-dispersive wave packets out of de Broglie phase waves that
jettisons the need for introducing an ad hoc uncertainty in particle velocity
to localize it. Key to this proposal is a Copernican inversion of the roles of
particle and observer. Rather than a single privileged observer associated
with the rest frame in Fig. 1(c), Mackinnon considered a continuum of
potential observers traveling at physically accessible velocities (from $-c$
to $c$). The wave-packet bandwidth $\Delta k$ that is established in a
particular reference frame is a result of the spread in the particle velocity
as observed in all these possible frames. Consequently, the particle can be
localized, and a unique wave-packet length scale identified, even when its
velocity is well-defined.

The physical setting envisioned by Mackinnon is depicted in Fig. 1(d), where
the particle moves at a velocity $v$ and an observer moves at $u$, both with
respect to a common rest frame in which the dBM wave packet is constructed.
Each potential observer records a different phase-wave frequency and
wavelength. The crucial step is that all potential observers travelling at
velocities $u$ ranging from $-c$ to $c$ report their observations to the
selected rest frame. These phase waves are superposed in this frame – after
accounting for Lorentz contraction and time dilation (Methods) – to yield a
wave packet uniquely identified by the particle rest mass $m_{\mathrm{o}}$ and
velocity $v$.

Consider first the simple scenario where the particle is at rest with respect
to the selected frame ($v\\!=\\!0$). Each observer reports to the common rest
frame a frequency
$\omega^{\prime}\\!=\\!\omega_{\mathrm{o}}/\sqrt{1-\beta_{u}^{2}}$ and a wave
number $k^{\prime}\\!=\\!-k_{\mathrm{o}}\beta_{u}/\sqrt{1-\beta_{u}^{2}}$,
where $\beta_{u}\\!=\\!\tfrac{u}{c}$. Accounting for time dilation results in
$\omega^{\prime}\rightarrow\omega\\!=\\!\omega_{\mathrm{o}}$, and accounting
for Lorentz contraction produces $k^{\prime}\rightarrow
k\\!=\\!-k_{\mathrm{o}}\beta_{u}$. Therefore, the frequency in the rest frame
based on the recordings of all the observers is
$\omega\\!=\\!\omega_{\mathrm{o}}$, just as in the case of a conventional de
Broglie phase wave, but the wave number now extends over the range from
$-k_{\mathrm{o}}$ to $k_{\mathrm{o}}$ as the observer velocity $u$ ranges from
$c$ to $-c$ [Fig. 1(e)]. In other words, the observer velocity $u$ serves as
an internal parameter that is swept to establish a new dispersion relationship
whose slope is zero, thus indicating a particle at rest
$\widetilde{v}\\!=\\!v\\!=\\!0$ Mackinnon (1978); Saari and Reivelt (2004).
The spectral representation of the support domain for this wave packet is a
horizontal line $\omega\\!=\\!\omega_{\mathrm{o}}$ in
$(k,\tfrac{\omega}{c})$-space delimited by the two light-lines
$k\\!=\\!\pm\tfrac{\omega}{c}$ [Fig. 1(e)]. In contradistinction to
conventional de Broglie wave packets, a physically motivated length scale
emerges for the dBM wave packet. The maximum spatial bandwidth is $\Delta
k\\!=\\!2k_{\mathrm{o}}$, which corresponds to a minimum wave-packet length
scale of $L_{\mathrm{min}}\\!\sim\\!\tfrac{\lambda_{\mathrm{o}}}{2}$, where
$\lambda_{\mathrm{o}}\\!=\\!\tfrac{2\pi}{k_{\mathrm{o}}}$. This can be viewed
as an ‘optical theorem’, whereby the dBM wave packet for a stationary particle
cannot be spatially localized below the associated de Broglie wavelength
$\lambda_{\mathrm{o}}$. Taking an equal-weight superposition across all the
wave numbers, the dBM wave packet is $\psi(z;t)\propto
e^{-i\omega_{\mathrm{o}}t}\mathrm{sinc}(\tfrac{\Delta k}{\pi}z)$, where
$\mathrm{sinc}(x)\\!=\\!\tfrac{\sin{\pi x}}{\pi x}$ Mackinnon (1978).

A similar procedure can be followed when $v\\!\neq\\!0$, whereupon the
frequency and wave number in the selected reference frame are
$\omega\\!=\\!\omega_{\mathrm{o}}(1-\beta_{v}\beta_{u})\big{/}\sqrt{1-\beta_{v}^{2}}$
and
$k\\!=\\!k_{\mathrm{o}}(\beta_{v}-\beta_{u})\big{/}\sqrt{1-\beta_{v}^{2}}$,
respectively (Methods). Because $v$ is fixed whereas $u$ extends from $-c$ to
$c$, a linear dispersion relationship between $\omega$ and $k$ is established,
$k\\!=\\!\tfrac{1}{\beta_{v}}(\tfrac{\omega}{c}-\tfrac{k_{\mathrm{o}}^{2}}{k_{1}})$,
where $k_{1}\\!=\\!k_{\mathrm{o}}/\sqrt{1-\beta_{v}^{2}}$. The slope of the
dBM dispersion relationship indicates that $\widetilde{v}\\!=\\!v$ as in
conventional de Broglie wave packets, but the dBM wave packet is now non-
dispersive, $\tfrac{d\widetilde{v}}{d\omega}\\!=\\!0$ [Fig. 1(f)]. The limits
on the spatial and temporal bandwidths for the dBM wave packet are $\Delta
k\\!=\\!2k_{1}$ and $\tfrac{\Delta\omega}{c}\\!=\\!\beta_{v}\Delta k$,
respectively, leading to a reduced characteristic length scale
$L_{\mathrm{min}}\\!\sim\\!\tfrac{\lambda_{\mathrm{o}}}{2}\sqrt{1-\beta_{v}^{2}}$
as a manifestation of Lorentz contraction; a faster particle is more tightly
localized. By assigning equal complex amplitudes to all the phase waves
associated with this moving particle, the propagation-invariant dBM wave
packet is $\psi(z;t)\propto e^{i\beta_{v}\Delta
k(z-\widetilde{v}t)}\mathrm{sinc}(\tfrac{\Delta k}{\pi}(z-\widetilde{v}t))$.
Crucially, unlike conventional de Broglie wave packets, the dBM wave packet is
not a solution to the Klein-Gordon equation, although a modified wave equation
can perhaps be constructed for it Mackinnon (1978).

### Optical de Broglie-Mackinnon wave packets in free space

Despite their intrinsic interest from a fundamental point of view, dBM wave
packets have remained to date theoretical entities. It has nevertheless been
recognized that optical waves in free space may provide a platform for their
construction Saari and Reivelt (2004); Zamboni-Rached and Recami (2008).
Because $(1+1)$D optical waves in free space are dispersion-free
($k\\!=\\!\tfrac{\omega}{c}$ and
$v_{\mathrm{ph}}\\!=\\!\widetilde{v}\\!=\\!c$), producing optical dBM wave
packets requires first adding a transverse coordinate $x$ to enlarge the field
dimensionality to $(2+1)$D. The dispersion relationship thus becomes
$k_{x}^{2}+k_{z}^{2}\\!=\\!(\tfrac{\omega}{c})^{2}$, which represents the
surface of a ‘light-cone’ Donnelly and Ziolkowski (1993); Yessenov  _et al._
(2022); here $k_{x}$ and $k_{z}$ are the transverse and longitudinal
components of the wave vector along $x$ and $z$, respectively. The spectral
support of any optical field corresponds to some region on the light-cone
surface [Fig. 2(a)]. For a fixed value of
$k_{x}\\!=\\!\pm\tfrac{\omega_{\mathrm{o}}}{c}$, we retrieve the axial
dispersion relationship for de Broglie phase waves
$\omega^{2}\\!=\\!\omega_{\mathrm{o}}^{2}+c^{2}k_{z}^{2}$. A convenient
parametrization of the field makes use of the propagation angle
$\varphi(\omega)$ with respect to the $z$-axis for the plane wave at a
frequency $\omega$, whereupon
$k_{x}(\omega)\\!=\\!\tfrac{\omega}{c}\sin{\varphi(\omega)}$ and
$k_{z}(\omega)\\!=\\!\tfrac{\omega}{c}\cos{\varphi(\omega)}$. Angular
dispersion is thus introduced into the $(2+1)$D field Torres  _et al._ (2010);
Fülöp and Hebling (2010), and its spectral support on the light-cone surface
is a one-dimensional (1D) trajectory. We take optical dBM wave packets to be
those whose axial dispersion relationship $\omega(k_{z})$ conforms to that of
a dBM wave packet. This requires that the projection of the spectral support
onto the $(k_{z},\tfrac{\omega}{c})$-plane be linear and delimited by the
light-lines $k_{z}\\!=\\!\pm\tfrac{\omega}{c}$. Indeed, the spectral
projections onto the $(k_{z},\tfrac{\omega}{c})$-plane in Fig. 2(a,b) coincide
with those in Fig. 1(e,f).

Consider first a monochromatic field $\omega\\!=\\!\omega_{\mathrm{o}}$ whose
spectral support is the circle at the intersection of the light-cone with a
horizontal iso-frequency plane [Fig. 2(a)]. This monochromatic field comprises
plane waves of the same frequency $\omega_{\mathrm{o}}$ that travel at angles
$\varphi$ extending from 0 to $2\pi$, whose axial wave numbers are
$k_{z}(\varphi)\\!=\\!\pm\sqrt{k_{\mathrm{o}}^{2}-k_{x}^{2}}\\!=\\!k_{\mathrm{o}}\cos\varphi$
and extend from $-k_{\mathrm{o}}$ to $k_{\mathrm{o}}$. This optical wave
packet [Fig. 2(a)] corresponds to the dBM wave packet for a particle in its
rest frame [Fig. 1(e)], and $\varphi$ serves as the new internal parameter to
be swept in order to produce the targeted dBM dispersion relationship,
corresponding to the observer velocity $u$ in Fig. 1(e). By setting the
spectral amplitudes equal for all the plane-wave components, we obtain
$\psi(x,z;t)\propto e^{-i\omega_{\mathrm{o}}t}\,\mathrm{sinc}(\tfrac{\Delta
k_{z}}{\pi}\sqrt{x^{2}+z^{2}})$, where $\Delta k_{z}\\!=\\!2k_{\mathrm{o}}$
[Fig. 2(a)]. Such a wave packet can be produced by a stationary, monochromatic
planar dipole placed at the origin of the $(x,z)$-plane. Observing this
optical field requires coherent field detectors arranged around the $2\pi$
angle subtended by the dipole, and then communicating the recorded
measurements to a central station. This procedure is therefore not dissimilar
in principle from that envisioned by Mackinnon for the dBM wave packet
associated with a stationary particle, in which the measurements recorded by
observers traveling at different velocities are communicated to the common
rest frame [Fig. 1(d)].

When the dipole moves at a velocity $v$ along the $z$-axis with respect to
stationary detectors encircling it, each constituent plane-wave undergoes a
different Doppler shift in the rest frame of the detectors. The field still
comprises plane waves travelling at angles $\varphi$ extending from 0 to
$2\pi$, but each plane wave now has a different frequency $\omega$.
Nevertheless, the new spectral support for the dBM wave packet on the light-
cone is related to that for the stationary monochromatic dipole. Indeed, the
Lorentz transformation associated with the relative motion between the source
and detectors tilts the horizontal iso-frequency spectral plane in Fig. 2(a)
by an angle $\theta$ with respect to the $k_{z}$-axis as shown in Fig. 2(b),
where $\tan{\theta}\\!=\\!\beta_{v}$ Bélanger (1986); Longhi (2004b); Saari
and Reivelt (2004); Kondakci and Abouraddy (2018), thus yielding a tilted
ellipse whose projection onto the $(k_{x},\tfrac{\omega}{c})$ is:

$\frac{k_{x}^{2}}{k_{\mathrm{o}}^{2}}+\frac{(\omega-
ck_{1})^{2}}{(\Delta\omega/2)^{2}}=1.$ (1)

The spectral projection onto the $(k_{z},\tfrac{\omega}{c})$-plane is now the
line
$k_{z}\\!=\\!k_{+}+\tfrac{\omega-\omega_{+}}{\widetilde{v}}\\!=\\!\tfrac{1}{\beta_{v}}(\tfrac{\omega}{c}-\tfrac{k_{\mathrm{o}}^{2}}{k_{1}})$,
where $\widetilde{v}\\!=\\!c\tan{\theta}\\!=\\!v$ is the wave-packet group
velocity along $z$,
$k_{+}\\!=\\!\tfrac{\omega_{+}}{c}\\!=\\!k_{\mathrm{o}}\sqrt{\tfrac{1+\beta_{v}}{1-\beta_{v}}}$,
and $k_{1}\\!=\\!k_{\mathrm{o}}/\sqrt{1-\beta_{v}^{2}}$. The spatial and
temporal bandwidths are related via
$\tfrac{\Delta\omega}{c}\\!=\\!\beta_{v}\Delta k_{z}$, where $\Delta
k_{z}\\!=\\!2k_{1}$. Each plane wave travels at a different direction in the
$(x,z)$-plane in such a way that their axial wave numbers $k_{z}$ reproduce
the dBM dispersion relationship [compare Fig. 1(f) to Fig. 2(b)]. By setting
the complex spectral amplitudes constant for all frequencies, we obtain the
dBM wave packet (with $\widetilde{v}<c$):

$\psi(x,z;t)\propto e^{i\beta_{v}\Delta
k_{z}(z-\widetilde{v}t)}\;\mathrm{sinc}\left(\frac{\Delta
k_{z}}{\pi}\sqrt{x^{2}+(z-\widetilde{v}t)^{2}}\right),$ (2)

Two parameters uniquely identify the optical dBM wave packet: the group
velocity $\widetilde{v}$ (corresponding to the particle velocity) and the wave
number $k_{\mathrm{o}}$ (corresponding to the particle mass). Furthermore, the
signature of the dBM wave packet in Eq. 2 is its circularly symmetric spatio-
temporal profile in $(x,t)$-space in any axial plane $z$. In contrast, all
other propagation-invariant wave packets that have been observed in free space
are X-shaped Saari and Reivelt (1997); Porras  _et al._ (2003b); Turunen and
Friberg (2010); Hernández-Figueroa  _et al._ (2014); Yessenov  _et al._ (2022)
and are not circularly symmetric. Indeed, truncating the spectrum of the
optical dBM wave packet obstructs the formation of the circularly symmetric
profile and gives rise instead to the more familiar X-shaped counterpart
Zamboni-Rached and Recami (2008); Yessenov  _et al._ (2022). The O-shaped
spatio-temporal profile as indicated by Eq. 2 can be observed only when the
full bandwidth – delimited by the light-lines – is included.

The field in the $(x,z)$-plane recorded by stationary detectors encircling the
dipole takes the form shown in Fig. 2(b), as pointed out recently in a thought
experiment by Wilczek Wilczek (2015). Despite the conceptual simplicity of
this optical scheme for producing dBM wave packets, it nevertheless faces
obvious experimental challenges. Encircling an optical dipole moving at a
relativistic speed with stationary detectors is far from practical
realizability. The more realistic configuration in which the detectors are
restricted to a small angular range within the paraxial regime centered on the
$z$-axis truncates the recorded field and precludes observing of the O-shaped
spatio-temporal profile Zamboni-Rached and Recami (2008); Yessenov  _et al._
(2022). For these reasons, it is not expected that the O-shaped dBM wave
packet can be observed using spatio-temporally structured optical fields in
free space.

### Optical de Broglie-Mackinnon wave packets in a dispersive medium

The necessity of including the entire bandwidth delimited by the intersection
of the dBM dispersion relationship with the free-space light-cone [Fig.
2(a-b)] presents insurmountable experimental obstacles. Producing paraxial dBM
wave packets necessitates confining the spectrum to a narrow range of values
of $k_{z}$ centered at a value $k_{z}\\!\sim\\!k_{\mathrm{o}}\\!>\\!0$.
Crucially, the linear spatio-temporal spectrum projected onto the
$(k_{z},\tfrac{\omega}{c})$-plane must remain delimited at both ends by the
light-line, to produce the circularly symmetric spatio-temporal wave-packet
profile. Clearly these requirements cannot be met in free space. Nevertheless,
this challenge can be tackled by exploiting the unique features of optical-
wave propagation in the presence of anomalous GVD. Specifically, the light-
cone structure is modified in presence of anomalous GVD so that the curvature
of the light-line has the same sign as that of the de Broglie dispersion
relationship [Fig. 2(c)]. In this case, imposing the characteristically linear
dBM dispersion relationship produces a spectral support domain on the
dispersive light-cone surface that satisfies all the above-listed
requirements: (1) $k_{z}\\!>\\!0$ is maintained throughout the entire span of
propagation angles $\varphi(\omega)$; (2) the field simultaneously remains
within the paraxial regime; and (3) the spectrum is delimited at both ends by
the light-line [Fig. 2(c)], thus yielding a wave packet having a circularly
symmetric spatio-temporal profile. The spectral support is in the form of an
ellipse at the intersection of the dispersive light-cone with a tilted
spectral plane. The center of this ellipse is displaced to a large value of
$k_{z}$, and the spectral projection onto the
$(k_{z},\tfrac{\omega}{c})$-plane is a line making an angle $\theta$ with the
$k_{z}$-axis. The resulting wave packet is propagation-invariant in the
dispersive medium and travels at a velocity
$\widetilde{v}\\!=\\!c\tan{\theta}$ independently of the physical parameters
of the dispersive medium.

In the anomalous-GVD regime, the wave number is given by
$k(\omega)\\!=\\!n(\omega)\omega/c=k(\omega_{\mathrm{o}}+\Omega)\\!\approx\\!n_{\mathrm{m}}k_{\mathrm{o}}+\tfrac{\Omega}{\widetilde{v}_{\mathrm{m}}}-\tfrac{1}{2}|k_{2\mathrm{m}}|\Omega^{2}+\cdots$;
where $n(\omega)$ is the refractive index, and the following quantities are
all evaluated at $\omega\\!=\\!\omega_{\mathrm{o}}$:
$n_{\mathrm{m}}\\!=\\!n(\omega_{\mathrm{o}})$ is the refractive index,
$\widetilde{v}_{\mathrm{m}}\\!=\\!1\big{/}\tfrac{dk}{d\omega}\big{|}_{\omega_{\mathrm{o}}}$
is the group velocity for a plane-wave pulse in the medium, and
$k_{2\mathrm{m}}\\!=\\!\tfrac{d^{2}k}{d\omega^{2}}\big{|}_{\omega_{\mathrm{o}}}\\!=\\!-|k_{2\mathrm{m}}|$
is the negative-valued anomalous GVD coefficient Saleh and Teich (2007). The
dispersion relationship in the medium $k_{x}^{2}+k_{z}^{2}\\!=\\!k^{2}$
corresponds geometrically to the surface of the modified dispersive light-cone
in Fig. 2(c). Similarly to the free-space scenario, we impose a spectral
constraint of the form
$k_{z}\\!=\\!n_{\mathrm{m}}k_{\mathrm{o}}+\tfrac{\Omega}{\widetilde{v}}\\!=\\!\tfrac{1}{\beta_{v}}\left\\{\tfrac{\omega}{c}-k_{\mathrm{o}}(1-n_{\mathrm{m}}\beta_{v})\right\\}$
in the medium, where $\Omega\\!=\\!\omega-\omega_{\mathrm{o}}$ and
$\widetilde{v}\\!=\\!c\tan{\theta}$ is the group velocity of the wave packet
[Fig. 2(c)]. The wave-packet spectrum as defined by this constraint is
delimited by the light-line at its two ends, both located however in the range
$k_{z}\\!>\\!0$, in contrast to the previous scenarios depicted in Fig. 1(e,f)
and Fig. 2(a,b); see Methods.

The spectral projections onto the $(k_{x},\tfrac{\omega}{c})$ and
$(k_{x},k_{z})$ planes of the spectral support on the dispersive light-cone
are ellipses (Methods):

$\frac{k_{x}^{2}}{k_{x,\mathrm{max}}^{2}}+\frac{(\omega-\omega_{\mathrm{c}})^{2}}{(\Delta\omega/2)^{2}}=1,\quad\frac{k_{x}^{2}}{k_{x,\mathrm{max}}^{2}}+\frac{(k_{z}-k_{\mathrm{c}})^{2}}{(\Delta
k_{z}/2)^{2}}=1,$ (3)

where the temporal bandwidth is
$\tfrac{\Delta\omega}{c}\\!=\\!2\tfrac{k_{\mathrm{o}}}{\sigma_{\mathrm{m}}}\tfrac{1-\beta_{v}^{\prime}}{\beta_{v}^{2}}\\!=\\!\beta_{v}\Delta
k_{z}$, $\sigma_{\mathrm{m}}\\!=\\!c\omega_{\mathrm{o}}|k_{2\mathrm{m}}|$ is a
dimensionless dispersion coefficient,
$\beta_{v}^{\prime}\\!=\\!\tfrac{\widetilde{v}}{\widetilde{v}_{\mathrm{m}}}$,
$k_{x,\mathrm{max}}\\!=\\!\tfrac{1}{2}\tfrac{\Delta\omega}{c}\sqrt{n_{\mathrm{m}}\sigma_{\mathrm{m}}}$,
$\omega_{\mathrm{c}}\\!=\\!\omega_{\mathrm{o}}-\Delta\omega/2$,
$k_{\mathrm{c}}\\!=\\!n_{\mathrm{m}}k_{\mathrm{o}}-\tfrac{\Delta k_{z}}{2}$,
$k_{x,\mathrm{max}}\\!\ll\\!n_{\mathrm{m}}k_{\mathrm{o}}$, $\Delta
k_{z}\\!\ll\\!n_{\mathrm{m}}k_{\mathrm{o}}$, and
$\Delta\omega\\!\ll\\!\omega_{\mathrm{o}}$. It is crucial to recognize that
the ellipse projected onto the $(k_{x},k_{z})$-plane does not enclose the
origin $(k_{x},k_{z})\\!=\\!(0,0)$, but is rather displaced to a central value
$k_{\mathrm{c}}\\!\gg\\!\Delta k_{z}$. Therefore, the optical field comprises
plane-wave components that propagate only in the forward direction within a
small angular range centered on the $z$-axis, and the field thus remains
within the paraxial domain. Nevertheless, because the spectrum is delineated
at both ends by the curved dispersive light-line, the resulting spatio-
temporal profile is circularly symmetric in any axial plane $z$. This wave
packet in the dispersive medium thus satisfies all the above-listed desiderata
for an optical dBM wave packet, but can be readily synthesized and observed in
contrast to its free-space counterparts. One difficulty, however, arises from
the form of $\varphi(\omega)$ in the dispersive medium, which differs
fundamentally from that in free space. Each frequency $\omega$ in a free-space
optical dBM wave packet is associated with two propagation angles
$\pm\varphi(\omega)$. However, each propagation angle $\varphi$ is associated
with a single frequency, so that $|\phi(\omega)|$ is one-to-one. In the
optical dBM wave packet in the dispersive medium, each $\omega$ is still
associated with two propagation angles $\pm\varphi(\omega)$; but
$\varphi(\omega)$ is now two-to-one, so that $\varphi(\omega)$ is folded back
on itself [Fig. 2(c)]. To synthesize such a field configuration, a synthesizer
capable of sculpting $\varphi(\omega)$ almost arbitrarily is required.

### Experimental confirmation

Setup. To construct the optical dBM wave packet in free space from a generic
pulsed beam in which the spatial and temporal degrees-of-freedom are
uncoupled, we introduce angular dispersion by assigning to each wavelength
$\lambda$ a particular pair of angles $\pm\varphi(\lambda)$, thereby coupling
the spatial and temporal degrees-of-freedom. We carry out this task using a
universal angular-dispersion synthesizer Hall and Abouraddy (2021), in which a
spatial light modulator (SLM) deflects each wavelength from a spectrally
resolved laser pulse at prescribed angles, as illustrated in Fig. 3 (Methods).
Because each wavelength $\lambda$ is deflected at $\varphi(\lambda)$
independently of all other wavelengths, $\varphi(\lambda)$ need not be one-to-
one. Indeed, it can readily be a two-to-one mapping as required for paraxial
optical dBM wave packets. The dBM wave packet is formed once all the
wavelengths are recombined by a grating to reconstitute the pulsed field. The
spatio-temporal spectrum of the synthesized wave packet is acquired by
operating on the spectrally resolved field with a spatial Fourier transform
and recording the intensity with a CCD camera. This measurement yields the
spatio-temporal spectrum projected onto the $(k_{x},\lambda)$-plane, from
which we can obtain the spectral projection onto the $(k_{z},\lambda)$-plane.
The spatio-temporal envelope $I(x;\tau)$ of the intensity profile at a fixed
axial plane $z$ is reconstructed in the frame travelling at $\widetilde{v}$
($\tau\\!=\\!t-z/\widetilde{v}$) via linear interferometry exploiting the
procedure developed in Refs. Kondakci and Abouraddy (2019); Yessenov  _et al._
(2019b); Bhaduri  _et al._ (2020) (Methods). The dispersive medium exploited
in our measurements is formed of a pair of chirped Bragg mirrors providing an
anomalous GVD coefficient of $k_{2\mathrm{m}}\\!\approx\\!-500$ fs2/mm and
$\widetilde{v}_{\mathrm{m}}\\!\approx\\!c$ (Methods).

Measurement results. We first verify the unique signature of dBM wave packets
in presence of anomalous GVD; namely, the O-shaped spatio-temporal intensity
profile at any axial plane after inculcating into the field the dBM dispersion
relationship. In Fig. 4 we verify three sought-after features: (1) The closed
elliptical spatio-temporal spectrum projected onto the
$(k_{x},\lambda)$-plane; (2) the linear spectral projection onto the
$(k_{z},\lambda)$-plane, indicating non-dispersive propagation in the
dispersive medium; and (3) the circularly symmetric spatio-temporal intensity
profile $I(x;\tau)$ reconstructed at a fixed axial plane ($z\\!=\\!30$ mm). In
Fig. 4(a) we plot the measurements for an optical dBM wave packet having a
group velocity $\widetilde{v}\\!=\\!0.9975c$. The temporal bandwidth is
constrained to a maximum value of $\Delta\lambda\\!\approx\\!16$ nm, and the
associated spatial bandwidth $\Delta k_{x}\\!\approx\\!0.03$ rad/$\mu$m, thus
resulting in a pulsewidth $\Delta T\\!\approx\\!200$ fs at $x\\!=\\!0$, and a
spatial profile width $\Delta x\\!\approx\\!38$ $\mu$m at $\tau\\!=\\!0$. The
spectral projection onto the $(k_{z},\lambda)$-plane is delimited at both ends
by the curved light-line of the dispersive medium. In other words, a larger
bandwidth is incompatible at this group velocity with propagation invariance
in the dispersive medium. Further increase in the bandwidth extends the
spectral projection below the dispersive light-line, which contributes to only
evanescent field components. The measured spatio-temporal profile $I(x;\tau)$
therefore has the smallest dimensions in space and time for a circularly
symmetric dBM wave packet compatible with the selected group velocity in the
medium.

To the best of our knowledge, this is the first observation of an O-shaped
spatio-temporal intensity profile for a dispersion-free wave packet in a
linear dispersive medium. Previous realizations of dispersion-free ST wave
packets in dispersive media (whether in the normal- or anomalous-GVD regimes)
revealed X-shaped spatio-temporal profiles Hall and Abouraddy (2022) similar
to those observed in free space Saari and Reivelt (1997); Kondakci and
Abouraddy (2017, 2019) or in non-dispersive dielectrics Bhaduri  _et al._
(2020). In these experiments, however, the wave packets were not delimited
spectrally by the dispersive-medium light-line, which is the prerequisite for
the realization of O-shaped optical dBM wave packets.

As mentioned earlier, two parameters characterize a dBM wave packet: the
velocity $v$ and the rest mass $m_{\mathrm{o}}$. The corresponding variables
associated with the optical dBM wave packet are $\widetilde{v}$ and
$\lambda_{\mathrm{o}}$, which can both be readily tuned in our experimental
arrangement by changing the functional dependence of $\varphi$ on $\lambda$.
In this way we can vary the first parameter; namely, the group velocity
$\widetilde{v}$. Increasing the group velocity from
$\widetilde{v}\\!=\\!0.9975c$ [Fig. 4(a)] to $\widetilde{v}\\!=\\!0.9985c$
[Fig. 4(b)] and then to $\widetilde{v}\\!=\\!0.999c$ [Fig. 4(c)] reduces the
maximum exploitable temporal bandwidth from $\Delta\lambda\\!\approx\\!16$ nm
to $\Delta\lambda\\!\approx\\!8$ nm and $\Delta\lambda\\!\approx\\!6$ nm,
respectively, while retaining the closed elliptic spectral projection onto the
$(k_{x},\lambda)$-plane, the linear spectral projection onto the
$(k_{z},\lambda)$-plane, and the associated O-shaped spatio-temporal profile
$I(x;\tau)$. The corresponding spatial bandwidths drop to $\Delta
k_{x}\\!\approx\\!0.023$ rad/$\mu$m and $\Delta k_{x}\\!\approx\\!0.017$
rad/$\mu$m, respectively. In all three dBM wave packets in Fig. 4, we retain a
fixed intersection with the dispersive light-line at
$\lambda_{\mathrm{o}}\\!\approx\\!1054$ nm (corresponding to a fixed particle
mass), such that reducing $\widetilde{v}$ decreases the wavelength of the
second intersection point. The second parameter, the wavelength
$\lambda_{\mathrm{o}}$ corresponding to particle rest mass $m_{\mathrm{o}}$
for de Broglie phase waves, can also be readily tuned [Fig. 5]. Here, the
maximum exploitable bandwidth changes as a result of shifting the value of
$\lambda_{\mathrm{o}}$ from $\lambda_{\mathrm{o}}\\!=\\!1054$ nm [Fig. 5(a)]
where $\Delta\lambda\\!=\\!16$ nm, to $\lambda_{\mathrm{o}}\\!=\\!1055$ nm
[Fig. 5(b)] where $\Delta\lambda\\!=\\!14$ nm, and then to
$\lambda_{\mathrm{o}}\\!=\\!1056$ nm [Fig. 5(c)] where
$\Delta\lambda\\!=\\!12$ nm. Once again, both the spatial and temporal widths
of the circularly symmetric O-shaped profile in the $(x,t)$-domain change
accordingly.

The Airy wave packet, as mentioned earlier, is the unique non-dispersive
solution to Schrödinger’s equation – no other waveform will do Unnikrishnan
and Rau (1996). Although Mackinnon obtained a particular ‘sinc’-function-
shaped wave packet Mackinnon (1978), this waveform is not unique. Indeed, the
sinc-function results from combining all the de Broglie phase waves with equal
weights. However, dBM wave packets can take on in principle arbitrary
waveforms by associating different magnitudes or phases with the plane-wave
components constituting it. We confirm in Fig. 6 that the spatio-temporal
profile $I(x;\tau)$ of optical dBM wave packets can be modified while
remaining propagation invariant in the dispersive medium. First, setting the
complex spectral amplitudes equal along the elliptical spectral support, we
obtain propagation-invariant circularly symmetric wave packets in the
dispersive medium [Fig. 6(a)]. Truncating the ellipse and eliminating the
plane wave components in the vicinity of $k_{x}\\!=\\!0$ disrupts the
formation of the full circular profile, but the wave packet nevertheless
propagates invariantly [Fig. 6(b)]. By introducing a $\pi$-step in the
spectral phase along $k_{x}$, a spatial null is formed along $x\\!=\\!0$ in
the profile of the dBM wave packet [Fig. 6(c)], whereas introducing the
$\pi$-phase-step along $\lambda$ produces a temporal null along $\tau\\!=\\!0$
[Fig. 6(d)]. Finally, alternating the phases between 0 and $\pi$ in the four
quadrants of the spatio-temporal spectral plane $(k_{x},\lambda)$ produces
spatial and temporal nulls along $x\\!=\\!0$ and $\tau\\!=\\!0$, respectively
[Fig. 6(e)]. Despite such variations in their spatio-temporal profiles, all
these optical dBM wave packets propagate invariantly in the dispersive medium.

## Discussion

The rapidly evolving versatile techniques for synthesizing optical fields
Forbes  _et al._ (2021); Yessenov  _et al._ (2022) played a critical role in
the realization of dBM wave packets as demonstrated here. This has helped
confirm the theoretical proposal made by Mackinnon almost 45 years ago for
constructing a non-dispersive wave packet from dispersive de Broglie phase
waves Mackinnon (1978). Furthermore, the experimental procedure implemented
here points to a general synthesis strategy that extends beyond the particular
scenario of dBM wave packets. The overarching theme is that novel dispersion
relationships for the axial propagation of a wave packet can be imposed by
first adding another dimension to the space, and then exploiting the new
dimension to tailor the dispersion relationship before spectral projection
back onto the original reduced-dimensionality space.

In the scenario studied here, we start with a $(1+1)$D physical wave in which
an axial dispersion relationship $\omega(k_{z})$ is enforced by the dynamics
of the wave equation. Increasing the dimensionality of the space from $(1+1)$D
to $(2+1)$D by including a transverse coordinate $x$ yields a new dispersion
relationship $\omega(k_{x},k_{z})$. In free space, optical wave packets are
subject to the constraint $\omega\\!=\\!ck_{z}$ in $(1+1)$D and
$\omega(k_{x},k_{z})\\!=\\!c\sqrt{k_{x}^{2}+k_{z}^{2}}$ in $(2+1)$D. Now, by
judiciously associating each transverse wave number $k_{x}$ with a particular
axial wave number $k_{z}$, a reduced-dimensional axial dispersion relationship
$\omega_{\mathrm{red.}}(k_{z})$ is obtained:
$\omega(k_{x},k_{z})\\!=\\!\omega(k_{x}(k_{z}),k_{z})\\!\mapsto\\!\omega_{\mathrm{red.}}(k_{z})$,
which can be engineered almost arbitrarily. In the experiment reported here,
we employed this strategy to produce a linear dispersion relationship
$\omega(k_{z})\\!=\\!(k_{z}-k_{\mathrm{o}})\widetilde{v}$ projected onto the
$(k_{z},\tfrac{\omega}{c})$-plane that deviates away from the light-line
$\omega\\!=\\!ck_{z}$. In presence of anomalous GVD, such a spatio-temporal
spectrum is delimited at both ends by the curved light-line in the dispersive
medium – thereby yielding the circular symmetric spatio-temporal profile
characteristic of dBM wave packets. Here, the transverse wave number $k_{x}$
played the role of the observer velocity $u$ in the physical configuration
envisioned by Mackinnon [Fig. 1(d)]. However, one may envision a variety of
other scenarios that can be facilitated by this general strategy. For example,
besides tuning the group velocity in free space, linear dispersive media, or
nonlinear optical materials and structures, one may produce accelerating wave
packets Yessenov and Abouraddy (2020); Hall  _et al._ (2022); Li and Kawanaka
(2020) whose group velocity changes with propagation in such media. These
features have been recently predicted to produce a host of new phenomena
related to two photon-emission Sloan  _et al._ (2022) and relativistic optics
Bliokh and Nori (2012); Caloz and Deck-Léger (2020).

Intriguingly, the strategy employed here is not constrained to optical waves.
Indeed, our approach to spatio-temporal structuring of the field is agnostic
with respect to the physical substrate, and can be implemented in principle
with acoustic waves, microwaves, surface plasmon polaritons Schepler  _et al._
(2020), electron beams, neutron beams, or other massive particles. In all
cases, an added spatial dimension can be exploited to override the intrinsic
dispersion relationship of the particular wave phenomenon, thus producing
novel propagation dynamics.

The dimensionality of the $(2+1)$D dBM wave packets synthesized here can be
further extended to the full-dimensional $(3+1)$D space of $(x,y,z;t)$ by
including the second transverse coordinate $y$. This can now be achieved in
light of very recent progress in producing so-called 3D ST wave packets that
are localized in all dimensions of $(3+1)$D space Guo  _et al._ (2021); Pang
_et al._ (2022); Yessenov  _et al._ (2021). Combining this new synthesis
methodology with the procedure outlined here for producing dBM wave packets in
the anomalous-GVD regime will yield spherically symmetric propagation-
invariant pulsed field structures. Such field configurations provide a
platform for exploring proposed topological structures associated with
polarization (spin texture) Guo  _et al._ (2021) without resorting to stereo-
projection onto a 2D plane. Moreover, such spherically symmetric optical dBM
wave packets are compatible with coupling to optical fibers and waveguides,
thus enabling new opportunities in optical communications, optical signal
processing, and nonlinear and quantum optics.

Finally, the ideal spectral constraint underlying optical dBM wave packets
implies an exact association between the spatial and temporal frequencies.
Such idealized wave packets consequently have infinite energy Sezginer (1985).
In any realistic system, however, a spectral uncertainty is inevitably
introduced into this association, resulting in a finite-energy wave packet
traveling for a finite distance over which it is approximately invariant
Kondakci  _et al._ (2019). In our experiments, this spectral uncertainty
arises from the finite spectral resolution of the diffraction grating employed
(Fig. 3), which is estimated to be $\approx\\!16$ pm, corresponding to a
propagation distance of $\approx\\!32$ m at a spectral tilt angle
$\theta=44.99^{\circ}$ Yessenov  _et al._ (2019b).

## References

  * Berry and Balazs (1979) M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
  * Greenberger (1980) D. M. Greenberger, “Comment on “Nonspreading wave packets”,” Am. J. Phys. 48, 256 (1980).
  * Siviloglou and Christodoulides (2007) G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
  * Zhang  _et al._ (2014) P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
  * Fu  _et al._ (2015) S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation dynamics of Airy water-wave pulses,” Phys. Rev. Lett. 115, 034501 (2015).
  * Voloch-Bloch  _et al._ (2013) N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
  * Kaminer  _et al._ (2015) I. Kaminer, J. Nemirovsky, M. Rechtsman, R. Bekenstein, and M. Segev, “Self-accelerating Dirac particles and prolonging the lifetime of relativistic fermions,” Nat. Phys. 11, 261–267 (2015).
  * Mackinnon (1978) L. Mackinnon, “A nondispersive de Broglie wave packet,” Found. Phys. 8, 157–176 (1978).
  * de Broglie (1925) L. de Broglie, “Recherches sur la théorie des quanta,” Ann. de Phys., 3, 22 (1925).
  * Cohen-Tannoudji  _et al._ (1991) C. Cohen-Tannoudji, B. Diu, and F. Laloe,  _Quantum Mechanics_ (Wiley, 1991).
  * Torres  _et al._ (2010) J. P. Torres, M. Hendrych, and A. Valencia, “Angular dispersion: an enabling tool in nonlinear and quantum optics,” Adv. Opt. Photon. 2, 319–369 (2010).
  * Fülöp and Hebling (2010) J. A. Fülöp and J. Hebling, “Applications of tilted-pulse-front excitation,” Recent Optical and Photonic Technologies , 207–230 (2010).
  * Saari and Reivelt (2004) P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
  * Zamboni-Rached and Recami (2008) M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
  * Yessenov  _et al._ (2022) M. Yessenov, L. A. Hall, K. L. Schepler, and A. F. Abouraddy, “Space-time wave packets,” arXiv:2201.08297 (2022).
  * Saleh and Teich (2007) B. E. A. Saleh and M. C. Teich,  _Principles of Photonics_ (Wiley, 2007).
  * Kondakci and Abouraddy (2016) H. E. Kondakci and A. F. Abouraddy, “Diffraction-free pulsed optical beams via space-time correlations,” Opt. Express 24, 28659–28668 (2016).
  * Parker and Alonso (2016) K. J. Parker and M. A. Alonso, “The longitudinal iso-phase condition and needle pulses,” Opt. Express 24, 28669–28677 (2016).
  * Wong and Kaminer (2017) L. J. Wong and I. Kaminer, “Ultrashort tilted-pulsefront pulses and nonparaxial tilted-phase-front beams,” ACS Photon. 4, 2257–2264 (2017).
  * Yessenov  _et al._ (2019a) M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Weaving the rainbow: Space-time optical wave packets,” Opt. Photon. News 30, 34–41 (2019a).
  * Porras  _et al._ (2003a) M. A. Porras, S. Trillo, C. Conti, and P. Di Trapani, “Paraxial envelope X waves,” Opt. Lett. 28, 1090–1092 (2003a).
  * Porras  _et al._ (2003b) M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. E 68, 016613 (2003b).
  * Longhi (2004a) S. Longhi, “Localized subluminal envelope pulses in dispersive media,” Opt. Lett. 29, 147–149 (2004a).
  * Porras and Di Trapani (2004) M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E 69, 066606 (2004).
  * Malaguti  _et al._ (2008) S. Malaguti, G. Bellanca, and S. Trillo, “Two-dimensional envelope localized waves in the anomalous dispersion regime,” Opt. Lett. 33, 1117–1119 (2008).
  * Malaguti and Trillo (2009) S. Malaguti and S. Trillo, “Envelope localized waves of the conical type in linear normally dispersive media,” Phys. Rev. A 79, 063803 (2009).
  * Hall and Abouraddy (2022) L. A. Hall and A. F. Abouraddy, “Canceling and inverting normal and anomalous group-velocity dispersion using space-time wave packets,” arXiv:2202.01148 (2022).
  * Kondakci and Abouraddy (2019) H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets of arbitrary group velocity in free space,” Nat. Commun. 10, 929 (2019).
  * Bhaduri  _et al._ (2020) B. Bhaduri, M. Yessenov, and A. F Abouraddy, “Anomalous refraction of optical spacetime wave packets,” Nat. Photon. 14, 416–421 (2020).
  * Hall and Abouraddy (2021) L. A. Hall and A. F. Abouraddy, “Universal angular-dispersion synthesizer,” arXiv:2109.13987 (2021).
  * Saari and Reivelt (1997) P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
  * Turunen and Friberg (2010) J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Prog. Opt. 54, 1–88 (2010).
  * Hernández-Figueroa  _et al._ (2014) H. E. Hernández-Figueroa, E. Recami, and M. Zamboni-Rached, eds.,  _Non-diffracting Waves_ (Wiley-VCH, 2014).
  * Chong  _et al._ (2010) A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
  * MacKinnon (1976) E. MacKinnon, “De Broglie’s thesis: A critical retrospective,” Am. J. Phys. 44, 1047–1055 (1976).
  * Espinosa (1982) J. M. Espinosa, “Physical properties of de Broglie’s phase waves,” Am. J. Phys. 50, 357–362 (1982).
  * Donnelly and Ziolkowski (1993) R. Donnelly and R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
  * Bélanger (1986) P. A. Bélanger, “Lorentz transformation of packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A 3, 541–542 (1986).
  * Longhi (2004b) S. Longhi, “Gaussian pulsed beams with arbitrary speed,” Opt. Express 12, 935–940 (2004b).
  * Kondakci and Abouraddy (2018) H. E. Kondakci and A. F. Abouraddy, “Airy wavepackets accelerating in space-time,” Phys. Rev. Lett. 120, 163901 (2018).
  * Wilczek (2015) F. Wilczek,  _A Beautiful Question: Finding Nature’s Deep Design_ (Penguin Press, 2015).
  * Yessenov  _et al._ (2019b) M. Yessenov, L. Mach, B. Bhaduri, D. Mardani, H. E. Kondakci, M. A. Alonso, G. A. Atia, and A. F. Abouraddy, “What is the maximum differential group delay achievable by a space-time wave packet in free space?” Opt. Express 27, 12443–12457 (2019b).
  * Kondakci and Abouraddy (2017) H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space-time beams,” Nat. Photon. 11, 733–740 (2017).
  * Unnikrishnan and Rau (1996) K. Unnikrishnan and A. R. P. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
  * Forbes  _et al._ (2021) A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured light,” Nat. Photon. 15, 253–262 (2021).
  * Yessenov and Abouraddy (2020) M. Yessenov and A. F Abouraddy, “Accelerating and decelerating space-time optical wave packets in free space,” Phys. Rev. Let. 125, 233901 (2020).
  * Hall  _et al._ (2022) L. A. Hall, M. Yessenov, and A. F. Abouraddy, “Arbitrarily accelerating space-time wave packets,” Opt. Lett. 47, 694–697 (2022).
  * Li and Kawanaka (2020) Z. Li and J. Kawanaka, “Velocity and acceleration freely tunable straight-line propagation light bullet,” Sci. Rep. 10, 11481 (2020).
  * Sloan  _et al._ (2022) J. Sloan, N. Rivera, J. D. Joannopoulos, and M. Soljačić, “Controlling two-photon emission from superluminal and accelerating index perturbations,” Nat. Phys. 18, 67–73 (2022).
  * Bliokh and Nori (2012) K. Y. Bliokh and F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86, 033824 (2012).
  * Caloz and Deck-Léger (2020) C. Caloz and Z.-L. Deck-Léger, “Spacetime metamaterials–Part I: General concepts,” IEEE Trans. Antennas Propag. 68, 1569–1582 (2020).
  * Schepler  _et al._ (2020) K. L Schepler, M. Yessenov, Y. Zhiyenbayev, and A. F Abouraddy, “Space–time surface plasmon polaritons: A new propagation-invariant surface wave packet,” ACS Photon. 7, 2966–2977 (2020).
  * Guo  _et al._ (2021) C. Guo, M. Xiao, M. Orenstein, and S. Fan, “Structured 3D linear space-time light bullets by nonlocal nanophotonics,” Light Sci. Appl. 10, 160 (2021).
  * Pang  _et al._ (2022) K. Pang, K. Zou, H. Song, M. Karpov, M. Yessenov, Z. Zhao, A. Minoofar, R. Zhang, H. Song, H. Zhou, X. Su, N. Hu, T. J. Kippenberg, A. F. Abouraddy, M. Tur, and A. E. Willner, “Synthesis of near-diffraction-free orbital-angular-momentum space-time wave packets having a controllable group velocity using a frequency comb,” Opt. Express 30, 16712–16724 (2022).
  * Yessenov  _et al._ (2021) M. Yessenov, J. Free, Z. Chen, E. G. Johnson, M. P. J. Lavery, M. A. Alonso, and A. F. Abouraddy, “Space-time wave packets localized in all dimensions,” arXiv:2111.03095 (2021).
  * Sezginer (1985) A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
  * Kondakci  _et al._ (2019) H. E. Kondakci, M. A. Alonso, and A. F. Abouraddy, “Classical entanglement underpins the invariant propagation of space–time wave packets,” Opt. Lett. 44, 2645–2648 (2019).

Data availability statement
The data that support the plots within this paper and other findings of this
study are available from the corresponding author upon reasonable request.

Acknowledgments
We thank M. Yessenov, K. L. Schepler, D. N. Christidoulides and A. Dogariu for
helpful discussions. This work was supported by the U.S. Office of Naval
Research (ONR) under contracts N00014-17-1-2458 and N00014-20-1-2789.
Author contributions

Correspondence and requests for materials should be addressed to A.F.A.
(email<EMAIL_ADDRESS>
Competing interests: The authors declare no competing interests.

## Methods

de Broglie phase waves. At rest, the frequency of the internal ‘clock’ and
that of the infinite-wavelength phase wave are the de Broglie frequency
$\omega_{\mathrm{o}}$. When the particle moves at a velocity $v$, the observed
frequency of the internal ‘clock’ in the rest frame is reduced to
$\omega\\!=\\!\omega_{\mathrm{o}}\sqrt{1-\beta_{v}^{2}}$ whereas that of the
phase wave is increased to
$\omega\\!=\\!\omega_{\mathrm{o}}/\sqrt{1-\beta_{v}^{2}}$, where
$\beta_{v}\\!=\\!v/c$ []. To obtain The phase velocity $v_{\mathrm{phh}}$ of
the phase wave, de Broglie proposed a ‘theory of phase harmony’, which
requires that the internal clock and the phase wave remain in phase for all
$t$ and at any $v$ []. The phase of the moving clock in the rest frame is
$\phi\\!=\\!\omega_{\mathrm{o}}t\sqrt{1-\beta_{v}^{2}}$, and that of the phase
wave is
$\phi\\!=\\!\omega_{\mathrm{o}}(t-\tfrac{z}{v_{\mathrm{ph}}})/\sqrt{1-\beta_{v}^{2}}$.
At time $t$, the particle has covered a distance $z\\!=\\!vt$, and equating
the phases yields $v_{\mathrm{ph}}\\!=\\!\tfrac{c^{2}}{v}\\!>\\!c$.
Alternatively, the Lorentz transformation of the proper time is
$t^{\prime}\\!=\\!(t-\tfrac{v}{c^{2}}z)/\sqrt{1-\beta_{v}^{2}}$,
$\omega_{\mathrm{o}}t^{\prime}\\!=\\!\omega t-kz\\!=\\!\omega
t-\tfrac{\omega}{v_{\mathrm{ph}}}z$, from which we again have
$v_{\mathrm{ph}}\\!=\\!\tfrac{c^{2}}{v}$.

Conventional de Broglie wave packets. The equations
$\omega\\!=\\!\omega_{\mathrm{o}}/\sqrt{1-\beta_{v}^{2}}$ and
$k\\!=\\!\tfrac{\omega}{v_{\mathrm{ph}}}\\!=\\!\tfrac{\omega}{c}\beta$ for the
phase wave yield the dispersion relationship
$k\\!=\\!\tfrac{1}{c}\sqrt{\omega^{2}-\omega_{\mathrm{o}}^{2}}$. A de Broglie
wave packet of finite bandwidth $\Delta\omega$ centered at
$\omega\\!=\\!\omega_{\mathrm{c}}$ requires including an uncertainty $\Delta
v$ in the particle velocity centered on the speed $v$ (which corresponds to
$\omega_{\mathrm{c}}$). The group velocity of the de Broglie wave packet is
thus
$\widetilde{v}\\!=\\!1\big{/}\tfrac{dk}{d\omega}\big{|}_{\omega_{\mathrm{c}}}\\!=\\!c\sqrt{1-(\tfrac{\omega_{\mathrm{o}}}{\omega_{\mathrm{c}}})^{2}}\\!=\\!c\beta_{v}\\!=\\!v$.

Formulation of de Broglie-Mackinnon (dBM) wave packets. Consider the
configuration depicted in Fig. 1(d), where the particle moves at a velocity
$v$ and an observer moves at a velocity $u$, both with respect to a selected
rest frame. Here the relative velocity of the particle with respect to the
observer is $\xi$, where
$\beta_{\xi}\\!=\\!\tfrac{\xi}{c}\\!=\\!\tfrac{\beta_{v}-\beta_{u}}{1-\beta_{v}\beta_{u}}$,
$\beta_{v}\\!=\\!\tfrac{v}{c}$, and $\beta_{u}\\!=\\!\tfrac{u}{c}$. According
to this observer, the frequency and wave number are
$\omega^{\prime}\\!=\\!\omega_{\mathrm{o}}\big{/}\sqrt{1-\beta_{\xi}^{2}}$ and
$k^{\prime}\\!=\\!k_{\mathrm{o}}\beta_{\xi}\big{/}\sqrt{1-\beta_{\xi}^{2}}$,
respectively. The crucial step proposed by Mackinnon is that all potential
observers with velocities $u$ ranging from $-c$ to $c$ report their
observations of $\omega^{\prime}$ and $k^{\prime}$ to the common rest frame,
where the wave packet is constructed epistemologically after accounting for
Lorentz contraction and time dilation. Following this prescription, it is
straightforward to show that

$\omega=\omega_{\mathrm{o}}\frac{1-\beta_{v}\beta_{u}}{\sqrt{1-\beta_{v}^{2}}},\quad
k=k_{\mathrm{o}}\frac{\beta_{v}-\beta_{u}}{\sqrt{1-\beta_{v}^{2}}}.$ (4)

Because $v$ is a fixed velocity whereas $u$ extends from $-c$ to $c$, a linear
dispersion relationship between $\omega$ and $k$ is established [Fig. 1(f)],

$k=\frac{1}{\beta_{v}}\left(\tfrac{\omega}{c}-k_{\mathrm{o}}\sqrt{1-\beta_{v}^{2}}\right).$
(5)

This line intersects with the light-line $k\\!=\\!\tfrac{\omega}{c}$ at
$k\\!=\\!k_{+}\\!=\\!k_{\mathrm{o}}\sqrt{\tfrac{1+\beta_{v}}{1-\beta_{v}}}$
(when $u\\!=\\!-c$), and with the light-line $k\\!=\\!-\tfrac{\omega}{c}$ at
$k\\!=\\!-k_{-}\\!=\\!-k_{\mathrm{o}}\sqrt{\tfrac{1-\beta_{v}}{1+\beta_{v}}}$
(when $u\\!=\\!c$), where
$k_{+}\\!=\\!\tfrac{\omega_{+}}{c}\\!>\\!k_{\mathrm{o}}$ and
$k_{-}\\!=\\!\tfrac{\omega_{-}}{c}\\!<\\!k_{\mathrm{o}}$ [Fig. 1(f)]. When
$u\\!=\\!0$, the associated de Broglie phase wave has
$\tfrac{\omega}{c}\\!=\\!k_{1}$ and $k\\!=\\!\beta_{v}k_{1}$, where
$k_{1}\\!=\\!\tfrac{k_{\mathrm{o}}}{\sqrt{1-\beta_{v}^{2}}}\\!>\\!k_{\mathrm{o}}$.
Another phase wave of interest is the one that retains the stationary
frequency $\omega\\!=\\!\omega_{\mathrm{o}}$, which occurs when
$\beta_{u}\\!=\\!\tfrac{1}{\beta_{v}}(1-\sqrt{1-\beta_{v}^{2}})\\!=\\!\tfrac{1}{\beta_{v}}(1-\tfrac{k_{\mathrm{o}}}{k_{1}})$,
and is associated with $k\\!=\\!k_{2}\\!=\\!k_{\mathrm{o}}\beta_{u}$. Finally,
the linear dispersion relationship has $k\\!=\\!0$ when $u\\!=\\!v$ and thus
$\tfrac{\omega}{c}\\!=\\!k_{\mathrm{o}}\sqrt{1-\beta_{v}^{2}}\\!=\\!\tfrac{k_{\mathrm{o}}^{2}}{k_{1}}\\!<\\!k_{\mathrm{o}}$.
Throughout, setting $v\\!=\\!0$ ($\beta_{v}\\!=\\!0$) in these relationships
yields the result in Fig. 1(e) for a dBM wave packet associated with a
particle in its rest frame.

Optical dBM wave packets in free space. For a monochromatic optical field at
$\omega\\!=\\!\omega_{\mathrm{o}}$, we have the dispersion relationship
$k_{x}^{2}+k_{z}^{2}\\!=\\!k_{\mathrm{o}}^{2}$, which is the circle at the
intersection of the free-space light-cone
$k_{x}^{2}+k_{z}^{2}\\!=\\!(\tfrac{\omega}{c})^{2}$ with the horizontal iso-
frequency plane $\omega\\!=\\!\omega_{\mathrm{o}}$. As described in the main
text, the full circle is the spectral support for the optical field produced
by a stationary planar dipole. When the source and detector are in relative
motion along the $z$-axis at a velocity $v$, the initially horizontal iso-
frequency-plane in $(k_{x},k_{z},\tfrac{\omega}{c})$-space is tilted by an
angle $\theta$ with respect to the $k_{z}$-axis,where
$\tan{\theta}\\!=\\!\beta_{v}$ Kondakci and Abouraddy (2018). The resulting
spectral constraint at the intersection with the light-cone (that conforms to
a dBM wave packet) is:

$k_{z}=k_{+}+\frac{\omega-\omega_{+}}{\widetilde{v}}=\frac{1}{\beta_{v}}\left(\frac{\omega}{c}-k_{\mathrm{o}}\sqrt{1-\beta_{v}^{2}}\right)=\frac{1}{\beta_{v}}\left(\frac{\omega}{c}-\frac{k_{\mathrm{o}}^{2}}{k_{1}}\right),$
(6)

where $k_{+}\\!=\\!\tfrac{\omega_{+}}{c}$ is the point on the light-line
$k_{z}\\!=\\!\tfrac{\omega}{c}$ intersecting with the tilted spectral plane,
and we make use of the same definitions of $k_{\mathrm{o}}$, $k_{+}$, and
$k_{1}$ as above for dBM wave packets. Eliminating $k_{z}$ from these two
relationships (the light-cone and the titled spectral plane) yields the
spectral projection onto the $(k_{x},\tfrac{\omega}{c})$-plane in the form of
an ellipse (Eq. 1) [Fig. 2(b)]. We can also obtain the spectral projection
onto the $(k_{x},k_{z})$-plane by eliminating $\omega$: substituting
$\tfrac{\omega}{c}\\!=\\!\beta_{v}k_{z}+k_{\mathrm{o}}\sqrt{1-\beta_{v}^{2}}$
from the spectral constraint into
$k_{x}^{2}+k_{z}^{2}\\!=\\!(\tfrac{\omega}{c})^{2}$ yields the ellipse
$\tfrac{k_{x}^{2}}{k_{\mathrm{o}}^{2}}+\tfrac{1}{(\Delta
k_{z}/2)^{2}}(k_{z}-\beta_{v}k_{1})^{2}\\!=\\!1$, where $\Delta
k_{z}\\!=\\!2k_{1}$, and $\beta_{v}k_{1}$ is the central axial wave number.
Substituting $k_{x}\\!=\\!\tfrac{\omega}{c}\sin{\varphi}$ and
$k_{z}\\!=\\!\tfrac{\omega}{c}\cos{\varphi}$, the propagation angle is
$\cos{\varphi(\omega)}\\!=\\!\\{1-\tfrac{\omega_{\mathrm{o}}}{\omega}\sqrt{1-\beta^{2}}\\}/\beta_{v}$,
$\omega\\!\neq\\!0$, which is not differentiable at $\varphi\\!=\\!0$ or
$\varphi\\!=\\!\pi$ – corresponding to the maximum and minimum points on the
ellipses in the $(k_{x},\tfrac{\omega}{c})$ or $(k_{x},k_{z})$ planes.

Optical dBM wave packets in presence of anomalous GVD. In presence of
anomalous GVD, the wave number in the medium expanded around
$\omega_{\mathrm{o}}$ is
$k\\!=\\!n(\omega)\omega/c\\!=\\!n_{\mathrm{m}}k_{\mathrm{o}}+\Omega/\widetilde{v}_{\mathrm{m}}-\tfrac{1}{2}|k_{2\mathrm{m}}|\Omega^{2}$;
here $\Omega\\!=\\!\omega-\omega_{\mathrm{o}}$ and $n(\omega)$ is the
frequency-dependent refractive index. The quantities $n_{\mathrm{m}}$,
$\widetilde{v}_{\mathrm{m}}$, and $k_{2\mathrm{m}}$ are all evaluated in the
medium at $\omega\\!=\\!\omega_{\mathrm{o}}$:
$n_{\mathrm{m}}\\!=\\!n(\omega_{\mathrm{o}})$ is the refractive index;
$\widetilde{v}\\!=\\!1\big{/}\tfrac{dk}{d\omega}\big{|}_{\omega_{\mathrm{o}}}$
is the group velocity, and
$k_{2\mathrm{m}}\\!=\\!\tfrac{d^{2}k}{d\omega^{2}}\big{|}_{\omega_{\mathrm{o}}}\\!=\\!-|k_{2\mathrm{m}}|$
is the negative-valued GVD coefficient in the anomalous dispersion regime. In
the small-angle (paraxial) approximation,

$k_{z}=\sqrt{k^{2}-k_{x}^{2}}\approx
n_{\mathrm{m}}k_{\mathrm{o}}+\frac{\Omega}{\widetilde{v}_{\mathrm{m}}}-\frac{1}{2}|k_{2\mathrm{m}}|\Omega^{2}-\frac{k_{x}^{2}}{2n_{\mathrm{m}}k_{\mathrm{o}}},$
(7)

and the light-line $k_{x}\\!=\\!0$) is now curved [Fig. 2(c)]. To produce a
dBM wave packet, we impose the spectral constraint

$k_{z}=n_{\mathrm{m}}k_{\mathrm{o}}+\frac{\Omega}{\widetilde{v}},$ (8)

which intersects with the light-line at two points: $\Omega\\!=\\!0$
($\omega\\!=\\!\omega_{\mathrm{o}}$) and
$\tfrac{\omega}{c}\\!=\\!-2k_{\mathrm{a}}\tfrac{1-\beta_{v}^{\prime}}{\beta_{v}^{\prime}}$;
here
$\beta_{v}^{\prime}\\!=\\!\tfrac{\beta_{v}}{\beta_{\mathrm{m}}}\\!=\\!\tfrac{\widetilde{v}}{\widetilde{v}_{\mathrm{m}}}$,
$\widetilde{v}\\!<\\!\widetilde{v}_{\mathrm{m}}$,
$k_{\mathrm{a}}\\!=\\!(c^{2}|k_{2\mathrm{m}}|\beta_{\mathrm{m}})^{-1}\\!=\\!\tfrac{\omega_{\mathrm{a}}}{c}$.
The maximum temporal bandwidth compatible with dispersion-free propagation in
this dispersive medium at a group velocity $\widetilde{v}$ is thus
$\tfrac{\Delta\omega}{c}\\!=\\!2k_{\mathrm{a}}\tfrac{1-\beta_{v}^{\prime}}{\beta_{v}^{\prime}}$,
and the corresponding maximum bandwidth of the axial wave number is $\Delta
k_{z}\\!=\\!2\tfrac{k_{\mathrm{a}}}{\beta_{\mathrm{m}}}\tfrac{1-\beta_{v}^{\prime}}{\beta_{v}^{\prime
2}}$, so that $\tfrac{\Delta\omega/c}{\Delta
k_{z}}\\!=\\!\beta_{\mathrm{m}}\beta_{v}^{\prime}\\!=\\!\beta_{v}$.

We can now obtain the spectral projections onto the
$(k_{x},\tfrac{\omega}{c})$ and $(k_{z},\tfrac{\omega}{c})$ planes for the
optical dBM wave packet in presence of anomalous GVD just as we did for their
counterparts in free space. By equating Eq. 7 and Eq. 8 we eliminate $k_{z}$
and obtain in the $(k_{x},\tfrac{\omega}{c})$-plane the ellipse
$\tfrac{k_{x}^{2}}{k_{x,\mathrm{max}}^{2}}+\tfrac{(\omega-\omega_{\mathrm{c}})^{2}}{(\Delta\omega/2)^{2}}\\!=\\!1$,
where the central frequency is
$\omega_{\mathrm{c}}\\!=\\!\omega_{\mathrm{o}}-\tfrac{\Delta\omega}{2}$,
$k_{x,\mathrm{max}}^{2}\\!=\\!\sigma_{\mathrm{m}}(\tfrac{\Delta\omega/2}{c})^{2}$,
and
$\sigma_{\mathrm{m}}\\!=\\!n_{\mathrm{m}}c\omega_{\mathrm{o}}|k_{2\mathrm{m}}|$
is a dimensionless GVD parameter. Similarly, we can obtain the spectral
projection onto the $(k_{x},k_{z})$-plane by substituting
$\Omega\\!=\\!\widetilde{v}(k_{z}-n_{\mathrm{m}}k_{\mathrm{o}})$ from the
spectral constraint in Eq. 8 into the light-cone in Eq. 7 to obtain the
ellipse:
$\tfrac{k_{x}^{2}}{k_{x,\mathrm{max}}^{2}}+\tfrac{(k_{z}-k_{\mathrm{c}})^{2}}{(\Delta
k_{z}/2)^{2}}\\!=\\!1$, where $k_{\mathrm{c}}$ is the center of the
$k_{z}$-span.

Details of the experimental setup and spectral measurements. The field
configuration shown in Fig. 2(c) is produced via the angular-dispersion
synthesizer depicted in Fig. 3 []. Starting with femtosecond laser pulses
(central wavelength $\lambda\\!=\\!1064$ nm, bandwidth
$\Delta\lambda\\!=\\!20$ nm, and pulsewidth $\Delta T\\!\approx\\!100$ fs;
Spark Lasers, Alcor), a diffraction grating (1200 lines/mm) resolves the
spectrum spatially, and a cylindrical lens (focal length $f\\!=\\!500$ mm)
collimates the wave front before incidence on a reflective, phase-only SLM
(Meadowlark, E19X12). The SLM deflects each wavelength $\lambda$ at angles
$\pm\varphi(\lambda)$ according to the elliptical spatio-temporal spectrum
given in Eq. 3. The wave front retro-reflected from the SLM returns to the
grating, and the optical dBM wave packet is formed. The spatio-temporal
spectrum is acquired by directing a portion of the spectrally resolved wave
front reflecting back from the SLM to a spatial Fourier transform (not shown
in Fig. 3 for simplicity), which thus yields the spatio-temporal spectral
projection onto the $(k_{x},\lambda)$-plane. We then obtain the spectral
projection onto the $(k_{z},\lambda)$-plane in the medium via the relationship
$k_{z}(\omega)\\!=\\!\sqrt{\\{(n(\omega)\tfrac{\omega}{c}\\}^{2}-k_{x}^{2}(\omega)}$.

Dispersive medium. The dispersive sample that we exploit comprises a pair of
chirped Bragg mirrors (Edmund 12-335) that provide anomalous group-delay
dispersion (GDD). By adjusting the separation between the two mirrors and the
incident angle of the wave packet, we can control the number of bounces off
the mirrors, thereby producing an anomalous-GVD coefficient of
$k_{2\mathrm{m}}\\!\approx\\!-500$ fs2/mm. Because the thickness of the
mirrors is negligible with respect to the free-space gap separating them, we
can thus have $\widetilde{v}_{\mathrm{m}}\\!\approx\\!c$
($\widetilde{n}_{\mathrm{m}}\\!\approx\\!1$).

Reconstruction of the spatio-temporal profiles of dBM wave packets. To
reconstruct the spatio-temporal intensity profile of a dBM wave packet
$I(x,z;\tau)$ at a fixed axial plane $z$, we make use of the interferometric
arrangement depicted schematically in Fig. 3. We bring together two wave
packets: the synthesized optical dBM wave packet, and a reference plane-wave
pulse taken from the initial laser pulse Kondakci and Abouraddy (2019). An
optical delay $\tau$ is placed in the path of the reference pulse. When the
dBM wave packet and the reference pulse overlap in space and time, we observe
high-visibility spatially resolved interference fringes at the CCD camera
placed in their common path. As we sweep the optical delay $\tau$ (thus
reducing the overlap between the dBM wave packet and the reference pulse), the
interference visibility drops. We make use of the recorded visibility along
$x$ and $tau$ to reconstruct the wave packet intensity profile $I(x;\tau)$.

Figure 1: de Broglie phase waves and wave packets, and de Broglie-Mackinnon
(dBM) wave packets. (a) In the rest frame of a particle, the internal ‘clock’
and the external ‘phase wave’ theorized by de Broglie both have the same
frequency $\omega_{\mathrm{o}}$. (b) When the particle moves at a velocity $v$
along $z$, the frequency of the internal clock in the rest frame decreases to
$\omega\\!=\\!\omega_{\mathrm{o}}\sqrt{1-\beta_{v}^{2}}$, whereas that of the
phase wave increases to
$\omega\\!=\\!\omega_{\mathrm{o}}/\sqrt{1-\beta_{v}^{2}}$. (c) The dispersion
relationship for de Broglie phase waves
$\omega^{2}\\!=\\!\omega_{\mathrm{o}}^{2}+c^{2}k^{2}$ plotted in
$(k,\tfrac{\omega}{c})$-space. The group velocity evaluated at
$\omega\\!=\\!\omega_{\mathrm{c}}$ is $\widetilde{v}\\!=\\!v$. Constructing a
localized de Broglie wave packet necessitates introducing an ad hoc
uncertainty in the particle velocity. (d) The physical setting for a dBM wave
packet. The particle travels at $v$ and the observer at $u$ (both along the
$z$-axis) with respect to a common rest frame. (e) The dispersion relationship
for a dBM wave packet in $(k,\tfrac{\omega}{c})$-space (lower panel)
corresponding to a stationary particle $v\\!=\\!0$, delimited by the light-
lines $|k|\\!=\\!\tfrac{\omega}{c}$. The observer velocity $u$ (upper panel)
is an internal parameter swept from $-c$ to $c$ to produce the dBM dispersion
relationship. (g) Same as (f) for $v\\!\neq\\!0$; here
$k_{+}\\!=\\!k_{\mathrm{o}}\sqrt{\tfrac{1+\beta_{v}}{1-\beta_{v}}}$,
$k_{-}\\!=\\!k_{\mathrm{o}}\sqrt{\tfrac{1-\beta_{v}}{1+\beta_{v}}}$,
$k_{1}\\!=\\!\tfrac{k_{\mathrm{o}}}{\sqrt{1-\beta_{v}^{2}}}$, and
$k_{2}\\!=\\!\tfrac{k_{\mathrm{o}}}{\beta_{v}}(1-\tfrac{k_{\mathrm{o}}}{k_{1}})$;
see Methods Figure 2: Optical de Broglie-Mackinnon (dBM) wave packets. From
left to right we depict the following: the light-cone in
$(k_{x},k_{z},\tfrac{\omega}{c})$-space intersecting with a spectral
constraint in the form of a plane; the spectral projection onto the
$(k_{z},\tfrac{\omega}{c})$-plane; the spectral projection onto the
$(k_{x},\tfrac{\omega}{c})$-plane; the propagation angle $\varphi(\omega)$;
and the real part of the spatio-temporal field profile $\psi(x,z;t)$ at a
fixed axial plane $z$. (a) A stationary monochromatic planar dipole in free
space resulting from the constraint $\omega\\!=\\!\omega_{\mathrm{o}}$. (b) A
moving planar dipole in free space corresponding to the constraint
$k_{z}\\!=\\!\tfrac{1}{\beta_{v}}(\tfrac{\omega}{c}-\tfrac{k_{\mathrm{o}}^{2}}{k_{1}})$.
(c) The field in a dispersive medium in the anomalous regime after imposing
the constraint
$k_{z}\\!=\\!\tfrac{1}{\beta_{v}}\\{\tfrac{\omega}{c}-k_{\mathrm{o}}(1-n_{\mathrm{m}}\beta_{v})\\}$.
Figure 3: Synthesizing and characterizing optical dBM wave packets. Starting
with femtosecond laser pulses, the space-time (ST) synthesis arrangement
associates each wavelength $\lambda$ with prescribed propagation angles
$\pm\varphi(\lambda)$ before traversing the dispersive medium Hall and
Abouraddy (2021). Interfering the synthesized optical dBM wave packets with
reference plane-wave pulses from the initial laser helps reconstruct the
spatio-temporal intensity profile of the dBM wave packets. Figure 4:
Observation of optical dBM wave packets in presence of anomalous GVD and
tuning their group velocity. In the first column we plot the measured spatio-
temporal spectrum projected onto the $(k_{x},\lambda)$-plane; the dotted curve
is the theoretical expectation based on Eq. 3. In the second column we plot
the spectral projection onto the $(k_{z},\lambda)$-plane using the data from
the first column; the dotted curve is the dispersive light-line. In the third
column we plot the spatio-temporal intensity profile $I(x;\tau)$ acquired in a
moving reference frame traveling at the group velocity of the wave packets;
the dotted circles are guides for the eye. (a) Measurements for an optical dBM
wave packet having a group velocity $\widetilde{v}\\!=\\!0.9975c$; (b)
$\widetilde{v}\\!=\\!0.9985c$; and (c) $\widetilde{v}\\!=\\!0.999c$. The
dimensionaless dispersion coefficient is
$\sigma_{\mathrm{m}}\\!=\\!c\omega_{\mathrm{o}}k_{2\mathrm{m}}=0.3$, and the
measurements are carried out at $z\\!=\\!30$ mm. Figure 5: Tuning the
equivalent rest-mass of an optical dBM wave packets. The columns correspond to
those in Fig. 4. The dimensionless dispersion coefficient is
$\sigma_{\mathrm{m}}\\!=\\!c\omega_{\mathrm{o}}k_{2\mathrm{m}}=0.3$, the
measurements are all acquired at $z\\!=\\!30$ mm, and the group velocity is
held fixed at $\widetilde{v}=0.9975c$. (a) The short-wavelength intersection
with the dispersive light-line is $\lambda_{\mathrm{o}}\\!=\\!1054$ nm; (b)
$\lambda_{\mathrm{o}}\\!=\\!1055$ nm; and (c)
$\lambda_{\mathrm{o}}\\!=\\!1056$ nm. Figure 6: Changing the spatio-temporal
structure of optical dBM wave packets. The columns from left to right are: the
measured spatio-temporal spectrum projected onto the $(k_{x},\lambda)$-plane,
along with the spectral phase; the measured spatio-temporal intensity profile
$I(x,z;\tau)$ at $z\\!=\\!15$ mm; the measured spatio-temporal intensity
profile $I(x,z;\tau)$ at $z\\!=\\!30$ mm; the calculated spatio-temporal
intensity profile $I(x,z;\tau)$ at fixed $z$; $I(x;\tau=0)$ at $z\\!=\\!30$
mm; and $I(x=0;\tau)$ at $z\\!=\\!30$ mm. The dimensionless dispersion
coefficient is Measurement of an O shaped spectrum with $v_{a}=0.9975c$ and
$\sigma_{\mathrm{m}}\\!=\\!c\omega_{\mathrm{o}}k_{2\mathrm{m}}=0.3$, and the
group velocity is held fixed at $\widetilde{v}\\!=\\!0.9975c$, corresponding
to Fig. 4(a). (a) The spectrum has uniform phase; (b) the spectrum has uniform
phase but its amplitude is truncated along $\lambda$; (c) a $\pi$-phase step
is introduced along $k_{x}$; (d) a $\pi$-phase step is introduced along
$\lambda$; and (e) the spectral phase is alternated between $0$ and $\pi$ in
the four quadrants of the $(k_{x},\lambda)$-plane.