# Theory of space-time supermodes in planar multimode waveguides

Abbas Shiri  CREOL, The College of Optics & Photonics, University of Central
Florida, Orlando, FL 32816, USA  Department of Electrical and Computer
Engineering, University of Central Florida, Orlando, FL 32816, USA  Kenneth L.
Schepler  CREOL, The College of Optics & Photonics, University of Central
Florida, Orlando, FL 32816, USA  Ayman F. Abouraddy  CREOL, The College of
Optics & Photonics, University of Central Florida, Orlando, FL 32816, USA
Department of Electrical and Computer Engineering, University of Central
Florida, Orlando, FL 32816, USA<EMAIL_ADDRESS>
###### Abstract

When an optical pulse is focused into a multimode waveguide or fiber, the
energy is divided among the available guided modes. Consequently, the
initially localized intensity spreads transversely, the spatial profile
undergoes rapid variations with axial propagation, and the pulse disperses
temporally. Space-time (ST) supermodes are pulsed guided field configurations
that propagate invariantly in multimode waveguides by assigning each mode to a
prescribed wavelength. ST supermodes can be thus viewed as spectrally
discrete, guided-wave counterpart of the recently demonstrated propagation-
invariant ST wave packets in free space. The group velocity of an ST supermode
is tunable independently – in principle – of the waveguide structure, group-
velocity dispersion is eliminated or dramatically curtailed, and the time-
averaged intensity profile is axially invariant along the waveguide in absence
of mode-coupling. We establish here a theoretical framework for studying ST
supermodes in planar waveguides. Modal engineering allows sculpting this
axially invariant transverse intensity profile from an on-axis peak or dip
(dark beam), to a multi-peak or flat distribution. Moreover, ST supermodes can
be synthesized using spectrally incoherent light, thus paving the way to
potential applications in optical beam delivery for lighting applications.

##  I Introduction

Space-time wave packets (STWPs) are pulsed beams endowed with a precise
spatio-temporal structure [1] that results in rigid transport (diffraction-
free and dispersion-free propagation) in linear media [2, 3, 4, 5]. By
assigning each spatial frequency underpinning the transverse spatial profile
to one temporal frequency underpinning the pulse profile [6, 7, 8, 9, 10, 11],
a host of unique attributes can be attained in the synthesized STWP, such as
self healing [12], tunable group velocity [13, 14, 15, 16, 17], and anomalous
refraction [18]. Historically, the first discovered STWP was Brittingham’s
focus-wave mode (FWM) [19], which is an example of ‘sideband’ STWPs, so-called
because low spatial frequencies are excluded from the spectrum on physical
grounds [20]. Subsequently, X-waves were identified and realized [21, 22, 23].
However, both sideband STWPs and X-waves are difficult to synthesize [22, 24,
25, 3], and any significant tuning of their attributes can be achieved only in
the non-paraxial regime while utilizing ultrabroad bandwidths [1]. More
recently, the newly identified family of ‘baseband’ STWPs has been pursued,
whose characteristics can be tuned over unprecedented scales while using
narrow bandwidths and remaining in the paraxial regime [11, 17, 26].

Until very recently, all experimental investigations of STWPs have been
confined to freely propagating fields [3, 4, 1], although a few theoretical
studies examined the propagation of X-waves in multimode cylindrical
waveguides [27, 28, 29], in addition to an early theoretical investigation of
FWMs in a single-mode fiber [30]. More recently, the impact of orbital angular
momentum (OAM) on FWMs in a single-mode fiber has been studied [31]. In light
of the above-mentioned difficulties in synthesizing X-waves and sideband STWPs
(such as FWMs) [1], it is unlikely that these proposals can be put to test.
However, recent studies have been directed at confining the more useful
baseband STWPs in waveguides. For example, hybrid guided space-time modes in
single-mode [32] or few-mode planar waveguides [33] have been demonstrated in
which the STWP structure is introduced along the unconfined dimension. In the
context of highly multimoded waveguides, the propagation and excitation of
STWPs via high-energy pulses in a step-index noninear multimode fiber or
waveguide has been investigated theoretically [34, 35, 36], and has very
recently been realized experimentally [37]. Additionally, STWPs can be
generated in a multimode planar, graded-index waveguide by endowing a segment
of the waveguide with an appropriately designed time-varying refractive index
[38].

These rapid developments in guided STWPs over the past two years raise a
natural question: are STWPs propagation invariant in multimode waveguides just
as they are in bulk media? Because guided modes have different propagation
constants and group velocities, the spatial profile of a conventional pulsed
multimode field changes erratically with propagation, even in absence of mode-
coupling. In contrast, a recent proof-of-principle experiment [39] has
indicated that an appropriately prepared, quasi-discrete-spectrum STWP in free
space can couple to a multimode step-index planar waveguide and maintain its
spatial profile along the waveguide length.

Here we develop the theory of STWPs in planar multimode optical waveguides in
absence of mode coupling. We denote such guided optical fields ‘ST
supermodes’. Ideal ST supermodes are formed of a superposition of
monochromatic waveguide modes each at a prescribed frequency, which is
selected to guarantee that the ST supermode as a whole propagates invariantly
at a tunable group velocity. However, ideal ST supermodes have a strictly
discretized spectrum and thus require infinite energy for their realization.
We relax here the monochromaticity constraint to introduce realistic finite-
energy ST supermodes, and also examine spectrally continuous ST supermodes. In
all three cases (ideal, finite-energy, and spectrally continuous), the
supermode group velocity is tunable, the impact of dispersion is reduced with
respect to a conventional pulsed multimode field comprising the same modes,
and the time-averaged intensity profile remains axially invariant as long as
the spectra associated with the different modes do not overlap. Furthermore,
the axially invariant intensity profile can be sculpted by varying the modal
contributions, thereby producing an on-axis peak or dip, a multi-peak
distribution, or a flat intensity profile. Moreover, we show that the axial
invariance of this intensity profile extends to spectrally incoherent fields,
which suggests potential applications in lighting. These results pave the way
towards investigating ST supermodes in optical fibers and conventional
waveguides in which light is confined along both transverse dimensions.

##  II overview of freely propagating space-time wave packets

We start by briefly reviewing STWPs in isotropic, non-dispersive media, and
restrict the transverse spatial profile to one dimension $x$, while holding
the field uniform along $y$, in anticipation of the treatment for ST
supermodes in a planar waveguide. The field for a generic scalar pulsed beam
(or wave packet) in a medium of refractive index $n$ is
$E(x,z;t)\\!=\\!e^{i(nk_{\mathrm{o}}z-\omega_{\mathrm{o}}t)}\psi(x,z;t)$,
where $\omega_{\mathrm{o}}$ is the carrier frequency,
$k_{\mathrm{o}}\\!=\\!\tfrac{\omega_{\mathrm{o}}}{c}$ the associated free-
space wave number, $c$ the speed of light in vacuum, and the angular spectrum
for the envelope $\psi(x,z;t)$ has the form:

$\psi(x,z;t)=\\!\int\\!\\!\\!\int
dk_{x}d\Omega\,\widetilde{\psi}(k_{x},\Omega)e^{ik_{x}x}e^{i(k_{z}-nk_{\mathrm{o}})z}e^{-i\Omega
t};$ (1)

here $\Omega\\!=\\!\omega-\omega_{\mathrm{o}}$, and the spatio-temporal
spectrum $\widetilde{\psi}(k_{x},\Omega)$ is the 2D Fourier transform of
$\psi(x,0;t)$, whose support on the surface of the light-cone
$k_{x}^{2}+k_{z}^{2}\\!=\\!(n\tfrac{\omega}{c})^{2}$ is thus a 2D domain.

In contrast, the spectral support for STWPs [Fig. 1] is a 1D curve on the
light-cone surface [6, 20]. The propagation invariance of STWPs requires that
these spectral curves are the intersection of the light-cone with a spectral
plane $\mathcal{P}(\theta)$ that is parallel to the $k_{x}$-axis.
Consequently, the spectral projection for any STWP onto the
$(k_{z},\tfrac{\omega}{c})$-plane is a straight line, thus signifying
dispersionless propagation.

Figure 1: Spectral representation of freely propagating STWPs in a non-
dispersive medium of refractive index $n$ ($n\\!=\\!1.5$ throughout). (a)
X-waves. We depict the spectral support for an X-wave at the intersection of
the light-cone $k_{x}^{2}+k_{z}^{2}\\!=\\!(n\tfrac{\omega}{c})^{2}$ with the
spectral plane $\mathcal{P}_{X}(\theta)$, along with its spectral projections
onto the $(k_{z},\tfrac{\omega}{c})$-plane, which is a straight line making an
angle $\theta$ with the $k_{z}$-axis, and the projection onto the
$(k_{x},\tfrac{\omega}{c})$-plane, which is a conic section [11]. The shaded
regions represent non-evanescent plane waves consistent with causal excitation
and propagation [20, 1]. Here $\omega_{\mathrm{o}}$ is an arbitrary
normalization frequency. (b) Same as (a) for sideband STWPs. The dashed parts
of the spectral projections are excluded because they correspond to negative
values of $k_{z}$. (c) Same as (a) for baseband STWPs.

There are three basic types of STWPs: X-waves, sideband STWPs, and baseband
STWPs [20], which are distinguished by the form of $\mathcal{P}(\theta)$. For
X-waves, the plane $\mathcal{P}_{\mathrm{X}}(\theta)$ passes through the
origin $\omega\\!=\\!k_{z}c\tan{\theta}$, with
$\tan^{-1}{(\tfrac{1}{n})}\\!<\\!\theta\\!<\\!\tfrac{\pi}{2}$ [Fig. 1(a)],
where $\theta$ (the spectral tilt angle) is the angle the plane makes with the
$k_{z}$-axis. The field for the X-wave is:

$E_{\mathrm{X}}(x,z;t)=\int\\!d\Omega\,\widetilde{\psi}(\omega)e^{ik_{x}x}e^{-i\omega(t-z/\widetilde{v})}\\!=\\!E_{\mathrm{X}}(x,0;t-z/\widetilde{v}),$
(2)

where $\widetilde{v}\\!=\\!c\tan{\theta}$ is both the group and phase velocity
of the rigidly transported X-wave, which is always superluminal
($\widetilde{v}\\!>\\!c/n$) [4]. In the luminal limit
$\widetilde{v}\\!=\\!c/n$, the X-wave degenerates into a plane-wave pulse
[20].

For sideband STWPs, the spectral plane $\mathcal{P}_{\mathrm{S}}(\theta)$ is
$\Omega\\!=\\!(k_{z}+nk_{\mathrm{o}})\widetilde{v}$, with group velocity
$\widetilde{v}\\!=\\!c\tan{\theta}$ and
$\tan^{-1}{(\tfrac{1}{n})}\\!\leq\\!\theta\\!<\\!\tfrac{\pi}{2}$, thus
encompassing the luminal ($\widetilde{v}\\!=\\!c/n$) and superluminal
$\widetilde{v}$ ($\widetilde{v}\\!>\\!c/n$) regimes [Fig. 1(b)]. The field is
given by
$E_{\mathrm{S}}(x,z;t)\\!=\\!e^{-i(nk_{\mathrm{o}}z+\omega_{\mathrm{o}}t)}\psi_{\mathrm{S}}(x,z;t)$,
so that the phase velocity is $-c/n$. It is clear from Fig. 1(b) that the
spatial frequencies $k_{x}$ in the vicinity of $k_{x}\\!=\\!0$ correspond to
negative values of $k_{z}$, which is incompatible with causal excitation and
propagation [40, 3, 1]. Consequently, the span of low spatial frequencies is
excluded from sideband STWPs on physical grounds. Both X-waves and sideband
STWPs require large numerical apertures and large bandwidths for their
realization if $\widetilde{v}$ is to depart from $c/n$. Hence these two types
of STWPs are difficult to produce experimentally [20, 1], and have found no
optical applications in the $\sim\\!3-4$ decades since their discovery [3].

Finally, for baseband STWPs the plane $\mathcal{P}_{\mathrm{B}}(\theta)$ is
$\Omega\\!=\\!(k_{z}-nk_{\mathrm{o}})\widetilde{v}$, with
$\widetilde{v}\\!=\\!c\tan{\theta}$ and $0\\!<\\!\theta\\!<\\!\pi$ [Fig.
1(c)]. Thus the group velocity of baseband STWPs can uniquely take on
subluminal, superluminal, or negative values [17]. The field is given by
$E_{\mathrm{B}}(x,z;t)\\!=\\!e^{i(nk_{\mathrm{o}}z-\omega_{\mathrm{o}}t)}\psi_{\mathrm{B}}(x,z;t)$,
so that the phase velocity is $c/n$. The angular spectrum for the envelopes of
sideband and baseband STWPs have the same form:

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

Both types of STWPs thus propagate invariantly at a group velocity
$\widetilde{v}\\!=\\!c\tan{\theta}$. However, there are two crucial
distinctions. First, sideband STWPs can be only luminal (FWMs) or
superluminal, whereas the group velocity of baseband STWPs can take on – in
principle – arbitrary values. Second, the spatial spectrum of baseband STWPs
is in the vicinity of $k_{x}\\!=\\!0$ [Fig. 1(c)], whereas that for sideband
STWPs [Fig. 1(b)] is displaced away from $k_{x}\\!=\\!0$ [20, 1].

##  III ST supermodes in planar waveguides

###  III.1 Spectral representation of planar-waveguide modes

Consider the symmetric step-index planar dielectric waveguide structure
depicted in Fig. 2(a), with core refractive index $n$ and cladding
$n_{\mathrm{c}}\\!<\\!n$. The index-guided field $E(x,z;t)$ propagates along
$z$, is confined along $x$, and is uniform along $y$. We refer to the axial
and transverse wave numbers as $\beta$ and $q$, respectively, so that the
light-cone surface for the core medium is
$\beta^{2}+q^{2}\\!=\\!(n\tfrac{\omega}{c})^{2}$. For the
$m^{\mathrm{th}}$-order mode ($m\\!=\\!0,1,2,\cdots$), the axial and
transverse wave numbers in the core are $\beta_{m}$ and $q_{m}$, respectively.
The spectral projection onto the $(\beta,\tfrac{\omega}{c})$-plane for any
modal dispersion relationship $\beta_{m}(\omega)$ is thus a 1D curve [Fig.
2(b)]. However, in contrast to STWPs, this spectral projection is not a
straight line, so that guided modes are always dispersive. For comparison, we
consider in the Appendix the analytically tractable case of a planar waveguide
formed of two perfect mirrors, in which case closed-form expressions can be
derived for the quantities we compute here.

Figure 2: (a) Schematic of a planar step-index waveguide with core and
cladding refractive indices $n$ and $n_{\mathrm{c}}$, respectively. (b) The
spectral support for the dispersion relationships $\beta_{m}(\omega)$
associated with the first 10 TE waveguide modes are 1D trajectories on the
light-cone surface $\beta^{2}+q^{2}\\!=\\!(n\tfrac{\omega}{c})^{2}$. We also
plot their spectral projections (solid black curves) onto the
$(q,\tfrac{\omega}{c})$ and $(\beta,\tfrac{\omega}{c})$ planes. The blue-
shaded regions (similarly to Fig. 1) correspond to propagating field
components in the bulk. (c) A pulsed localized optical field at the input of a
multimode waveguide couples to multiple modes over a bandwidth $\Delta\omega$.
The inset depicts the corresponding spectral projection onto the
$(q,\tfrac{\omega}{c})$-plane. (d) The spatio-temporal intensity profiles
$I(x;t)$ for individual pulsed modes at $z\\!=\\!0$;
$\lambda_{\mathrm{o}}\\!=\\!1.55$ $\upmu$m, $\Delta T\\!=\\!150$ fs, and
$\Delta\lambda\\!\approx\\!23$ nm. (e) The spatio-temporal intensity
$I(x,z;\tau)$, with $\tau\\!=\\!t-nz/c$, for the pulsed multimode field
comprising an equal-weighted superposition of the first 16 modes in a frame
moving at $v\\!=\\!c/n$ at $z\\!=\\!0$, 5, and 10 mm. (f) The time-averaged
intensity $I(x,z)$ of the field in (e). The axial planes selected in (e) are
identified by dashed arrows.

###  III.2 Conventional pulsed multimode guided fields

In a multimode waveguide, the field is a superposition of modes
$E(x,z;t)\\!=\\!\sum_{m}A_{m}E_{m}(x,z;t)$, where $A_{m}$ is the modal
amplitude, normalized such that $\sum_{m}|A_{m}|^{2}\\!=\\!1$, and
$E_{m}(x,z;t)$ is the field for the $m^{\mathrm{th}}$ mode, normalized such
that $\iint\\!dxdt\,|E(x,z;t)|^{2}\\!=\\!1$ in absence of absorption or
radiation leakage. At a fixed frequency $\omega_{\mathrm{o}}$, the
monochromatic modal fields are
$E_{m}(x,z;t)\\!=\\!u_{m}(x;\omega_{\mathrm{o}})e^{i\\{\beta_{m}(\omega_{\mathrm{o}})z-\omega_{\mathrm{o}}t\\}}$,
where $u_{m}(x;\omega_{\mathrm{o}})$ is the transverse spatial mode profile.
We consider throughout sufficiently small bandwidths $\Delta\omega$ so that
one may ignore the frequency dependence of the modal profiles,
$u_{m}(x;\omega)\\!\approx\\!u_{m}(x)$, which are orthonormal
$\int\\!dx\,u_{m}(x)u_{\ell}^{*}(x)\\!=\\!\delta_{m\ell}$. The spectral
support for this monochromatic multimode field is a collection of points on
the light-cone surface at the intersection of the horizontal iso-frequency
plane $\omega\\!=\\!\omega_{\mathrm{o}}$ with modal dispersion curves
$\beta_{m}(\omega)$ shown in Fig. 2(b). The intensity of this monochromatic
multimode field is:

$I(x,z)=|E(x,z;t)|^{2}=|\sum_{m}A_{m}u_{m}(x)e^{i\beta_{m}(\omega_{\mathrm{o}})z}|^{2}.$
(4)

Starting with a localized field at $z\\!=\\!0$, the axially varying relative
phase factor $e^{i\beta_{m}(\omega_{\mathrm{o}})z}$ produces erratic changes
in the transverse intensity profile along $z$.

If instead a pulse (width $\Delta T$ and bandwidth
$\Delta\omega\\!\sim\\!\tfrac{1}{\Delta T}$ centered at $\omega_{\mathrm{o}}$)
is spatially localized at $z\\!=\\!0$, the pulse energy once coupled to the
waveguide is typically distributed among a multiplicity of modes, and we
assume, for simplicity, that the excited modes share the same bandwidth
$\Delta\omega$ [Fig. 2(c)]. The individual pulsed modes can be expressed as:

$E_{m}(x,z;t)=e^{i\\{\beta_{m}(\omega_{\mathrm{o}})z-\omega_{\mathrm{o}}t\\}}u_{m}(x)\psi_{m}(z;t),$
(5)

where $\psi_{m}(z;t)$ is an axial modal envelope. By expanding
$\beta_{m}(\omega)$ into a Taylor series around $\omega_{\mathrm{o}}$,
$\beta_{m}(\omega)\\!=\\!\beta_{m}(\omega_{\mathrm{o}}+\Omega)=\beta_{m}(\omega_{\mathrm{o}})+\frac{\Omega}{\widetilde{v}_{m}}+\frac{1}{2}\eta_{m}\Omega^{2}+\cdots$,
we can write $\psi_{m}(z;t)$ as follows:

$\psi_{m}(z;t)=\int_{\Delta\omega}\\!\\!d\Omega\,\widetilde{\psi}(\Omega)\,e^{-i\Omega(t-z/\widetilde{v}_{m})}e^{i\eta_{m}\Omega^{2}z/2};$
(6)

here $\Omega\\!=\\!\omega-\omega_{\mathrm{o}}$, $\widetilde{\psi}(\Omega)$ is
the complex spectrum of the input pulse, the integration domain is its full
bandwidth $\Delta\omega$,
$\widetilde{v}_{m}\\!=\\!1\big{/}\tfrac{d\beta_{m}}{d\omega}\big{|}_{\omega_{\mathrm{o}}}$
is the group velocity of the $m^{\mathrm{th}}$-mode, its phase velocity is
$v_{m}\\!=\\!\tfrac{\omega_{\mathrm{o}}}{\beta_{m}(\omega_{\mathrm{o}})}$, and
$\eta_{m}\\!=\\!\tfrac{d^{2}\beta_{m}}{d\omega^{2}}\big{|}_{\omega_{\mathrm{o}}}$
is the group-velocity dispersion (GVD) coefficient [41]. We normalize the
spectrum so that $\int\\!dt|\psi_{m}(z;t)|^{2}\\!=\\!1$. In absence of GVD
($\eta_{m}\\!=\\!0$), the envelope has a temporal width of $\Delta T$
traveling rigidly at a group velocity $\widetilde{v}_{m}$. In presence of GVD,
the envelope disperses temporally with a chirp parameter
$a_{m}\\!=\\!\eta_{m}(\Delta\omega)^{2}z$ [41]. We plot in Fig. 2(d) examples
of the spatio-temporal intensity profiles for individual pulsed modes at the
waveguide entrance $I(x;t)\\!=\\!|E_{m}(x,z\\!=\\!0;t)|^{2}$ for
$m\\!=\\!0,1,2,3$, which are separable with respect to $x$ and $t$. Although
the spatio-temporal profile $I_{m}(x,z;t)\\!=\\!|E_{m}(x,z;t)|^{2}$ for a
single pulsed mode changes along $z$ due to dispersion, the time-averaged
intensity $I_{m}(x,z)\\!=\\!\int\\!dt\,I_{m}(x,z;t)\\!=\\!|u_{m}(x)|^{2}$ is
independent of $z$.

The initially localized separable pulsed multimode field at $z\\!=\\!0$ is a
superposition of such pulsed modes. The field subsequently undergoes complex
propagation dynamics, in part due to the relative phase factors
$e^{i\beta_{m}(\omega_{\mathrm{o}})z}$, similarly to the monochromatic
multimode fields described above. Additionally, the pulsed multimode field
comprises modal envelopes traveling at different group velocities
$\widetilde{v}_{m}$ (modal GVD), each separately undergoing dispersive
spreading with a chirp parameter $a_{m}$ (waveguide GVD), and potentially
chromatic GVD (if $n$ is wavelength-dependent). An example is shown in Fig.
2(e) of an equal-weight superposition of the first 16 modes. At $z\\!=\\!0$,
the input field is localized along $x$ (beam width $\Delta x\\!=\\!5.6$
$\upmu$m) and along $t$ (pulse width $\Delta T\\!=\\!150$ fs). Subsequently,
the pulsed multimode field spreads temporally and spatially [Fig. 2(e)].

The time-averaged intensity for this pulsed multimode field,
$I(x,z)\\!=\\!\int\\!dt\,|E(x,z;t)|^{2}$, is:

$I(x,z)=\sum_{m,\ell}A_{m}A_{\ell}^{*}u_{m}(x)u_{\ell}^{*}(x)\psi_{m\ell}(z),$
(7)

where the mutual envelope $\psi_{m\ell}(z)$ is:

$\psi_{m\ell}(z)=\int\\!d\Omega\,|\widetilde{\psi}(\Omega)|^{2}\,e^{i\Omega(\widetilde{n}_{m}-\widetilde{n}_{\ell})z/c}e^{i(\eta_{m}-\eta_{\ell})\Omega^{2}z/2};$
(8)

here $\widetilde{n}_{m}\\!=\\!c/\widetilde{v}_{m}$ is the modal group index.
The intensity $I(x,z)$ is initially localized in the vicinity of $x\\!=\\!0$
at $z\\!=\\!0$ but expands rapidly to the waveguide interfaces, and the
transverse profile subsequently varies erratically along $z$ [Fig. 2(f)]. This
is a generic feature of all multimode waveguides of sufficiently large size.
We proceed to show that ST supermodes that combine these same modes across the
same temporal bandwidth $\Delta\omega$ nevertheless retain their intensity
profile invariantly along the waveguide.

###  III.3 Ideal ST supermodes

As shown in Fig. 1, the spectral support for a freely propagating STWP is the
1D curve at the intersection of the light-cone with a plane
$\mathcal{P}(\theta)$. An STWP propagates invariantly without diffraction or
dispersion in linear media [42, 43, 44, 45, 46, 47] at a group velocity
$\widetilde{v}\\!=\\!c\tan{\theta}$, which can be changed – in principle –
independently of the refractive index by tuning $\theta$. In a similar vein,
the spectral support for a waveguide mode is a 1D curve on the light-cone
surface. However, in contrast to STWPs, guided modes are dispersive, and their
group velocity is determined by the waveguide structure. Therefore, pulsed
single-mode and multimode guided fields do not propagate invariantly.

Figure 3: Ideal ST supermodes. (a) Spectral projection onto the
$(\beta,\tfrac{\omega}{c})$-plane of the modal dispersion curves
$\beta_{m}(\omega)$ and the spectral plane $\mathcal{P}(\theta)$; here
$\theta\\!=\\!20^{\circ}$. The spectral representation of an ideal supermode
comprises the intersection points at the frequencies $\omega_{m}$ (represented
by green dots). (b) Projected temporal and spatial spectra for an ideal ST
supermode onto the $\omega$-axis and the $q$-axis. (c) Spatio-temporal
intensity profile $I(x,z;\tau)$ for an ideal ST supermode at a fixed axial
plane, obtained by an equal-weight superposition of the first 8 even-parity
modes ($m\\!=\\!0,2,\cdots,14$), with $\Delta T\\!=\\!280$ fs,
$\Delta\lambda\\!\approx\\!22.6$ nm, and $\Delta x\\!=\\!2.8$ $\upmu$m. (d)
Time-averaged intensity $I(x,z)$ for the ideal ST supermode in (c).

An ideal ST supermode is a pulsed multimode guided field that nevertheless
propagates invariantly along the waveguide. This unique behavior is achieved
by superposing a multiplicity of monochromatic modes, each at a frequency
$\omega_{m}$ selected such that the supermode envelope travels unchanged at a
group velocity $\widetilde{v}$ that is distinct from those of any of the
underlying modes, and which can be tuned – in principle – independently of the
waveguide structure. The field for the ST supermode can thus be expressed as
follows:

$E_{\mathrm{ST}}(x,z;t)=\sum_{m}A_{m}E_{\mathrm{ST},m}(x,z;t),$ (9)

where the contribution from each mode is:

$E_{\mathrm{ST},m}(x,z;t)=u_{m}(x)e^{i\\{\beta_{m}(\omega_{m})z-\omega_{m}t\\}},$
(10)

and the frequency $\omega_{m}$ is at the intersection of the modal dispersion
curve $\beta_{m}(\omega)$ with a spectral plane $\mathcal{P}(\theta)$.
Consequently, the spectrum of an idealized ST supermode is discretized.

For ST supermodes based on baseband STWPs (referred to henceforth as baseband
ST supermodes), the spectral plane $\mathcal{P}_{\mathrm{B}}(\theta)$ is
$\omega\\!=\\!\omega_{\mathrm{o}}+(\beta-nk_{\mathrm{o}})\widetilde{v}$, where
$\widetilde{v}\\!=\\!c\tan{\theta}$. Consequently, the axial wave number for
any mode is
$\beta_{m}(\omega_{m})\\!=\\!(\omega_{m}-\omega_{\mathrm{o}})/\widetilde{v}+nk_{\mathrm{o}}$,
and the ST supermode field is
$E_{\mathrm{ST}}(x,z;t)\\!=\\!e^{i(nk_{\mathrm{o}}z-\omega_{\mathrm{o}}t)}\psi_{\mathrm{ST}}(x,z;t)$,
where the overall axially invariant envelope is given by:

$\psi_{\mathrm{ST}}(x,z;t)=\sum_{m}u_{m}(x)e^{-i(\omega_{m}-\omega_{\mathrm{o}})(t-z/\widetilde{v})}=\psi_{\mathrm{ST}}(x,0;t-z/\widetilde{v}).$
(11)

The phase velocity is $v\\!=\\!c/n$ and the group velocity is
$\widetilde{v}\\!=\\!c\tan{\theta}$. The spectral support for an ideal
baseband ST supermode [Fig. 3(a)] comprises a group of points (each
representing a monochromatic mode) at the intersection of
$\mathcal{P}_{\mathrm{B}}(\theta)$ with the modal dispersion curves
$\beta_{m}(\omega)$ evaluated for the waveguide in Fig. 2(a).

Alternatively, an ST supermode can be based on sideband STWPs, in which case
the frequencies $\omega_{m}$ of the contributing modes are determined by
intersecting the modal dispersion curves with the spectral plane
$\mathcal{P}_{\mathrm{S}}(\theta)$,
$\omega\\!=\\!\omega_{\mathrm{o}}+(\beta+nk_{\mathrm{o}})\widetilde{v}$. The
field for such a sideband ST supermode is
$E_{\mathrm{ST}}(x,z;t)\\!=\\!e^{-i(nk_{\mathrm{o}}z+\omega_{\mathrm{o}}t)}\psi_{\mathrm{ST}}(x,z;t)$,
so the phase velocity is $v\\!=\\!-c/n$, and the envelope
$\psi_{\mathrm{ST}}(x,z;t)$ is given by Eq. 11, except that the values of
$\omega_{m}$ and the range of values for $m$ will be different from those of a
baseband ST supermode sharing the same values of $\theta$ and $\Delta\omega$.
Finally, X-wave ST supermodes are formed by utilizing the plane
$\mathcal{P}_{\mathrm{X}}(\theta)$ given by $\omega\\!=\\!\beta\widetilde{v}$,
in which case:

$E_{\mathrm{ST}}(x,z;t)=\sum_{m}u_{m}(x)e^{-i\omega_{m}(t-z/\widetilde{v})}=E_{\mathrm{ST}}(x,0;t-z/\widetilde{v}).$
(12)

Despite the differences between baseband, sideband, and X-wave ST supermodes,
the spatio-temporal intensity is axially invariant
$I(x,z;t)\\!=\\!|E_{\mathrm{ST}}(x,z;t)|^{2}\\!=\\!I(x,0;t-z/\widetilde{v})$,
and their idealized spectra are discrete [Fig. 3(b)] at frequencies
$\omega_{m}$ extending over the bandwidth $\Delta\omega$. However, this ideal
limit corresponds to infinite energy; that is,
$\int\\!dxdt\,|E_{\mathrm{ST}}(x,z;t)|^{2}$ is not bounded, and thus cannot be
realized in practice. We will lift this restriction below in two ways: first
by relaxing the monochromaticity constraint and allowing a finite but small
bandwidth $\delta\omega$ (the spectral uncertainty) for each mode centered at
the ideal modal frequencies $\omega_{m}$, and then by examining the opposite
limit of ST supermodes with continuous spectra.

Figure 4: Comparison between index-guided ST supermodes based on baseband
STWPs, sideband STWPs, and X-waves in a planar multimode waveguide. In all
cases $d\\!=\\!50$ $\upmu$m, $n\\!=\\!1.5$, and $n_{\mathrm{c}}\\!=\\!1$. (a)
Intersection of the modal lines $\beta_{m}(\omega)$, represented by the grey-
shaded zone, with a spectral plane (thin red dotted line at
$\theta\\!=\\!40^{\circ}$) in the $(\beta,\frac{\omega}{c})$-plane. The
horizontal and vertical axes are normalized with respect to
$k_{d}\\!=\\!\tfrac{\pi}{d}$. (b) Same as (a) after zooming in onto a
bandwidth $\Delta\lambda\\!=\\!50$ nm at a carrier wavelength of
$\lambda_{\mathrm{o}}\\!=\\!1.55$ $\upmu$m, corresponding to the yellow
horizontal zone between two dotted lines in (a). (c) Same as (b), but
projected onto the $(q,\frac{\omega}{c})$-plane. Figure 5: (a) Maximum number
of modes $M_{\mathrm{max}}$ accessible to the baseband ST supermodes in Fig. 4
with the spectral tilt angle $\theta$, at bandwidths $\Delta\lambda\\!=\\!10$,
50, 100, and 200 nm; $\lambda_{\mathrm{o}}\\!=\\!1.55$ $\upmu$m,
$n\\!=\\!1.5$, $n_{\mathrm{c}}\\!=\\!1$, and $d\\!=\\!50$ $\upmu$m. (b,c) Same
as (a), but for (b) sideband and (c) X-wave ST supermodes. In these two cases,
$\theta$ is limited by the core and cladding light-lines, from
$\theta\\!\approx\\!34^{\circ}$ to $45^{\circ}$.

We plot in Fig. 3(c) the spatio-temporal intensity profile
$I(x,z;t)\\!=\\!|E_{\mathrm{ST}}(x,z;t)|^{2}$ for an ideal baseband ST
supermode at a fixed axial plane. This profile is independent of axial
position $I(x,z;t)\\!=\\!I(x,0;t-z/\widetilde{v})$, and is formed of a central
X-shaped spatio-temporal structure that resembles that of a freely propagating
STWP. The temporal width $\Delta T$ of the peak of the X-shaped feature is
determined by the temporal bandwidth ($\Delta
T\\!\sim\\!\tfrac{1}{\Delta\omega}$), and its spatial width $\Delta x$ is
determined by the spatial bandwidth ($\Delta x\\!\sim\\!\tfrac{1}{\Delta q}$).
However, a field formed of the superposition of monochromatic modes will
extend indefinitely in time around the central localized feature [Fig. 3(c)].
A related phenomenon was recently observed for free STWPs after discretizing
the spectrum in the context of space-time Talbot effects [48, 49].
Nevertheless, the intensity distribution is not periodic in time because the
frequencies $\omega_{m}$ are not equally spaced. We show below that relaxing
the monochromaticity constraint reduces the overall temporal extent of the ST
supermode around the central X-shaped feature.

The group velocity of the ST supermode $\widetilde{v}\\!=\\!c\tan{\theta}$
(group index $\widetilde{n}\\!=\\!\cot{\theta}$) is independent of the
waveguide structure and is distinct from the group velocities of the
contributing modes $\widetilde{v}_{m}$. This group velocity is the speed of
the central feature in Fig. 3(c), which can in principle be subluminal
($\widetilde{v}\\!<\\!c/n$), superluminal ($\widetilde{v}\\!>\\!c/n$), or even
negative-valued. This group velocity can be tuned in the same waveguide by
tuning $\theta$, which entails changing the frequencies $\omega_{m}$ assigned
to the modes.

In the ideal limit considered here, all effects of GVD (whether modal,
waveguide, or chromatic) have strictly speaking been eliminated. Because each
contributing mode is monochromatic, waveguide GVD (and, likewise, chromatic
GVD) is eliminated. Furthermore, the spectral projection for the ST supermode
onto the $(\beta,\tfrac{\omega}{c})$-plane must lie along a straight line,
thereby indicating a fixed group velocity $\widetilde{v}$, and thus
elimination of modal GVD.

Finally, the time-averaged intensity
$I(x,z)\\!=\\!\int\\!dt\,|E_{\mathrm{ST}}(x,z;t)|^{2}$ can be readily shown to
be:

$I(x,z)=\sum_{m}|A_{m}|^{2}|u_{m}(x)|^{2}=I(x).$ (13)

In other words, the time-averaged intensity is fixed along the waveguide [Fig.
3(d)], in contrast to conventional pulsed multimode fields having the same
contributing modes and bandwidth $\Delta\omega$ [Eq. 7, Fig. 2(f)]. By varying
the modal weights $|A_{m}|^{2}$, one may indeed sculpt the axially invariant
transverse intensity profile $I(x)$, which we explore further below.

###  III.4 Comparison of baseband, sideband, and X-wave ST supermodes

Any of the three types of STWPs (baseband, sideband, or X-waves) can in
principle be utilized to construct an ST supermode. Indeed, early theoretical
efforts considered making use of X-waves [27, 28, 29] and sideband FWMs [30]
for this purpose. However, just as these two types of STWPs have not proven
amenable to physical realization in free space at optical frequencies [1],
they face similarly insurmountable technical hurdles for contributing to ST
supermodes. We verify here that baseband STWPs provide the only practical path
towards realizing ST supermodes, and that they are more versatile with respect
to tuning the attributes of the ST supermode.

To elucidate the reason why baseband STWPs succeed where X-waves and sideband
STWPs come up short, we consider the maximum number of waveguide modes
$M_{\mathrm{max}}$ that can contribute to an ST supermode over a fixed
temporal bandwidth $\Delta\omega$. Small $M_{\mathrm{max}}$ curtails the
advantages conferred by ST supermodes, and the situation approaches that of
attempting to couple to single mode or few modes in a highly multimoded
waveguide. In Fig. 4 we examine $M_{\mathrm{max}}$ for baseband, sideband, and
X-wave ST supermodes, while holding the following key parameters fixed:
$\theta\\!=\\!40^{\circ}$ and $\Delta\omega\\!=\\!2.15\tfrac{\pi c}{d}$, which
corresponds to $\Delta\lambda\\!=\\!50$ nm at
$\lambda_{\mathrm{o}}\\!=\\!1.55$ $\upmu$m and $d\\!=\\!50$ $\upmu$m. We
select a superluminal group velocity
$\widetilde{v}\\!\approx\\!0.84c\\!>\\!c/1.5$ because the superluminal regime
is common to all three types of STWPs, whereas baseband STWPs uniquely span
the subluminal and negative-$\widetilde{v}$ regimes. Indeed, for sideband and
X-wave ST supermodes we have
$\tfrac{1}{n}\\!<\\!\tan{\theta}\\!<\\!\tfrac{1}{n_{\mathrm{c}}}$
($33.7^{\circ}\\!<\theta\\!<\\!45^{\circ}$) when $n\\!=\\!1.5$ and
$n_{\mathrm{c}}\\!=\\!1$). This limitation is absent from baseband ST
supermodes where the range for $\theta$ remains
$0^{\circ}\\!<\\!\theta\\!<\\!180^{\circ}$ as for their freely propagating
counterparts.

We identify in Fig. 4(a) the intersections of the spectral planes with the
dispersion curves $\beta_{m}(\omega)$ projected onto the
$(\beta,\tfrac{\omega}{c})$-plane. In Fig. 4(b) we zoom in onto the selected
spectral window to evaluate $M_{\mathrm{max}}$. Although all three ST
supermodes have the same $\theta$, the baseband ST supermode has a distinct
advantage because its spectral support is in general closer to the light-line
of the core medium. Consequently, the spectral plane intersects with the
dispersion curves of a larger number of guided modes than in the case of the
sideband or the X-wave ST supermode. Moreover, the lowest-order modes are
associated with the baseband ST supermode (starting from $m\\!=\\!0$), whereas
those associated with the other two types are high-order modes, as is clear in
the spectral projections onto the $(q,\tfrac{\omega}{c})$-plane [Fig. 4(c)].

This conclusion is highlighted further in Fig. 5 where we plot
$M_{\mathrm{max}}$ with $\theta$ at different bandwidths for all three types
of ST supermodes. In all cases, $M_{\mathrm{max}}$ increases as $\theta$
deviates away from the luminal condition ($\widetilde{n}\\!=\\!n$). Moreover,
$M_{\mathrm{max}}$ increases with bandwidth at fixed $\theta$. For any
selection of parameters, $M_{\mathrm{max}}$ is always significantly larger for
the baseband case when compared with the sideband or X-wave cases.
Additionally, the baseband ST supermode enables spanning a much wider range of
values for $\theta$. Therefore, baseband ST supermodes can have access to a
larger number of modes $M_{\mathrm{max}}$ for a fixed $\Delta\omega$ and
$\theta$, they can make use of the low-order modes, and thus require a smaller
numerical aperture than that for sideband or X-wave ST supermodes. In light of
these findings, we proceed to examine solely baseband ST supermodes, which are
likely to be the only ones realizable in practice [39].

##  IV Finite-energy ST supermodes

As mentioned above, the ideal limit for any ST supermode – whereupon each
contributing mode is associated with a single frequency $\omega_{m}$ – cannot
be attained in practice because it requires infinite energy for its
realization. Instead, any realizable ST supermode must feature a ‘spectral
uncertainty’: a finite bandwidth $\delta\omega$ associated with each mode
centered at the ideal frequencies $\omega_{m}$. However, unlike the
conventional pulsed multimode field, the bandwidth assigned to each mode is
small $\delta\omega\\!\ll\\!\Delta\omega$ [compare Fig. 6(a) with Fig. 2(c)].
To construct a finite-energy ST supermode, we start – just as for an ideal ST
supermode – by determining the discrete modal frequencies $\omega_{m}$ at the
intersection of the modal dispersion curves $\beta_{m}(\omega)$ with the
baseband spectral plane $\omega\\!=\\!\omega_{\mathrm{o}}+(\beta-
nk_{\mathrm{o}})\widetilde{v}$, and then associate a finite bandwidth
$\delta\omega$ with each mode (assumed for simplicity to be independent of
$m$), rather than taking purely monochromatic modes. Therefore, the spectrum
is not discrete, and the modal contributions are
$E_{m}(x,z,t)\\!=\\!e^{i\\{\beta_{m}(\omega_{m})z-\omega_{m}t\\}}u_{m}(x)\psi_{m}(z,t)$,
where $\psi_{m}(z,t)$ is the axial modal envelope. As such, the envelope for
the finite-energy ST supermode appears at first sight to be similar to the
conventional pulsed multimode field in Eq. 5. However, there are several
crucial differences between the two guided pulsed multimode fields. First,
whereas the carrier frequency is $\omega_{\mathrm{o}}$ for the pulsed modes in
a conventional field, it is $\omega_{m}$ for each mode in the ST supermode.
Second, the integration in Eq. 5 for the conventional field is carried out
over the full bandwidth $\Delta\omega$ (centered at $\omega_{\mathrm{o}}$),
whereas for the ST supermode it extends over only $\delta\omega$ (centered at
$\omega_{m}$); see Fig. 6(a). Consequently, the temporal extent of
$\psi_{m}(z;t)$ for the ST supermode is $\Delta
T_{\mathrm{p}}\\!\sim\\!\tfrac{1}{\delta\omega}\\!\gg\\!\Delta T$ [Fig. 6(b)].
Third, the Taylor expansion for $\beta_{m}(\omega)$ for each mode is centered
at a different frequency $\omega_{m}$ rather than the same frequency
$\omega_{\mathrm{o}}$. However, in most cases the change in values of
$\widetilde{v}_{m}$ and $\eta_{m}$ caused by this frequency shift are
negligible.

Figure 6: (a) The spectral support for a finite-energy baseband ST supermode
(the small green domains) at the intersection of the modal dispersion curves
$\beta_{m}(\omega)$ for the waveguide in Fig. 2(a) with a tilted spectral
plane $\omega\\!=\\!\omega_{\mathrm{o}}+(\beta-nk_{\mathrm{o}})\widetilde{v}$.
The narrow spectra (bandwidth $\delta\omega$) associated with the modes do not
overlap; compare to the spectral support for a conventional pulsed multimode
field in Fig. 2(c). (b) Spatio-temporal intensity profiles for the same first
four individual modes in Fig. 2(d), but each with bandwidth 0.2 nm (18-ps
pulse widths). (c) Spatio-temporal profile of a superluminal ST supermode
($\theta\\!=\\!75^{\circ}$) formed of the first 8 even-symmetry modes with
equal weights. The intensity profile is shown at three axial planes. The
X-shaped central peak has a temporal width of 280 fs
($\Delta\lambda\\!\approx\\!22$ nm at $\lambda_{\mathrm{o}}\\!=\\!1.55$
$\upmu$m. (d) The time-averaged intensity $I(x,z)$ is invariant along the
propagation direction. The axial planes selected in (c) are identified by
dashed arrows.

Of course, the attributes of finite-energy ST supermodes also deviate from
their ideal counterparts. First, realistic ST supermodes are not strictly
propagation invariant. At $z\\!=\\!0$ the profile still has the central
X-shaped feature at $t\\!=\\!0$; however, the profile no longer extends
indefinitely along $t$, and is instead limited to a temporal width $\Delta
T_{\mathrm{p}}\\!\sim\\!\tfrac{1}{\delta\omega}$ determined by the axial
envelope. Indeed, at $z\\!=\\!0$ the envelope for the $m^{\mathrm{th}}$ mode
is
$\psi_{m}(0;t)\\!=\\!\int_{\delta\omega}\\!d\Omega\widetilde{\psi}(\Omega)e^{-i\Omega
t}$, which is independent of $m$. Therefore, all the modal envelopes initially
overlap, but subsequently walk off with propagation along $z$. Each envelope
propagates at a group velocity $\widetilde{v}_{m}$ so that modal GVD broadens
the superposed envelope, and causes walk-off with respect to the central
X-shaped feature of the ST supermode that travels instead at a group velocity
$\widetilde{v}\\!=\\!c\tan{\theta}$. Moreover, each envelope $\psi_{m}(z;t)$
undergoes waveguide GVD (and potentially chromatic GVD) with a chirp parameter
$a_{m}\\!=\\!\eta_{m}(\delta\omega)^{2}z$, which further contributes to
deforming the finite-energy ST supermode with propagation [Fig. 6(c)].

Figure 7: Changing the intensity profile of ST supermodes by modal-weight
engineering. The value of the parameters for the waveguide and the ST
supermodes are the same as in Fig. 6. (a) Superposing the first 8 odd-parity
modes with equal weights results in a null at the center of the intensity
profile. (b) Combining the first 16 modes of both odd and even modes creates a
uniform field across the waveguide. (c) Superposing modes indexed by $3m+2$
creates two symmetric off center peaks. (d) Combining the first 8 even-parity
modes with equal weights but random relative phases selected independently
over $[0,2\pi]$. Although the central X-shaped feature is no longer visible in
the spatio-temporal profile $I(x;\tau)$ of the ST supermode at any $z$, the
time-averaged intensity is nevertheless the same as that in Fig. 6(d).

Nevertheless, the chirp parameter is reduced with respect to a conventional
pulsed multimode field by a factor $(\tfrac{\Delta\omega}{\delta\omega})^{2}$,
which can be quite significant. For example, in the proof-of-principle
experiment in [39] the values were $\Delta\lambda\\!\approx\\!1.7$ nm and
$\delta\lambda\\!\approx\\!42$ pm, so that GVD is reduced by a factor of
$\approx\\!1600$. Therefore, both waveguide and chromatic GVD can be
drastically reduced in an ST supermode with respect to a conventional pulsed
multimode field having the same total bandwidth $\Delta\omega$. However, the
modal GVD experienced by the finite-energy ST supermode and the conventional
field are the same if the same set of modes contribute to them. Therefore, in
practice, ST supermodes can help reduce dispersive effects in those scenarios
dominated by chromatic or waveguide GVD, but not those in which modal GVD is
the dominant factor.

Although the spatio-temporal intensity profile for finite-energy ST supermodes
are not propagation invariant [Fig. 6(c)], their time-averaged intensity
$I(x,z)\\!=\\!\int\\!dt\,|E(x,z;t)|^{2}$ can nevertheless be axially invariant
[Fig. 6(d)]. It is straightforward to show that axial invariance of $I(x,z)$
for a finite-energy ST supermode is guaranteed as long as the spectral domains
associated with the contributing modes do not overlap, which is easily
achieved when $\delta\omega\\!\ll\Delta\omega$. In this case, the time-
averaged intensity $I(x,z)\\!=\\!\sum_{m}|A_{m}|^{2}|u_{m}(x)|^{2}$ retains
the same form as in an ideal ST supermode. Axial variation in $I(x,z)$ occurs
only when the modal bandwidths overlap spectrally.

Figure 8: Spectrally continuous ST supermodes. (a) ST supermode spectrum
projected onto the $(\beta,\tfrac{\omega}{c})$-plane. The full input spectrum
of the pulse coupled to the waveguide is accommodated by the waveguide modes
selected. Instead of assigning a constant narrow bandwidth to each mode that
is smaller than the mode spacing, the assigned spectra are contiguous between
the successive modes. The top-left inset shows the spectrum projected onto the
$(q,\tfrac{\omega}{c})$-plane, and the bottom-right inset shows the overall
spectral magnitude ($k_{d}\\!=\\!\tfrac{\pi}{d}$). (b) The time-averaged
intensity profile $I(x,z)$ with $\theta\\!=\\!75^{\circ}$ and selecting the
first 8 even-parity modes. (c) $I(x;\tau)$ at different axial planes; only the
central X-shaped feature initially appears, which undergoes deformation along
$z$ because of modal dispersion. (d) Applying a random phase to each frequency
spreads the field out temporally and randomises the intensity distribution.
Nevertheless, the time-averaged intensity remains that in (b).

###  IV.1 Sculpting the transverse spatial profile

Equation 13, which applies to both ideal and finite-energy ST supermodes,
indicates that the propagation-invariant time-averaged spatial intensity
profile in the waveguide can be tuned by varying the modal weights in the ST
supermode. In Fig. 6(d), we made use of equal-weighted modes with even-parity
symmetry ($m\\!=\\!0,2,4,\cdots$), which results in an on-axis peak atop an
approximately flat background. Alternatively, one may produce a propagation-
invariant on-axis dip by selecting modes with odd-parity symmetry
($m\\!=\\!1,3,5,\cdots$) as shown in Fig. 7(a). The evolution of the spatio-
temporal profile resembles that in Fig. 6(d) except that an intensity null is
always maintained at $x\\!=\\!0$, thereby resulting in an on-axis dip in the
time-averaged intensity at $I(x\\!=\\!0,z)$. This unique dark-beam profile in
a multimode waveguide offers opportunities for atom guidance in hollow-core
waveguides and in reducing unwanted nonlinear interactions in the transport of
high-energy pulses.

Alternatively, by selecting equal-weighted modes with both even- and odd-
parity symmetry ($m\\!=\\!0,1,2,3\cdots$), we obtain a flat intensity profile
across the waveguide cross section, as shown in Fig. 7(b). Such a
configuration can perhaps be useful in lighting applications (see below for a
discussion of spectrally incoherent fields). A further example is shown in
Fig. 7(c) in which a symmetric double-peaked structure is realized by
selecting a subset of even- and odd-parity modes indexed by $3m+2$. Using
Fourier analysis, more complex intensity profiles can be synthesized by
carefully controlling the modal weights $|A_{m}|^{2}$. In general, however,
only even-parity intensity distributions can be produced in a symmetric
waveguide using this procedure.

###  IV.2 Impact of spectral coherence

Examining Eq. 13 shows that the relative modal phases do not impact the
intensity profile $I(x,z)$. This indicates that random phases introduced
between the modal weights (which can arise when using a spectrally incoherent
source rather than a coherent pulse) will not induce axial changes as long as
the relative magnitudes are maintained constant. We confirm this conclusion in
Fig. 7(d) where we introduce random phases selected with a uniform probability
from the span $[0,2\pi]$ to the modes comprising the ST supermode. The
intensity $I(x,z)$ is identical to the coherent field in Fig. 6(d), where the
same constitutive modes were in phase. However, the spatio-temporal profile
$I(x,z;t)$ is of course impacted by these random phases. This result indicates
that even spectrally incoherent light can be used to produce ST supermodes.

##  V Spectrally continuous ST supermodes

Neither ideal ST supermodes (where the spectrum is discretized at frequencies
$\omega_{m}$) nor finite-energy ST supermodes (where narrow spectra
$\delta\omega$ are centered at $\omega_{m}$) utilize the full spectrum
$\Delta\omega$. Therefore, this overall approach to synthesizing ST supermodes
is not spectrally nor energy efficient. Surprisingly, one may indeed construct
spectrally continuous ST supermodes while retaining some of the advantages of
their ideal counterparts. The general approach is illustrated in Fig. 8(a),
where the full spectrum $\Delta\omega$ is divided amongst $M$ spectrally
contiguous modes, each of bandwidth
$\Delta\omega_{m}\\!\sim\\!\tfrac{\Delta\omega}{M}$. As long as the modal
spectra do not overlap, the time-averaged intensity remains
$I(x,z)\\!=\\!\sum_{m}|A_{m}|^{2}|u_{m}(x)|^{2}\\!=\\!I(x,0)$, which is
axially invariant as shown in Fig. 8(b).

Of course, the time-resolved spatio-temporal profile $I(x,z;t)$ is not
invariant along $z$. Interestingly, as a consequence of the absence of gaps in
its spectrum, the temporal extent of the ST supermode is now reduced to
encompass only the central X-shaped feature, and thus has a similar profile to
its freely propagating counterpart [Fig. 8(c)]. Furthermore, this finite-
energy ST supermode undergoes dispersion with propagation, with waveguide and
chromatic GVD reduced by a factor $\sim\\!M^{2}$, which can be significant.
Just as in the case of finite-energy ST supermodes examined above, modal GVD
is the same as that for a conventional pulsed multimode field comprising the
same modes. Therefore, spectrally continuous ST supermodes provide an
advantage with regards to dispersion reduction without sacrificing power, in
addition to the axial invariance of the time-averaged intensity.

The axial invariance of $I(x,z)$ extends to spectrally incoherent light as
well, as shown in Fig. 8(d). Here we assign independent randomly selected
phases with a uniform probability over $[0,2\pi]$, with each phase assigned to
a spectral window of width $\delta\lambda\\!=\\!30$ pm. Consequently, the
field spreads temporally with no discernible spatio-temporal features as
expected for an incoherent field, in contrast to the coherent scenario in Fig.
8(c). Nevertheless, the time-averaged intensity $I(x,z)$, which is here the
quantity of interest, remains axially invariant as shown in Fig. 8(b). This
class of spectrally continuous ST supermodes has not yet been demonstrated
experimentally.

##  VI Discussion

Although ST supermodes can be considered guided counterparts of freely
propagating STWPs, there are nevertheless significant differences between
them. First, whereas the spectrum of ideal ST supermodes is discrete, that for
an ideal STWP is continuous. Second, the maximum propagation distance
$L_{\mathrm{max}}$ of realistic STWPs in free space is limited by the spectral
uncertainty, whereas that for ST supermodes is determined by the spectral
overlap between the underlying modes. Therefore, even in presence of spectral
uncertainty in a realistic ST supermode (the finite spectrum associated with
each mode), its transverse intensity profile can remain invariant
indefinitely. Third, the spectral discretization results in an extended
spatio-temporal structure for ST supermodes around a central X-shaped feature,
whereas freely propagating STWPs comprise only an X-shaped feature. The impact
of spectral discretization on broadening the spatio-temporal profile occurs
even in free STWPs as confirmed recently in the context of ST Talbot effects
[48, 49]. On the other hand, we have shown here that taking a spectrally
continuous ST supermode has the same spatio-temporal profile as its free STWP
counterpart. Finally, whereas the spatio-temporal profile of a free STWP
remains invariant until $L_{\mathrm{max}}$ is reached, the corresponding
profile for an ST supermode is deformed because of modal dispersion even
though the time-averaged intensity profile is invariant. Where the impact of
waveguide and chromatic GVD can be significantly reduced in a realistic ST
supermode, the impact of modal dispersion remains.

The theoretical treatment provided here extends beyond the constraints we have
imposed on our analysis. First, although we focused here on coherent pulsed
light, the main conclusions remain valid for broadband, spectrally incoherent
continuous-wave radiation. The same conclusion was reached for free STWPs (see
[50, 51, 3] for theoretical treatments and [52, 53] for experimental
realizations). This opens up new opportunities for delivering broadband
incoherent light with controllable intensity profiles over multimode fibers
for lighting applications. Second, with regards to polarization, all our
calculations here pertain to TM-polarized modes, but the analysis applies
equally to TE and mixed polarization modes. Third, we have considered here
index-guided ST supermodes. Nevertheless, this basic concept applies to any
other guidance mechanism; e.g., we outline in the Appendix the theory of ST
supermodes in planar-mirror waveguides, and [54] describes the scenario of a
graded-index planar waveguide. Finally, our analysis assumed planar waveguides
with light confined along one transverse dimension. However, the analysis may
be extended to encompass ST supermodes in conventional waveguides in which
light is confined in both transverse dimensions, and also to fiber-guided ST
supermodes. Recent theoretical [55, 56] and experimental [57, 58, 59] work on
the synthesis of STWPs localized in all dimensions now brings this possibility
closer to experimental realization.

A recent breakthrough [37] was achieved with respect to the nonlinear
synthesis of ST supermodes in a multimode optical fiber via a high-energy
excitation pulse [60]. However, because the spectrum associated with each mode
in the ST supermode is large, the defining characteristics of ST supermodes as
emerging from our analysis here are not expected to be observable. Finally, we
note that the structure of ST supermodes is similar to that of the field in a
waveguide moving at relativistic speeds [61]. This can be understood by
recognizing that STWPs can be obtained from the Lorentz transformation of
monochromatic beams [62, 63, 7, 64]. The study of STWPs and ST supermodes can
thus enrich our understanding of relativistic interactions between optical
fields and photonic devices.

##  VII Conclusions

In conclusion, we have presented a theoretical treatment for ST supermodes in
planar multimode waveguides, which are the guided counterparts to freely
propagating STWPs. In contrast to conventional pulsed multimode fields in
waveguides or fibers, ST supermodes retain their time-averaged intensity
profile indefinitely. Moreover, the group index of the ST supermode can be
tuned above or below the group indices of its constitutive modes. It is
expected that the theoretical treatment developed here and recently reported
experimental results [39] can be extended to realizing ST supermodes in
multimode fibers or conventional waveguides in which the field is confined in
both transverse dimensions.

Figure 9: Planar-mirror waveguide. (a) Schematic of planar-mirror waveguide
structure. (b) The modal dispersion curves are 1D curves on the surface of the
light-cone $q^{2}+\beta^{2}\\!=\\!(n\tfrac{\omega}{c})^{2}$, where $q$ and
$\beta$ are the transverse and axial wave numbers, respectively. (c) The
spectral support for a baseband ST supermode is the intersection of a tilted
plane $\mathcal{P}(\theta)$ with the modal dispersion curves, which
corresponds to a collection of points (shown by green dots) in the
$(q,\tfrac{\omega}{c})$-plane and (d) in the
$(\beta,\tfrac{\omega}{c})$-plane. Here $\theta\\!=\\!60^{\circ}$ and
$n\\!=\\!1.5$, which corresponds to a superluminal ST supermode.

Funding. U.S. Office of Naval Research (ONR) N00014-17-1-2458 and
N00014-20-1-2789.

## Appendix: ST supermodes in planar-mirror waveguides

Figure 10: Maximum number of modes accessible in a planar-mirror waveguide
with $\theta$ for (a) baseband ST supermodes at bandwidths
$\Delta\lambda\\!=\\!10$, 50, 100, and 200 nm;
$\lambda_{\mathrm{o}}\\!=\\!1.55$ $\upmu$m, $n\\!=\\!1.5$, and $d\\!=\\!50$
$\upmu$m. (b) Same as (a), but for X-wave ST supermodes. For X-waves, $\theta$
is restricted to the superluminal region and $\theta\\!=\\!90^{\circ}$.

For sake of comparison to the index-guided ST supermodes described in the main
text, we consider their counterpart in a planar waveguide formed of two
perfect parallel mirrors [Fig. 9(a)]. The separation between the two mirrors
along $x$ is $d$, and the refractive index of the medium is $n$ (assumed to be
independent of frequency). The quantized transverse wave number $q_{m}$ for
the $m^{\mathrm{th}}$ mode ($m\\!=\\!1,2,3,\cdots$) is
$q_{m}\\!=\\!m\tfrac{\pi}{d}\\!=\\!k\sin{\varphi_{m}}\\!=\\!mk_{\mathrm{c}}$,
where $k\\!=\\!nk_{o}=\tfrac{n\omega}{c}$,
$\omega_{\mathrm{c}}\\!=\\!\tfrac{\pi c}{nd}$ is the cut-off frequency,
$\lambda_{\mathrm{c}}\\!=\\!2d$ is the cut-off wavelength in the waveguide,
$k_{\mathrm{c}}\\!=\\!\tfrac{2\pi}{\lambda_{\mathrm{c}}}\\!=\\!n\tfrac{\omega_{\mathrm{c}}}{c}$,
$\varphi_{m}$ is the frequency-dependent bounce angle of a plane wave relative
to the $z$-axis [41], and the axial wave number is
$\beta_{m}\\!=\\!\sqrt{n^{2}k_{\mathrm{o}}^{2}-q_{m}^{2}}\\!=\\!k\cos{\varphi_{m}}$
where $k_{\mathrm{o}}\\!=\\!\tfrac{2\pi}{\lambda_{\mathrm{o}}}$ and
$\lambda_{\mathrm{o}}$ is the vacuum wavelength. The mode profiles are
$u_{m}(x)\\!\propto\\!\cos{(q_{m}x)}$ for odd $m$ and
$u_{m}(x)\\!\propto\\!\sin({q_{m}x)}$ for even $m$, are independent of
$\omega$, and orthonormal $\int\\!dx\,u_{m}(x)u_{n}^{*}(x)\\!=\\!\delta_{nm}$.
The group velocity at $\omega\\!=\\!\omega_{\mathrm{o}}$ is
$\widetilde{v}_{m}\\!=\\!1\big{/}\tfrac{d\beta_{m}}{d\omega}\big{|}_{\omega_{\mathrm{o}}}\\!=\\!\tfrac{c}{n}\cos{\\{\varphi_{m}(\omega_{\mathrm{o}})\\}}$,
which is always subluminal. The associated GVD coefficient is
$\eta_{m}\\!=\\!\tfrac{d^{2}\beta_{m}}{d\omega^{2}}\big{|}_{\omega_{\mathrm{o}}}\\!=\\!-\tfrac{n^{2}m^{2}\pi^{2}}{c^{2}d^{2}\beta^{3}_{m}}\\!=\\!-\tfrac{n}{c\omega_{\mathrm{o}}}\tan^{2}{\varphi_{m}}\sec{\varphi_{m}}$;
the GVD of each mode is always anomalous (negative-valued $\eta_{m}$), and
increases in absolute value with modal index;
$\varphi_{m}\\!\rightarrow\\!90^{\circ}$ for higher $m$, whereby $|\eta_{m}|$
increases. In general, the temporally resolved and time-averaged intensity
profiles have similar forms to those for index-guided ST supermodes.

###  VII.1 X-wave ST supermodes

The frequencies $\omega_{m}$ for an X-wave ST supermode in the planar-mirror
waveguide are at the intersection of the dispersion curves $\beta_{m}(\omega)$
with the plane $\omega\\!=\\!\beta\widetilde{v}$, where
$\widetilde{v}\\!=\\!c\tan{\theta}$, $\theta$ is the spectral tilt angle, and
we define a group index
$\widetilde{n}\\!=\\!c/\widetilde{v}\\!=\\!\cot{\theta}\\!<\\!n$. The
intersection points are at
$\omega_{m}\\!=\\!m\omega_{\mathrm{c}}\tfrac{n}{\sqrt{n^{2}-\widetilde{n}^{2}}}$
and
$\beta_{m}\\!=\\!mk_{\mathrm{c}}\tfrac{\widetilde{n}}{\sqrt{n^{2}-\widetilde{n}^{2}}}$.
The group velocity for any mode
$\widetilde{v}_{m}\\!=\\!\tfrac{c}{n}\sqrt{1-(\tfrac{q_{m}}{n\omega_{\mathrm{c}}/c})^{2}}\\!=\\!\widetilde{v}_{\mathrm{g}}$
is independent of $m$, is subluminal, and
$\widetilde{v}\cdot\widetilde{v}_{\mathrm{g}}\\!=\\!(\tfrac{c}{n})^{2}$;
whereas the ST supermode is superluminal at $\widetilde{v}$. Nevertheless, the
GVD parameters for the modes $\eta_{m}$ remain different. Moreover, the phase
velocity of the modes
$v_{\mathrm{ph}}\\!=\\!\tfrac{\omega_{m}}{\beta_{m}}\\!=\\!\widetilde{v}$ is
equal for all the modes, and is equal to the group velocity for the ST
supermode.

Because the frequencies of waveguide modes $\omega_{m}$ contributing to this
ST supermode are harmonics of the fundamental frequency $\omega_{\mathrm{c}}$
($\omega_{m}\\!\propto\\!m\omega_{\mathrm{c}}$), a large bandwidth is required
to produce an ST supermode with a reasonable number of underlying modes (see
[27, 28, 29]). This makes the synthesis of ST supermodes based on X-waves
difficult, just as in their index-guiding counterparts.

###  VII.2 Baseband ST supermodes

For baseband ST supermodes the modal frequencies $\omega_{m}$ are at the
intersection of the dispersion curves $\beta_{m}(\omega)$ with the plane
$\omega\\!=\\!\omega_{\mathrm{o}}+(\beta-nk_{\mathrm{o}})c\tan{\theta}$,

$\frac{n\omega_{m}}{c}=\frac{k}{(\widetilde{n}-n)}\left\\{\widetilde{n}\pm
n\sqrt{1+\left(\frac{m\lambda_{o}}{2nd}\right)^{2}\frac{n-\widetilde{n}}{n+\widetilde{n}}}\right\\},$
(14)

with $\widetilde{v}\\!=\\!c\tan{\theta}$ and
$\widetilde{n}\\!=\\!\cot{\theta}$, which can be superluminal, subluminal or
even negative-valued. There is no limit on $m$ in the superluminal regime
($\widetilde{n}\\!<\\!n$), in the subluminal regime ($\widetilde{n}\\!>\\!n$)
real solutions for $\omega_{m}$ require that
$m<\tfrac{2nd}{\lambda_{\mathrm{o}}}\sqrt{\frac{|n-\widetilde{n}|}{n+\widetilde{n}}}$.

###  VII.3 Sideband ST supermodes

We now consider sideband ST wave packets whose modal centers lie in the
($\beta,\tfrac{\omega}{c})$-plane along the line
$\tfrac{\omega_{m}}{c}\\!=\\!k_{\mathrm{o}}+(\beta_{m}(\omega_{m})+nk_{\mathrm{o}})\tan{\theta}$.
The group velocity of the ST supermode is $\widetilde{v}\\!=\\!c\tan{\theta}$.
Similar to ST supermodes based on X-waves, those based on sideband STWPs are
also only superluminal.

$\frac{n\omega_{m}}{c}=\frac{k}{(\widetilde{n}+n)}\left\\{\widetilde{n}\pm
n\sqrt{1+\left(\frac{m\lambda_{\mathrm{o}}}{2nd}\right)^{2}\frac{n+\widetilde{n}}{n-\widetilde{n}}}\right\\}.$
(15)

In general, like the X-wave, a sideband ST supermode requires a large
bandwidth that is not practical in optics.

###  VII.4 Comparison of X-wave, baseband, and sideband ST supermodes

To conclude this review of planar-mirror waveguide supermode properties we
compare the number of modes that can be accessed for baseband and X-wave ST
wave packets. Mode frequency equations for X-waves and baseband waves (given
in Appendix A and B) can be used to find the number of modes, $\Delta m$, that
are present for a given fixed bandwidth $\Delta\omega$. Fig. 10 shows
calculations of the number of modes available for supermode propagation at a
given spectral tilt angle and bandwidth. In all cases baseband supermodes have
many more modes available than X-wave supermodes. We also note that baseband
modes have low $m$ values while X-waves must use high $m$ value modes (i.e.,
they are highly nonparaxial).

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