# Spectral reorganization of space-time wave packets in presence of normal
group-velocity dispersion

Layton A. Hall1 Ayman F. Abouraddy1,∗ 1CREOL, The College of Optics &
Photonics, University of Central Florida, Orlando, FL 32816, USA
∗Corresponding author<EMAIL_ADDRESS>
###### Abstract

Space-time wave packets (STWPs) are pulsed beams that propagate invariantly
(without diffraction or dispersion) in linear media. The behavior of STWPs in
free space is now well-established, and recently their propagation invariance
was confirmed in both the normal and anomalous dispersion regimes. However,
yet-to-be-observed rich dynamics of spectral reorganization have been
predicted to occur in the presence of normal group-velocity dispersion (GVD).
Indeed, propagation invariance in the normal-GVD regime is compatible with
spatio-temporal spectra that are X-shaped, hyperbolic, parabolic, elliptical,
or even separable along the spatial and temporal degrees-of-freedom. These
broad varieties of field structures can be classified in a two-dimensional
space parameterized by the group velocity of the STWP and its central axial
wave number. Here we observe the entire span of spectral reorganization for
STWPs in the paraxial regime in normally dispersive ZnSe at a wavelength $\sim
1$ $\mu$m with STWPs of on-axis pulse width of $\sim\\!200$ fs. By tuning the
group velocity and central axial wave number of the STWP, we observe
transitions in the structure of the spatio-temporal spectrum and verify the
associated change in its intensity profile. These results lay the foundation
for initiating novel phase-matched nonlinear processes using STWPs in
dispersive media.

##  I Introduction

Space-time wave packets (STWPs) are a class of spatio-temporally structured
pulsed optical fields that are propagation-invariant in linear media [1, 2, 3,
4, 5, 6]. To date, STWPs have been experimentally studied predominantly in
free space [7] and non-dispersive dielectrics [8], in which a host of useful
characteristics of STWPs have been verified, including: tunable group
velocities [9, 4, 10] and acceleration [11, 12, 13, 14, 15, 16, 17], self-
healing [18], and anomalous refraction [19, 20]. Moreover, almost arbitrary
dispersion profiles can be introduced in free space [21, 22], which has
enabled the observation of the space-time Talbot effect [23].

Although the propagation invariance of STWPs in linear dispersive media has
been pursued theoretically [24, 25, 26, 27, 28], the only experimental test
has been the propagation of X-waves in normally dispersive silica glass [29,
30] (subsequent attempts [31, 32] with STWPs could not confirm this effect).
Extensive studies have investigated the generation of STWPs in nonlinear
dispersive media [33, 34], but their propagation invariance was not verified.
However, rapidly evolving developments in preparing STWPs via universal
angular-dispersion synthesis [35] has very recently made it possible to
confirm dispersion-free propagation of STWPs in dispersive media – in presence
of either normal or anomalous GVD [36].

In addition to propagation invariance, a rich spatio-temporal structural
dynamics has been predicted to accompany STWPs in dispersive media, which have
not been observed to date. In free space, although elliptic, hyperbolic, or
parabolic spatio-temporal spectra can occur as the STWP group velocity is
tuned, nevertheless, they are all well-approximated in the paraxial regime by
a parabola, and their intensity profiles usually show no accompanying
variation (except in presence of spectral-phase modulation [37, 38]). In
contrast, Malaguti and Trillo (MT henceforth) have predicted fascinating
changes in the spatio-temporal spectra of STWPs in the presence of normal GVD
[39] (see also [40], in addition to earlier [24, 25, 26, 27, 28] and recent
work [41]). Here, a wide range of spatio-temporal spectra are compatible with
dispersion-free propagation in the normal-GVD regime: including elliptic,
parabolic, hyperbolic, X-shaped, and separable spectra – all clearly
observable in the paraxial regime. Tuning the STWP parameters lead to
crossovers between these different structures and thus ‘spectral
reorganization’ of the STWP. Once the medium characteristics are fixed, two
key parameters dictate these dynamics: the STWP group velocity and its central
axial wave number. Tuning one of these parameters or both leads to dramatic
changes in the spectral structure of the STWP if it is to remain dispersion-
free in the medium. Indeed, tuning these parameters is predicted to drive
structural reorganization in the STWP spectrum and spatio-temporal intensity
profile [39].

Here, we observe for the first time the broad range of predicted spectral
reorganization of propagation-invariant STWPs taking place in presence of
normal GVD as the STWP group velocity and central axial wave number are
continuously tuned within the paraxial regime. Starting with generic
femtosecond pulses at a wavelength $\lambda_{\mathrm{o}}\\!\approx\\!1$
$\mu$m, we synthesize STWPs with on-axis pulsewidths in the range
$\sim\\!200-450$ fs (bandwidths of $8-16$ nm), which are propagation invariant
in normally dispersive ZnSe. By tuning the group velocity and central axial
wave number, we span the two-dimensional (2D) parameter space proposed by MT
(which we refer to as the MT-plane), and reveal the wealth of spatio-temporal
spectral structures and profiles for STWPs that are compatible with
propagation invariance in ZnSe; including elliptic, parabolic, hyperbolic
(with gaps along the spatial or temporal spectral axes), X-shaped, and even
approximately separable spatio-temporally spectra. In light of the importance
of dispersion in the phase-matching of a wide variety of nonlinear optical
processes, we expect that these results will be useful in producing new
nonlinear and quantum optical effects in bulk media and in planar nonlinear
cavities and waveguides [42, 43, 44, 45, 41].

##  II Theory of Propagation-invariant space-time wave packets in presence of
normal GVD

###  II.1 Spectral representation of conventional pulsed beams on the light-
cone surface

In a medium of refractive index $n(\omega)$, the dispersion relationship for a
monochromatic plane wave is $k_{x}^{2}+k_{z}^{2}\\!=\\!k^{2}$, where
$k(\omega)\\!=\\!n(\omega)\tfrac{\omega}{c}$ is the wave number, $\omega$ is
the angular temporal frequency, $c$ is the speed of light in vacuum, and
$k_{x}$ and $k_{z}$ are the transverse and longitudinal components of the wave
vector along the transverse $x$ and axial $z$ coordinates, respectively (we
hold the field uniform along $y$ for simplicity). We expand
$k(\omega)\\!=\\!k(\omega_{\mathrm{o}}+\Omega)$ to second order in
$\Omega\\!=\\!\omega-\omega_{\mathrm{o}}$ (where $\omega_{\mathrm{o}}$ is a
fixed frequency):
$k(\omega)\\!=\\!n_{\mathrm{m}}k_{\mathrm{o}}+\tfrac{\Omega}{\widetilde{v}_{\mathrm{m}}}+\tfrac{1}{2}k_{2}\Omega^{2}$,
where $n_{\mathrm{m}}\\!=\\!n(\omega_{\mathrm{o}})$,
$k_{\mathrm{o}}\\!=\\!\tfrac{\omega_{\mathrm{o}}}{c}$,
$\widetilde{v}_{\mathrm{m}}\\!=\\!1\big{/}\tfrac{dk}{d\omega}\big{|}_{\omega_{\mathrm{o}}}$
is the group velocity of a plane-wave pulse in the medium, and
$k_{2}\\!=\\!\tfrac{d^{2}k}{d\omega^{2}}\big{|}_{\omega_{\mathrm{o}}}$ is the
GVD coefficient (positive-valued for normal GVD) [46]. Writing the field as
$E(x,z;t)\\!=\\!e^{i(n_{\mathrm{m}}k_{\mathrm{o}}z-\omega_{\mathrm{o}}t)}\psi(x,z;t)$,
the envelope is:

$\psi(x,z;t)\\!=\\!\\!\iint\\!dk_{x}d\Omega\widetilde{\psi}(k_{x},\Omega)e^{i(k_{x}x-\tfrac{k_{x}^{2}z}{2n_{\mathrm{m}}k_{\mathrm{o}}})}e^{-i\Omega(t-\tfrac{z}{\widetilde{v}_{\mathrm{m}}})}e^{ik_{2}\Omega^{2}z/2},$
(1)

where the spatio-temporal spectrum $\widetilde{\psi}(k_{x},\Omega)$ is the
Fourier transform of $\psi(x,0;t)$, and the envelope undergoes diffractive
spatial spreading and dispersive temporal broadening with propagation along
$z$. With respect to the geometric cone
$k_{x}^{2}+k_{z}^{2}\\!=\\!(\tfrac{\omega}{c})^{2}$ associated with
propagation in free space [Fig. 1(a)], the light-cone
$k_{x}^{2}+k_{z}^{2}\\!=\\!(n(\omega)\tfrac{\omega}{c})^{2}$ in presence of
normal GVD is curved inwardly [Fig. 1(c)], a fact that undergirds many of the
phenomena described below. For a conventional pulsed beam (Eq. 1), the
spectral support for $\widetilde{\psi}(k_{x},\Omega)$ on the light-cone
surface is a 2D domain, corresponding to finite spatial and temporal
bandwidths $\Delta k_{x}$ and $\Delta\omega$, respectively.

###  II.2 Spectral representation of STWPs on the light-cone surface

The impact of both diffraction and dispersion can be halted in a STWP by
imposing a strict association between $k_{x}$ and $\Omega$, so that the
spectral support on the light-cone surface becomes a 1D curve rather than a 2D
domain [47, 48, 49, 1]:
$\widetilde{\psi}(k_{x},\Omega)\\!\rightarrow\\!\widetilde{\psi}(\Omega)\delta(k_{x}-k_{x}(\Omega))$.
The association $k_{x}\\!=\\!k_{x}(\Omega)$ enforces the following constraint
on the axial wave number [48]:

$k_{z}(\Omega)=b+\frac{\Omega}{\widetilde{v}},$ (2)

where $\widetilde{v}$ is the group velocity of the STWP in the dispersive
medium ($\widetilde{v}$ may be higher or lower than
$\widetilde{v}_{\mathrm{m}}$), and $b\\!<\\!n_{\mathrm{m}}k_{\mathrm{o}}$ is
the axial wave number at $\omega\\!=\\!\omega_{\mathrm{o}}$
($\Omega\\!=\\!0$), referred to henceforth as the central axial wave number.
Writing $b\\!=\\!n_{\mathrm{m}}k_{\mathrm{o}}\cos{\varphi_{\mathrm{o}}}$,
$\varphi_{\mathrm{o}}$ is the angle the frequency component
$\omega_{\mathrm{o}}$ makes with the $z$-axis, and
$b\\!<\\!n_{\mathrm{m}}k_{\mathrm{o}}$ therefore indicates an angular or
directional offset at $\omega_{\mathrm{o}}$. The constraint on $k_{z}$ in Eq.
2 confines the STWP spectral support on the dispersive light-cone surface to
its intersection with a spectral plane $\mathcal{P}$ defined by Eq. 2. This
plane is parallel to the $k_{x}$-axis and makes an angle $\theta$ with respect
to the $k_{z}$-axis, where $\widetilde{v}\\!=\\!c\tan{\theta}$. This
intersection is in general a conic section (if terms up to second order in
$\Omega$ are retained), and its projection onto the
$(k_{z},\tfrac{\omega}{c})$-plane is a straight line, indicating absence of
dispersion of any order for the STWP in the dispersive medium [Fig. 1(c)].

Figure 1: (a) The spectral support of a free-space STWP is the intersection of
the light-cone $k_{x}^{2}+k_{z}^{2}\\!=\\!(\tfrac{\omega}{c})^{2}$ with a
tilted spectral plane $\mathcal{P}$. (b) Possible spectral projections onto
the $(k_{x},\Omega)$-plane in free space. (c) The light-cone
$k_{x}^{2}+k_{z}^{2}\\!=\\!(n(\omega)\tfrac{\omega}{c})^{2}$ in presence of
normal GVD and the spectral support of a STWP at its intersection with
$\mathcal{P}$. (d) Possible spectral projections onto the
$(k_{x},\Omega)$-plane in presence of normal GVD.

In presence of normal GVD, a wide variety of geometric structures can occur
for the spatio-temporal spectra (or ‘spectra’ for brevity), including: an
ellipse; a hyperbola (both branches of which can be accessible in the paraxial
regime) with a gap along $k_{x}$ or along $\Omega$; a parabola; an X-shaped
spectrum; or a spectrum that is separable in terms of $k_{x}$ and $\Omega$
formed of a pair of parallel straight lines [Fig. 1(d)]. In contrast, the
spectra in free space can only be an ellipse, a hyperbola (only a single
branch of which is accessible for $\omega\\!>\\!0$), a parabola, or a straight
line ($\mathcal{P}$ is tangential to the light-cone); see Fig. 1(b). In the
paraxial regime, the portion of the spectrum in free space in the vicinity of
$k_{x}\\!=\\!0$ (so-called ‘baseband’ STWPs [49]) can always be approximated
by a parabola [1].

###  II.3 Representation in the MT-plane

Figure 2: (a) In the MT-plane spanned by $\varphi_{\mathrm{o}}$ (the angular
offset at $\omega\\!=\\!\omega_{\mathrm{o}}$) and the STWP group velocity
$\widetilde{v}$, each point determines a unique trio of coefficients
$\alpha_{0}$, $\alpha_{1}$, and $\alpha_{2}$ (Eq. 4). The horizontal line
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}\\!<\\!\widetilde{v}_{\mathrm{m}}$
is the locus of $\alpha_{2}\\!=\\!0$ (parabolic spectra), separating the plane
into an upper region $\alpha_{2}\\!>\\!0$ (hyperbolic spectra), and a lower
region $\alpha_{2}\\!<\\!0$ (elliptical spectra). (b) Representation on the
dispersive light-cone of the STWPs identified by points (i)-(iv) in (a). (c)
The curve
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})$ is
the locus of $\alpha_{1}\\!=\\!0$. The point
$\varphi_{\mathrm{o}}\\!=\\!\varphi_{\mathrm{B}}$
($\alpha_{1}\\!=\\!\alpha_{2}\\!=\\!0$) at
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}\\!=\\!\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{B}})$
corresponds to a dispersion-free pulsed Bessel beam that is separable with
respect to space and time. (d) Spectral representation of the STWPs identified
by points (i)-(iv) in (c). (e) The domain $\alpha_{2}\\!>\\!0$ of hyperbolic
spectra is subdivided by means of the curves $\widetilde{v}_{\mathrm{max}}$
and $\widetilde{v}_{\mathrm{min}}$ corresponding to $\Delta\\!=\\!0$, along
which the spectrum is X-shaped. In the shaded regions where $\Delta\\!>\\!0$,
the hyperbolas have an $\Omega$-gap, and a $k_{x}$-gap in the region between
$\widetilde{v}_{\mathrm{min}}$ and $\widetilde{v}_{\mathrm{max}}$ where
$\Delta\\!<\\!0$. (f) Spectral representation of the STWPs identified by
points (i)-(iv) in (e).

The field in the dispersive medium after applying the constraint in Eq. 2 is
$E(x,z;t)\\!=\\!e^{i(bz-\omega_{\mathrm{o}})}\psi(x,z;t)$, where:

$\psi(x,z;t)=\\!\int\\!\\!d\Omega\;\widetilde{\psi}(\Omega)e^{ik_{x}(\Omega)x}e^{-i\Omega(t-\tfrac{z}{\widetilde{v}})}=\psi(x,0;t-z/\widetilde{v}),$
(3)

which represents a STWP traveling rigidly (without dispersion or diffraction)
at a group velocity $\widetilde{v}$. The dispersion relationship for
propagation invariance in the medium is
$k_{x}^{2}+(b+\tfrac{\Omega}{\widetilde{v}})^{2}\\!=\\!(n_{\mathrm{m}}k_{\mathrm{o}}+\tfrac{\Omega}{\widetilde{v}_{\mathrm{m}}}+\tfrac{1}{2}k_{2}\Omega^{2})^{2}$,
and retaining terms up to second order in $\Omega$ yields:

$k_{x}^{2}(\Omega)=\alpha_{0}+2\alpha_{1}\Omega+\alpha_{2}\Omega^{2},$ (4)

where the coefficients $\alpha_{0}$, $\alpha_{1}$, and $\alpha_{2}$ are given
by:

$\displaystyle\alpha_{0}$ $\displaystyle=$
$\displaystyle(n_{\mathrm{m}}k_{\mathrm{o}}\sin{\varphi_{\mathrm{o}}})^{2},$
$\displaystyle\alpha_{1}$ $\displaystyle=$ $\displaystyle
n_{\mathrm{m}}k_{\mathrm{o}}\cos{\varphi_{o}}\left(\frac{1}{\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})}-\frac{1}{\widetilde{v}}\right),$
$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\widetilde{v}_{\mathrm{e}}^{2}}-\frac{1}{\widetilde{v}^{2}};$
(5)

where
$\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})\\!=\\!\widetilde{v}_{\mathrm{m}}\cos{\varphi_{\mathrm{o}}}$,
$\widetilde{v}_{\mathrm{e}}\\!=\\!\widetilde{v}_{\mathrm{m}}\cos{\varphi_{\mathrm{B}}}$,
and
$\tan^{2}{\varphi_{\mathrm{B}}}\\!=\\!n_{\mathrm{m}}k_{\mathrm{o}}k_{2}\widetilde{v}_{\mathrm{m}}^{2}$
(the physical significance of these terms will become clear shortly). It is
crucial to appreciate that Eq. 4 is not a perturbative expansion of $k_{x}$ in
terms of $\Omega$ (although we do truncate the equation at second order in
$\Omega$). Indeed, any of the terms in Eq. 4 can vanish or dominate the sum.
In fact, $k_{x}$ may not even be differentiable with respect to $\Omega$ at
$\Omega\\!=\\!0$.

Once the material properties $n_{\mathrm{m}}$, $\widetilde{v}_{\mathrm{m}}$,
and $k_{2}$ are determined at $\omega_{\mathrm{o}}$, the structure of the
dispersive light-cone is fixed. Only two independent parameters –
$\widetilde{v}$ and $\varphi_{\mathrm{o}}$ – remain to determine the values of
the coefficients $\alpha_{0}$, $\alpha_{1}$, and $\alpha_{2}$. Therefore, all
classes of propagation-invariant STWPs in the normal-GVD regime can be
uniquely mapped to a 2D parameter space spanned by $\widetilde{v}$ and
$\varphi_{\mathrm{o}}$, which we refer to as the MT-plane [39]. Each point in
this space identifies a class of STWPs associated with a particular functional
form of $k_{x}(\Omega)$ in Eq. 4. The horizontal axis at
$\widetilde{v}\\!=\\!0$ represents monochromatic beams, whereas the vertical
axis at $\varphi_{\mathrm{o}}\\!=\\!0$
($b\\!=\\!n_{\mathrm{m}}k_{\mathrm{o}}$) corresponds to STWPs whose spectra
pass through the point $(k_{x},\Omega)\\!=\\!(0,0)$.

In contrast, free-space STWPs are uniquely identified by the single parameter
$\widetilde{v}$, and hence the fewer spectral structures accessible in that
case [7] [compare Fig. 1(b) with Fig. 1(d)]. We proceed to describe the
changes in the spectral structure for STWPs across the MT-plane.

###  II.4 Escape velocity

A critical class of STWPs are those whose group velocity is equal to the
‘escape’ velocity
$\widetilde{v}_{\mathrm{e}}\\!=\\!\widetilde{v}_{\mathrm{m}}\cos{\varphi_{\mathrm{B}}}\\!<\\!\widetilde{v}_{\mathrm{m}}$,
whereupon the coefficient $\alpha_{2}$ vanishes. The locus of STWPs in the MT-
plane for which $\alpha_{2}\\!=\\!0$ is thus the horizontal line
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}$ [Fig. 2(a)], which divides
the MT-plane into two regions: the spectrum is a hyperbola in the upper region
($\widetilde{v}\\!>\\!\widetilde{v}_{\mathrm{e}}$ and $\alpha_{2}\\!>\\!0$),
and is an ellipse in the lower region
($\widetilde{v}\\!<\\!\widetilde{v}_{\mathrm{e}}$ and $\alpha_{2}\\!<\\!0$).
Along $\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}$ where
$\alpha_{2}\\!=\\!0$, the spectrum
$k_{x}^{2}(\Omega)\\!=\\!\alpha_{\mathrm{o}}+2\alpha_{1}\Omega$ is a parabola
[Fig. 2(b)]. Leaving the paraxial regime, the peak of the parabola reaches
$\Omega\\!\rightarrow\\!\infty$ when
$\varphi_{\mathrm{o}}\\!=\\!\varphi_{\mathrm{B}}$, whereupon the spectrum
becomes a pair of vertical lines. Subsequently, when
$\varphi_{\mathrm{o}}\\!>\\!\varphi_{\mathrm{B}}$, the orientation of the
parabola flips, and its peak lies at $\Omega\\!>\\!0$. However, these dynamics
around $\varphi_{\mathrm{o}}\\!=\\!\varphi_{\mathrm{B}}$ occur typically in
the non-paraxial regime.

###  II.5 Tilted-pulse-front velocity

A second constraint of interest is that resulting in
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})\\!=\\!\widetilde{v}_{\mathrm{m}}\cos{\varphi_{\mathrm{o}}}$,
where $\alpha_{1}\\!=\\!0$ and is represented by the curved path in Fig. 2(c).
The form of this group velocity
$\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})$ is that associated with
so-called tilted pulse fronts (TPFs) in the dispersive medium [50, 51]. A TPF
results from introducing conventional angular dispersion into a pulsed field
(as occurs when it traverses a grating or a prism [52, 53]).

A wide variety of spectra arise along this curve. At
$\varphi_{\mathrm{o}}\\!=\\!0$ and
$\widetilde{v}_{\mathrm{TPF}}(0^{\circ})\\!=\\!\widetilde{v}_{\mathrm{m}}\\!>\\!\widetilde{v}_{\mathrm{e}}$
($\alpha_{2}\\!>\\!0$), we have $\alpha_{0}\\!=\\!\alpha_{1}\\!=\\!0$ and thus
$k_{x}\\!=\\!\pm\sqrt{\alpha_{2}}\Omega$, so that the spectrum is X-shaped and
formed of two straight lines symmetrically tilted and intersecting at
$(k_{x},\Omega)\\!=\\!(0,0)$, which is associated with $\mathcal{P}$
tangential to the light-cone at $\omega_{\mathrm{o}}$ [Fig. 2(d)]. The special
case of an X-shaped spectrum was studied for free-space STWPs where it was
demonstrated that they experience anomalous-GVD [54] (and thus may propagate
dispersion-free in the normal-GVD regime).

As $\varphi_{\mathrm{o}}$ increases in the $\alpha_{2}\\!>\\!0$ region
($\widetilde{v}\\!>\\!\widetilde{v}_{\mathrm{e}}$), the X-shaped spectrum
evolves into a hyperbola with a gap along $k_{x}$, and whose center is at
$(k_{x},\Omega)\\!=\\!(0,0)$ [Fig. 2(d)]. Upon reaching the point
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}$ at
$\varphi_{\mathrm{o}}\\!=\\!\varphi_{\mathrm{B}}$ whereupon
$\alpha_{1}\\!=\\!\alpha_{2}\\!=\\!0$ and $k_{x}\\!=\\!\pm
n_{\mathrm{m}}k_{\mathrm{o}}\sin{\varphi_{\mathrm{B}}}$, the hyperbolic
curvature vanishes and the spectrum is formed of two parallel straight lines,
thus decoupling $k_{x}$ and $\Omega$ [Fig. 2(d)]. The envelope
$\psi_{\mathrm{B}}(x,z;t)\\!\propto\\!\cos{(k_{x}x)}\psi_{t}(t-z/\widetilde{v}_{\mathrm{e}})$
becomes separable in space and time, corresponding to a cosine wave in the
transverse plane and an arbitrary pulse profile in time propagating
dispersion-free at a group velocity $\widetilde{v}_{\mathrm{e}}$. When both
transverse dimensions are taken into consideration, the transverse profile
takes the form of a Bessel beam
$\psi_{\mathrm{B}}(r,z;t)\\!\propto\\!J_{0}(k_{r}r)\psi_{t}(t-z/\widetilde{v}_{\mathrm{e}})$,
where $r$ is the radial coordinate and $k_{r}$ a fixed radial wave number.
This STWP in presence of normal GVD has been studied theoretically [55, 56,
57, 58] and is typically called a pulsed Bessel beam [59]. To the best of our
knowledge, it has not been realized experimentally to date because
$\varphi_{\mathrm{B}}$ is typically in the non-paraxial domain (e.g.,
$\varphi_{\mathrm{B}}\\!\approx\\!19.2^{\circ}$ for ZnSe at
$\lambda_{\mathrm{o}}\\!\approx\\!1$ $\mu$m). However, prior to reaching
$\varphi_{\mathrm{B}}$, the hyperbolic curvature is low and the STWP is
approximately separable (see below). When
$\widetilde{v}_{\mathrm{TPF}}\\!<\\!\widetilde{v}_{\mathrm{e}}$ and
$\alpha_{2}\\!<\\!0$ ($\varphi_{\mathrm{o}}\\!>\\!\varphi_{\mathrm{B}}$), the
spectrum is an ellipse whose center is $(k_{x},\Omega)\\!=\\!(0,0)$; the
ellipse is deep within the non-paraxial regime in the vicinity of
$\omega_{\mathrm{o}}$ [Fig. 2(d)].

###  II.6 Increasing the group velocity at zero angular offset

The physical meaning of the escape velocity $\widetilde{v}_{\mathrm{e}}$ can
be understood geometrically by following the evolution of the spectrum as we
increase $\widetilde{v}$ while holding $\varphi_{\mathrm{o}}\\!=\\!0^{\circ}$
($\alpha_{0}\\!=\\!0$). At $\widetilde{v}\\!=\\!0$, $\mathcal{P}$ is a
horizontal iso-frequency plane that intersects with the light-cone in a circle
(a monochromatic beam). As $\widetilde{v}$ increases, the intersection becomes
an ellipse with vertex at $(k_{x},\Omega)\\!=\\!(0,0)$, while the opposing
vertex moves downwards along the $\Omega$-axis with further increase in
$\widetilde{v}$ until $\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}$
($\alpha_{2}\\!=\\!0$), at which point $\mathcal{P}$ ‘escapes’ from re-
intersecting with the light-cone, and the spectrum becomes a parabola; hence
the appellation ‘escape velocity’.

In the region
$\widetilde{v}_{\mathrm{e}}\\!<\\!\widetilde{v}\\!<\\!\widetilde{v}_{\mathrm{m}}$
($\alpha_{2}\\!>\\!0$ but $\alpha_{1}\\!<\\!0$), the spectrum is a hyperbola
with a gap along $\Omega$; the vertex of the lower branch is
$(k_{x},\Omega)\\!=\\!(0,0)$ and that of the upper is
$(k_{x},\Omega)\\!=\\!(0,2\tfrac{|\alpha_{1}|}{\alpha_{2}})$. Increasing
$\widetilde{v}$ further results in the upper branch approaching the lower
branch until $\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{m}}$, whereupon
$\alpha_{1}\\!=\\!0$ and $k_{x}\\!=\\!\pm\sqrt{\alpha_{2}}\Omega$; the
spectrum becomes X-shaped as described above. Further increasing
$\widetilde{v}\\!>\\!\widetilde{v}_{\mathrm{m}}$ produces once again a
hyperbolic spectrum with a gap along $\Omega$, but the upper-branch vertex is
$(k_{x},\Omega)\\!=\\!(0,0)$, whereas that of the lower is
$(k_{x},\Omega)\\!=\\!(0,-2\tfrac{\alpha_{1}}{\alpha_{2}})$, which moves
downwards with further increase in $\widetilde{v}$.

###  II.7 Hyperbolic spectra

The domain $\widetilde{v}\\!>\\!\widetilde{v}_{\mathrm{e}}$
($\alpha_{2}\\!>\\!0$) where the spectrum is hyperbolic is divided into three
distinct regions in terms of the sign of the determinant
$\Delta\\!=\\!\alpha_{1}^{2}-\alpha_{0}\alpha_{2}$ of Eq. 4 [Fig. 2(e)].

The constraint $\Delta\\!=\\!0$ is satisfied along two curves that branch away
from the point
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{m}},0^{\circ})$
in the MT-plane, and are identified as $\widetilde{v}_{\mathrm{min}}$ and
$\widetilde{v}_{\mathrm{max}}$ in Fig. 2(e), where:

$\widetilde{v}_{\mathrm{min}}=\widetilde{v}_{\mathrm{m}}\frac{\cos{\varphi_{\mathrm{B}}}}{\cos{(\varphi_{\mathrm{o}}-\varphi_{\mathrm{B}})}},\;\;\;\widetilde{v}_{\mathrm{max}}=\widetilde{v}_{\mathrm{m}}\frac{\cos{\varphi_{\mathrm{B}}}}{\cos{(\varphi_{\mathrm{o}}+\varphi_{\mathrm{B}})}}.$
(6)

Along either curve we have
$k_{x}\\!=\\!\pm\sqrt{\alpha_{2}}(\Omega+\tfrac{\alpha_{1}}{\alpha_{2}})$,
which is X-shaped with center at
$\Omega\\!=\\!-\tfrac{\alpha_{1}}{\alpha_{2}}$. Changing
$\varphi_{\mathrm{o}}$ along the $\widetilde{v}_{\mathrm{max}}$-curve results
in $\widetilde{v}$ increasing monotonically, with the center of the X-spectrum
moving downwards along the $\Omega$-axis [Fig. 2(f)]. Along the
$\widetilde{v}_{\mathrm{min}}$-curve, on the other hand, $\widetilde{v}$
initially decreases with $\varphi_{\mathrm{o}}$ until
$\varphi_{\mathrm{o}}\\!=\\!\varphi_{\mathrm{B}}$, after which $\widetilde{v}$
increases with $\varphi_{\mathrm{o}}$. In the first portion of the
$\widetilde{v}_{\mathrm{min}}$-curve
($\varphi_{\mathrm{o}}\\!<\\!\varphi_{\mathrm{B}}$), the center of the
X-spectrum moves upwards along $\Omega$ [Fig. 2(f)]. The center reaches
infinity asymptotically at $\varphi_{\mathrm{o}}\\!=\\!\varphi_{\mathrm{B}}$,
and the spectrum becomes separable with respect to $k_{x}$ and $\Omega$. In
the second portion of the $\widetilde{v}_{\mathrm{min}}$-curve
($\varphi_{\mathrm{o}}\\!>\\!\varphi_{\mathrm{B}}$), the center of the
X-spectrum starts at $\Omega\\!\rightarrow\\!-\infty$ and moves upwards along
$\Omega$ with increase in $\varphi_{\mathrm{o}}$.

The region corresponding to a positive determinant $\Delta\\!>\\!0$ is formed
of three disconnected areas in the MT-plane [Fig. 2(e)]. These are the domains
external to the two curves $\widetilde{v}_{\mathrm{min}}$ and
$\widetilde{v}_{\mathrm{max}}$. In this case, we can write
$k_{x}^{2}\\!=\\!\alpha_{2}(\Omega-\Omega_{+})(\Omega-\Omega_{-})$, and the
spectrum is hyperbola with a gap along $\Omega$ whose vertices are
$\Omega_{\pm}\\!=\\!\tfrac{-\alpha_{1}\pm\sqrt{\Delta}}{\alpha_{2}}$ [Fig.
2(f)].

Finally, the regime where the determinant is negative $\Delta\\!<\\!0$ is the
single connected domain enclosed by the branches
$\widetilde{v}_{\mathrm{min}}$ and $\widetilde{v}_{\mathrm{max}}$ [Fig. 2(e)].
There are no real zeros for
$\alpha_{0}+2\alpha_{1}\Omega+\alpha_{2}\Omega^{2}\\!=\\!0$ in this domain,
and the spectrum takes the form of a hyperbola with a gap along $k_{x}$ [Fig.
2(f)].

##  III Experimental verification of spectral reorganization for space-time
wave packets

We proceed to verify the predictions outlined above concerning the spectral
reorganization of STWPs in presence of normal GVD by synthesizing in free
space STWPs that are propagation invariant once coupled to normally dispersive
ZnSe. The ZnSe sample is in the form of stacked 1-inch-diameter discs each of
thickness 5 mm (Thorlabs; WG71050) to a maximum thickness of 30 mm. The ZnSe
Sellmeier equation is
$n^{2}(\lambda)\\!=\\!4+\frac{1.9\lambda^{2}}{\lambda^{2}-0.113}$ ($\lambda$
in units of $\mu$m) [60]. We select the operating wavelength
$\lambda_{\mathrm{o}}\\!\approx\\!1$ $\mu$m, at which ZnSe is characterized by
$\widetilde{v}_{\mathrm{m}}\\!\approx\\!0.39c$ (phase index
$n_{\mathrm{m}}\\!\approx\\!2.49$ and group index
$\widetilde{n}_{\mathrm{m}}\\!\approx\\!2.57$), and GVD coefficient
$k_{2}\\!\approx\\!607$ fs2/mm (corresponding to
$\varphi_{\mathrm{B}}\\!\approx\\!19.2^{\circ}$).

We synthesize the STWPs starting from generic femtosecond pulses of width 100
fs at $\lambda_{\mathrm{o}}\\!\sim\\!1$ $\mu$m by associating with each
wavelength $\lambda$ a prescribed spatial frequency $k_{x}(\lambda)$ according
to Eq. 4, where $\lambda$ is the free-space wavelength. This is achieved by
spatially resolving the pulse spectrum and then utilizing a spatial light
modulator (SLM) to impart to each wavelength the appropriate propagation angle
with respect to the $z$-axis. Because each wavelength is addressed
independently, arbitrary forms of the relationship $k_{x}(\lambda)$ can be
produced with such a strategy [35]. This setup therefore allows us to readily
produce STWPs with any target values of $\widetilde{v}$ and
$\varphi_{\mathrm{o}}$ within the paraxial regime by changing the 2D phase
distribution imparted by the SLM to the spatially resolved wave front (see
Appendix for details).

Figure 3: Trajectories in the paraxial MT-plane
($\varphi_{\mathrm{o}}\\!<\\!1.5^{\circ}$) traced in our experiments. The
points along the curves indicate the experimental values selected for
$\widetilde{v}$ and $\varphi_{\mathrm{o}}$.

At normal incidence onto the planar surface between free space and ZnSe, the
transverse wave number $k_{x}$ is conserved because of translational
invariance along $x$, and the optical frequency $\omega$ is invariant.
Consequently the $(k_{x},\lambda)$-spectrum is a refractive invariant;
however, the $(k_{z},\lambda)$-spectrum is not. In the medium
$k_{z}\\!=\\!\sqrt{(n(\omega)\tfrac{\omega}{c})^{2}-k_{x}^{2}}$ is linear in
$\Omega$ in accordance with Eq. 2 for all values of $\widetilde{v}$ and
$\varphi_{\mathrm{o}}$, but in free space
$k_{z}\\!=\\!\sqrt{(\tfrac{\omega}{c})^{2}-k_{x}^{2}}$ is not. In other words,
the synthesized STWP is dispersion-free (to all orders) in ZnSe, but it
undergoes GVD in free space. The change in the STWP group velocity upon
transition from free space to the dispersive medium is governed by the law of
refraction derived in [61, 62].

The result is a STWP with profile $I(x,z;t)$ that is propagation invariant in
ZnSe, $I(x,z;t)\\!\rightarrow\\!I(x,0;t-z/\widetilde{v})\\!=\\!I(x;\tau)$,
where $\tau\\!=\\!t-z/\widetilde{v}$ is time in the moving frame of the STWP.
For each STWP we measure the spectral projection onto the
$(k_{x},\lambda)$-plane, from which we obtain the corresponding projection
onto the $(k_{z},\lambda)$-plane. We also reconstruct the spatio-temporal
intensity profile $I(x,z;\tau)\\!=\\!|\psi(x,z;\tau)|^{2}$ at fixed axial
planes $z$ in the medium (see Appendix for details).

Figure 4: Measured spectra and profiles for propagation invariant STWPs in
presence of normal GVD while varying $\varphi_{\mathrm{o}}$ and holding
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}$. The first row is the
spectrum projected onto the $(k_{x},\lambda)$-plane; the second row is the
spectrum projected onto the $(k_{z},\lambda)$-plane in the medium; and the
third row is the spatio-temporal intensity profile $I(x;\tau)$ measured at a
fixed axial plane $z\\!=\\!30$ mm in ZnSe. Here $\tau\\!=\\!t-z/\widetilde{v}$
is time measured in the moving frame of the STWP. The white curve along the
bottom horizontal axis is the on-axis pulse profile $I(0;\tau)$ of width
$\Delta T$. The white curve along the left vertical axis is the beam profile
at the pulse center $I(x;0)$ of width $\Delta x$. (a)
$\varphi_{\mathrm{o}}\\!=\\!0^{\circ}$; (b)
$\varphi_{\mathrm{o}}\\!=\\!0.6^{\circ}$; and (c)
$\varphi_{\mathrm{o}}\\!=\\!1.2^{\circ}$. Figure 5: Measured spectra and
profiles for propagation-invariant STWPs in presence of normal GVD while
varying $\varphi_{\mathrm{o}}$ along the curve $\widetilde{v}_{\mathrm{max}}$
[Fig. 2(e)]. The rows have the same structure as in Fig. 4. The curves in the
second row panels are the dispersive light-line in the medium
($k_{x}\\!=\\!0$), and the shaded area corresponds to evanescent waves. (a)
$\varphi_{\mathrm{o}}\\!=\\!0^{\circ}$; (b)
$\varphi_{\mathrm{o}}\\!=\\!0.15^{\circ}$; (c)
$\varphi_{\mathrm{o}}\\!=\\!0.3^{\circ}$; (d)
$\varphi_{\mathrm{o}}\\!=\\!0.6^{\circ}$; and (e)
$\varphi_{\mathrm{o}}\\!=\\!1.2^{\circ}$. Figure 6: Measured spectra and
profiles for propagation-invariant STWPs in presence of normal GVD while
varying $\varphi_{\mathrm{o}}$ along the curve $\widetilde{v}_{\mathrm{min}}$
[Fig. 2(e)]. The rows have the same structure as in Fig. 4. (a)
$\varphi_{\mathrm{o}}\\!=\\!0^{\circ}$; (b)
$\varphi_{\mathrm{o}}\\!=\\!0.15^{\circ}$; (c)
$\varphi_{\mathrm{o}}\\!=\\!0.3^{\circ}$; (d)
$\varphi_{\mathrm{o}}\\!=\\!0.6^{\circ}$; and (e)
$\varphi_{\mathrm{o}}\\!=\\!1.2^{\circ}$.

The synthesized STWPs have a bandwidth in the range $\Delta\lambda\\!=\\!8-16$
nm at a wavelength $\lambda_{\mathrm{o}}\\!\approx\\!1062$ nm, corresponding
to pulsewidths of $\Delta T\\!\approx\\!200-450$ fs for the on-axis profile
$I(0;\tau)$; we denote the pulsewidth at the beam center $\Delta T$. The beam
width $\Delta x$ at the pulse center $I(x;0)$ is determined by the spatial
bandwidth $\Delta k_{x}$, which is related to the temporal bandwidth
$\Delta\lambda$ through the STWP parameters $\widetilde{v}$ and
$\varphi_{\mathrm{o}}$ (via Eq. 4). In the measured spectra, we plot the
results with $\lambda$ decreasing upwards along the vertical axis to retain
the orientation of the curves associated with $\Omega$ as shown in Fig. 2.
Furthermore, we plot the spectra projected onto the $(k_{z},\lambda)$-plane
with $k_{z}-n_{\mathrm{m}}k_{\mathrm{o}}-\Omega/\widetilde{v}_{\mathrm{m}}$
along the horizontal axis for clarity. Although these spectra are linear only
in the $(k_{z},\Omega)$-plane, they remain approximately linear in the
$(k_{z},\lambda)$-plane because $\Delta\lambda\\!\ll\\!\lambda_{\mathrm{o}}$.
We plot the profiles at the maximum ZnSe sample length of 30 mm, but the
profile remains invariant at shorter ZnSe lengths [Fig. 10].

We trace along several of the crucial trajectories in the MT-plane that bring
out the rich dynamics of spectral reorganization: varying
$\varphi_{\mathrm{o}}$ while $\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}$
[Fig. 4], $\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{max}}$ [Fig. 5];
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{min}}$ [Fig. 6],
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})$ [Fig.
7], and varying $\widetilde{v}$ while holding
$\varphi_{\mathrm{o}}\\!=\\!1^{\circ}$ [Fig. 8]. We identify in Fig. 3 the
points along those trajectories. In our experiments we remain within the
paraxial limit with $\varphi_{\mathrm{o}}\\!<\\!2^{\circ}$, which in turn
requires that $\widetilde{v}$ remains close to $\widetilde{v}_{\mathrm{m}}$.
Throughout, the measured STWP spectra and profiles are in excellent agreement
with calculations based on the theoretical model described in the previous
Section without any fitting parameters.

Figure 7: Measured spectra and profiles for propagation-invariant STWPs in
presence of normal GVD while varying $\varphi_{\mathrm{o}}$ and simultaneously
maintaining
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})$. The
rows have the same structure as in Fig. 4. (a)
$\varphi_{\mathrm{o}}\\!=\\!0^{\circ}$; (b)
$\varphi_{\mathrm{o}}\\!=\\!0.15^{\circ}$; (c)
$\varphi_{\mathrm{o}}\\!=\\!0.3^{\circ}$; (d)
$\varphi_{\mathrm{o}}\\!=\\!0.6^{\circ}$; and (e)
$\varphi_{\mathrm{o}}\\!=\\!1.2^{\circ}$.

###  III.1 Escape velocity

In ZnSe at $\lambda_{\mathrm{o}}\\!\approx\\!1$ $\mu$m, the escape velocity is
$\widetilde{v}_{\mathrm{e}}\\!\approx\\!0.944\widetilde{v}_{\mathrm{m}}\\!\approx\\!0.367c$.
In our first experiment [Fig. 4], we hold $\widetilde{v}$ fixed at
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{e}}$ ($\alpha_{2}\\!=\\!0$) and
vary $\varphi_{\mathrm{o}}$ from $0^{\circ}$ to $1.2^{\circ}$. The spectrum is
a parabola that is symmetric around $k_{x}\\!=\\!0$ and whose vertex along the
$\Omega$-axis moves from $\Omega\\!=\\!0$ upwards as $\varphi_{\mathrm{o}}$
increases. The STWP profile remains X-shaped throughout. Because the temporal
bandwidth $\Delta\lambda$ and the spatial bandwidth $\Delta k_{x}$ both remain
almost constant while $\varphi_{\mathrm{o}}$ is varied, the on-axis pulsewidth
$\Delta T$ and the beam width $\Delta x$ at the pulse center in turn remain
almost constant.

###  III.2 Hyperbolic spectra with positive and negative determinants

Next, we vary $\varphi_{\mathrm{o}}$ and simultaneously tune $\widetilde{v}$
to produce STWPs represented by the points along the curves corresponding to
$\widetilde{v}_{\mathrm{max}}$ and $\widetilde{v}_{\mathrm{min}}$ (Eq. 6)
identified in Fig. 3. The spectra for the STWPs associated with
$\widetilde{v}_{\mathrm{max}}$ [Fig. 5] are quite similar to those associated
with $\widetilde{v}_{\mathrm{min}}$ [Fig. 6] except that they are inverted
along the $\lambda$-axis. At $\varphi_{\mathrm{o}}\\!=\\!0^{\circ}$ where
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{m}}$ (a point common to both
$\widetilde{v}_{\mathrm{max}}$ and $\widetilde{v}_{\mathrm{min}}$), the
spectrum is X-shaped with center at $(k_{x},\Omega)\\!=\\!(0,0)$. Increasing
$\varphi_{\mathrm{o}}$ retains the overall structure except that the
intersection point moves away from $(k_{x},\Omega)\\!=\\!(0,0)$, and at
$\varphi_{\mathrm{o}}\\!>\\!0.3^{\circ}$, the intersection point exits the
accessible spectral range. The temporal bandwidth remains fixed throughout
($\Delta\lambda\\!\approx\\!16$ nm), so that the on-axis pulsewidth $\Delta T$
also remains fixed. However, the reorganization in the spatio-temporal
spectral structure leads to a noticeable change in the spatial beam profile.
First, as $\varphi_{\mathrm{o}}$ increases [Fig. 5(b,c) and Fig. 6(b,c)], the
spatial bandwidth $\Delta k_{x}$ increases, and thus the beam width $\Delta x$
becomes narrower. However, with $\varphi_{\mathrm{o}}\\!>\\!0.3^{\circ}$ [Fig.
5(d,e) and Fig. 6(d,e)], the spatial spectrum is no longer ‘baseband’, and is
instead centered at a non-zero-valued spatial frequency $k_{x,\mathrm{o}}$.
The region in the vicinity of $k_{x}\\!=\\!0$ is eliminated from the spatial
spectrum, so that the envelope of the beam profile at $\tau\\!=\\!0$ is
accompanied by spatial fringes whose period decreases with
$\varphi_{\mathrm{o}}$. Furthermore, broadening of the spatial envelope width
$\Delta x$ occurs because the spatial bandwidth $\Delta k_{x}$ decreases with
$\varphi_{\mathrm{o}}$ [Fig. 5(d,e) and Fig. 6(d,e)].

###  III.3 TPF group velocity

Next, we examine the STWPs whose spectra are represented by points along the
curve
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})$,
where $\alpha_{1}\\!=\\!0$, over the span from
$\varphi_{\mathrm{o}}\\!=\\!0^{\circ}$ to $1.2^{\circ}$. At
$\varphi_{\mathrm{o}}\\!=\\!0^{\circ}$, $\alpha_{0}\\!=\\!\alpha_{1}\\!=\\!0$,
$\alpha_{2}\\!>\\!0$, and $k_{x}\\!=\\!\pm\sqrt{\alpha_{2}}\Omega$, which is
an X-shaped spectrum and the profile $I(x;\tau)$ is clearly X-shaped [Fig.
7(a)]. Increasing $\varphi_{\mathrm{o}}$ results in the X-shaped spectrum
morphing into a hyperbola with a gap along the $k_{x}$-axis. The profile
$I(x;\tau)$ is X-shaped, but whose branches have started to lose their
distinctiveness, and spatial oscillations appear [Fig. 7(b)]. The curvature of
the hyperbola drops with increasing $\varphi_{\mathrm{o}}$, so that $k_{x}$
and $\Omega$ decouple, and the profile $I(x;\tau)$ is no longer X-shaped [Fig.
7(c-e)]. Rather, it becomes approximately separable along space and time,
$\psi(x,0;t)\\!\approx\\!\psi_{x}(x)\psi_{t}(t)$, with $\psi_{x}(x)$
approaching a cosine function $\psi_{x}(x)\\!\approx\\!\cos{k_{x}x}$, and
$\psi_{t}(t)$ a transform-limited temporal pulse corresponding to the
bandwidth $\Delta\lambda$. This spatio-temporal separability becomes exact
when $\varphi_{\mathrm{o}}\\!=\\!\varphi_{\mathrm{B}}$, which occurs in the
non-paraxial regime. The measurements in Fig. 7(d,e) thus approximate in the
transverse dimension the predicted propagation invariance of pulsed Bessel
beams [56, 57, 58, 59]. Note that the on-axis pulse width $\Delta T$ is
constant throughout because $\Delta\lambda$ is maintained fixed.

Figure 8: Measured spectra and profiles for propagation-invariant STWPs in
presence of normal GVD while varying $\widetilde{v}$ and holding
$\varphi_{\mathrm{o}}\\!=\\!1^{\circ}$. The rows have the same structure as in
Fig. 4. (a) $\widetilde{v}\\!=\\!0.99\widetilde{v}_{\mathrm{min}}$; (b)
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{min}}$; (c)
$\widetilde{v}\\!=\\!(\widetilde{v}_{\mathrm{min}}+\widetilde{v}_{\mathrm{max}})/2$;
(d) $\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{max}}$; and (e)
$\widetilde{v}\\!=\\!1.01\widetilde{v}_{\mathrm{max}}$. Here
$\widetilde{v}_{\mathrm{min}}\\!=\\!0.99\widetilde{v}_{\mathrm{m}}$,
$\widetilde{v}_{\mathrm{max}}\\!=\\!1.01\widetilde{v}_{\mathrm{m}}$, and
$\widetilde{v}_{\mathrm{m}}\\!\approx\\!0.39c$.

###  III.4 Changing the group velocity at fixed angular offset

Finally, we examine the spectral reorganization as we tune $\widetilde{v}$
while holding $\varphi_{\mathrm{o}}$ fixed. At
$\varphi_{\mathrm{o}}\\!=\\!1^{\circ}$ [Fig. 3], realizing the group velocity
$\widetilde{v}\\!=\\!0.99\widetilde{v}_{\mathrm{min}}\\!>\\!\widetilde{v}_{\mathrm{e}}$
results in a hyperbolic spectrum with gap along $\Omega$, although only one
branch of it fits in the selected spectral window. Increasing $\widetilde{v}$
to $\widetilde{v}_{\mathrm{min}}$ results in an X-spectrum whose center is
outside the spectral window. Whereas $\Delta\lambda$ remains fixed, this
spectral reorganization is accompanied by a decrease in $\Delta k_{x}$ and a
displacement of the spatial-spectrum center to a non-zero spatial frequency.
Therefore, the beam width $\Delta x$ increases, and is accompanied by rapid
spatial oscillations [Fig. 8(b)]. Furthermore, the tilt angle of the two
branches of the X-shaped profile (the so-called pulse-front tilt [50]) changes
dramatically as a result of the change in curvature of the spatio-temporal
spectrum [63]. For
$\widetilde{v}_{\mathrm{min}}\\!<\\!\widetilde{v}\\!<\\!\widetilde{v}_{\mathrm{max}}$,
the spectrum is a hyperbola with gap along $k_{x}$. The hyperbolic curvature
here is small, so that the spectrum is approximately separable along $k_{x}$
and $\Omega$, and the profile is also approximately separable with respect to
space and time (a cosine wave along $x$ and a pulse whose width is determined
by $\Delta\lambda$ along $t$); see Fig. 8(c). Increasing $\widetilde{v}$
further reverses these changes and recapitulated the earlier spectral
reorganization but with the spectra flipped along the $\lambda$-axis [Fig.
8(d,e)].

##  IV Discussion

The STWPs that we have studied are all undergirded by angular dispersion [53,
52]; that is, each frequency $\omega$ travels in the medium at a different
angle $\varphi(\omega)$ with respect to the $z$-axis, with
$k_{z}(\omega)\\!=\\!\tfrac{\omega}{c}\cos{\varphi(\omega)}$ and
$k_{x}(\omega)\\!=\\!\tfrac{\omega}{c}\sin{\varphi(\omega)}$. To achieve
dispersion-free propagation in the normal-GVD regime, the angular dispersion
introduced into the field must yield anomalous GVD that counterbalances the
normal GVD in the medium. It is well-established that angular dispersion does
indeed introduce anomalous GVD as needed here [64, 65, 66]. However, as we
recently uncovered [63, 22, 67], the proof provided by Martinez, Gordon, and
Fork [64] tacitly assumes that the angular dispersion is differentiable; that
is, $\tfrac{d\varphi}{d\omega}$ is defined everywhere. These field
configurations are typically called TPFs, and correspond to the curve
$\widetilde{v}\\!=\\!\widetilde{v}_{\mathrm{TPF}}(\varphi_{\mathrm{o}})$ in
Fig. 2(c). We have shown that angular dispersion can only yield normal GVD
(thus allowing propagation invariance in the anomalous GVD regime) when
$\tfrac{d\varphi}{d\omega}$ is not defined at some frequency (non-
differentiable angular dispersion) [22, 67]. However, non-differentiable
angular dispersion can also produce dispersion-free STWPs in the normal-GVD
regime. Many of the STWPs examined here correspond to non-differentiable
angular dispersion; e.g., all the hyperbolic spectra with a gap along $\Omega$
in which $\tfrac{d\varphi}{d\Omega}$ is not be defined at the vertices of the
two branches of the hyperbola. More theoretical work is needed to fully
identify the regions of differentiable and non-differentiable angular
dispersion in the MT-plane in presence of normal GVD.

We have identified one point in the MT-plane at
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{e}},\varphi_{\mathrm{B}})$
that corresponds to a pulsed Bessel beam $I_{\mathrm{B}}(x;\tau)$ that
separates into a product of spatial and temporal profiles
$I_{\mathrm{B}}\\!\propto\\!J_{0}^{2}(k_{r}r)|\psi_{t}(t-z/\widetilde{v}_{\mathrm{e}})|^{2}$.
Although the spectrum is separable with respect to $k_{x}$ and $\Omega$, the
field nevertheless incorporates angular dispersion: $k_{x}\\!=\\!\pm
n(\omega)\tfrac{\omega}{c}\sin{\varphi(\omega)}$. Each frequency $\omega$
travels at an angle $\varphi(\omega)$ with respect to the $z$-axis in the
medium in such a way that $k_{x}(\Omega)$ is held constant. Another
propagation-invariant separable wave packet is the Bessel-Airy wave packet
[68] in which the pulse profile is an Airy function, which accelerates in a
dispersive medium (whether normal or anomalous GVD). In contrast, the pulsed
Bessel beam can have an arbitrary temporal waveform; however, this applies
only in the normal-GVD regime, and there are no anomalous-GVD counterparts.

Finally, we emphasize that all the spatio-temporal profiles depicted in Fig. 4
through Fig. 8 are propagation invariant – they are all independent of $z$ in
the medium; see Fig. 10.

##  V Conclusion

In conclusion, we have explored the rich dynamics of spectral reorganization
for propagation-invariant STWPs in presence of normal GVD. Two parameters
identify each family of STWPs, the group velocity and the central axial wave
number. By varying these two parameters, a wide range of spatio-temporal
spectra are produced that are compatible with propagation invariance in the
dispersive medium, including X-shaped, hyperbolic (with gaps along the
spatial-frequency or temporal-frequency axes), parabolic, or elliptic, or
separable spatio-temporal spectra. All these spectra result from the
intersection of the dispersive light-cone with a spectral plane that is tilted
and displaced with respect to the light-cone. We have experimentally validated
these dynamics of spectral reorganization for STWPs in the near-infrared at a
wavelength $\sim\\!1$ $\mu$m with pulses of width $\sim\\!200-450$ fs and
bandwidth $\Delta\lambda\\!\approx\\!8-16$ nm in the paraxial regime using
ZnSe in its normal-GVD regime. These results pave the way to exploiting STWPs
in initiating nonlinear optical processes by accessing new forms of phase-
matching that are engendered by their unique spatio-temporal spectral
structures.

Figure 9: (a) Schematic of the universal angular-dispersion synthesizer used
to prepare the STWPs reported on here. (b-e) Examples of 2D phase patterns
$\Phi(\lambda,x)$ imparted by the SLM to the incident spectrally resolved wave
front. The phase patterns produce STWPs corresponding to the following points
in the MT-plane: (b)
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{m}},0^{\circ})$,
(c)
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(1.001\widetilde{v}_{\mathrm{m}},0^{\circ})$,
(d)
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{TPF}},0.2^{\circ})$,
and (e)
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{e}},0^{\circ})$.
Figure 10: Verifying the propagation invariance of the STWPs in presence of
normal GVD over the course of 30-mm propagation distance in ZnSe. (a) The STWP
corresponding to the point
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{m}},0^{\circ})$
and (b)
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{e}},0^{\circ})$
in the MT-plane.

## Appendix: Experimental details

The setup used to synthesize and characterize the STWPs utilizes the universal
angular-dispersion synthesizer described in [35] as shown in Fig. 9(a). Plane-
wave pulses (width $\approx\\!100$ fs, bandwidth $\approx\\!25$ nm, at a
central wavelength of $\lambda_{\mathrm{o}}\\!\approx\\!1064$ nm) are
spectrally resolved via a diffraction grating (1200 lines/mm), collimated by a
cylindrical lens (focal length $f\\!=\\!500$ mm), and modulated by a phase-
only spatial light modulator (SLM; Meadowlark, E19X12) placed at the focal
plane of the lens. Each wavelength is confined to a column on the SLM where it
acquires the phase
$\Phi(x,\lambda)\\!=\\!\pm\frac{2\pi}{\lambda}\sin{\\{\varphi^{\prime}(\lambda)\\}}x$,
where $\varphi^{\prime}(\lambda)$ is a deflection angle for $\lambda$ with
respect to the $z$-axis in free space (and is related to the angle
$\varphi(\lambda)$ in the medium via Snell’s law). The retro-reflected wave
front returns to the grating whereupon the STWP is reconstituted. Examples of
the 2D phase distributions imparted to the spectrally resolved wave fronts are
shown in Fig. 9(b-e). The phase distribution in Fig. 9(b) corresponds to the
X-shaped spectrum associated with STWP represented by the point
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{m}},0^{\circ})$
in the MT-plane. Moving to the point
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(1.001\widetilde{v}_{\mathrm{m}},0^{\circ})$
in Fig. 9(c), the spectrum is hyperbolic with a gap along the $\Omega$-axis,
which is manifest in the portion of the phase distribution that is nulled. In
contrast, the phase distribution in Fig. 9(d) is associated with STWP
represented by the point
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{TPF}},0.2^{\circ})$,
which corresponds to a hyperbolic spectrum but with the gap along $k_{x}$
rather than $\Omega$. Finally, the phase distribution Fig. 9(e) is associated
with STWP represented by the point
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{e}},0^{\circ})$,
which corresponds to a parabolic spectrum at the escape velocity. Here $k_{x}$
changes monotonically with $\lambda$.

The spatio-temporal spectrum in the $(k_{x},\lambda)$-plane is acquired by
introducing a spatial Fourier transform in the path of the spectrally resolved
wave front and recording the intensity with a CCD camera [7]. From the
spectrum in the $(k_{x},\lambda)$-plane, we obtain the associated spectrum in
the $(k_{z},\lambda)$-plane for the field in the medium from
$k_{z}^{2}\\!=\\!(n(\omega)\frac{\omega}{c})^{2}-k_{x}^{2}$, which is to be
compared with the theoretical expectation in Eq. 2. The spatio-temporal
intensity profile of the STWP $I(x,z;\tau)$ is reconstructed at a given axial
plane $z$ by interfering it with the initial laser pulses as a reference [10,
69, 19]; see Fig. 9(a). When the reference pulse and the STWP overlap in space
and time, the visibility of spatially resolved fringes recorded by a CCD
camera helps reconstruct the STWP profile as we sweep an optical delay $\tau$
placed in the reference path.

We verify in Fig. 10 the propagation invariance of the STWPs in ZnSe. We
reconstruct the intensity profile $I(x,z;\tau)$ at three axial planes:
$z\\!=\\!0$ before entrance to the ZnSe sample; $z\\!=\\!15$ mm; and
$z\\!=\\!30$ mm. We plot the results for two different STWPs corresponding to
the points
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{m}},0^{\circ})$
and
$(\widetilde{v},\varphi_{\mathrm{o}})\\!=\\!(\widetilde{v}_{\mathrm{e}},0^{\circ})$
in the MT-plane. In both cases, the profile remains unchanged with propagation
in the dispersive medium.

## ACKNOWLEDGMENTS

The authors acknowledge the support of the Office of Naval Research (ONR,
Grant Nos. N00014-17-1-2458 and N00014-20-1-2789).

The authors declare no conflicts of interest.

## DATA AVAILABILITY

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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