2023

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Sixian You

1]Department of Electrical Engineering and Computer Science, Massachusetts
Institute of Technology, Cambridge, MA, USA

2]Department of Brain and Cognitive Sciences, Massachusetts Institute of
Technology, Cambridge, MA, USA

# Spectral-temporal-spatial customization via modulating multimodal nonlinear
pulse propagation

Tong Qiu Honghao Cao Kunzan Liu Li-Yu Yu Manuel Levy Eva Lendaro Fan Wang
<EMAIL_ADDRESS>[  [

###### Abstract

Multimode fibers (MMFs) have recently reemerged as attractive avenues for
nonlinear effects due to their high-dimensional spatiotemporal nonlinear
dynamics and scalability for high power. High-brightness MMF sources with
effective control of the nonlinear processes would offer new possibilities for
a wide range of applications from high-power fiber lasers, to bioimaging and
chemical sensing, and to novel physics phenomena. Here we present a simple yet
effective way of controlling nonlinear effects at high peak power levels: by
leveraging not only the spatial but also the temporal degrees of freedom of
the multimodal nonlinear pulse propagation in step-index MMFs using a
programmable fiber shaper. This method represents the first method that
enables modulation and optimization of multimodal nonlinear pulse propagation,
achieving high tunability and broadband high peak power. Its potential as a
nonlinear imaging source is further demonstrated by applying the MMF source to
multiphoton microscopy, where widely tunable two-photon and three-photon
imaging is achieved with adaptive optimization. These demonstrations highlight
the effectiveness of directly modulating multimodal nonlinear pulse
propagation to enhance the high-dimensional customization and optimize the
high spectral brightness of MMF output. These advancements provide new
possibilities for technology advances in nonlinear optics, bioimaging,
spectroscopy, optical computing, and material processing.

##  1 Introduction

Multimode fibers (MMFs) have become renewed focuses of interest for
investigating multimode nonlinear optics, owing to their versatile degrees of
freedom for guided waves and their potential for high power scalability
wright2022nonlinear ; krupa2019multimode ; wright2022physics ;
krupa2016observation ; krupa2018spatiotemporal ; leventoux20213d ;
shalaby2014visible ; zhou2015half ; lopez2016visible ; krupa2016spatiotemporal
; eftekhar2017versatile ; rishoj2019soliton . Over the last 5–10 years,
significant progress has been made in studying and controlling nonlinear
effects in MMFs, presenting unprecedented opportunities for a plethora of
applications in optical sensing, imaging, manipulation, and computing. For
example, the application of MMFs to fiber lasers could permit the development
of low-cost light sources with dramatically higher pulse energy and average
power due to the larger mode areas of the MMFs wright2017spatiotemporal ;
teugin2019spatiotemporal ; wei2020harnessing ; ding2021spatiotemporal . More
importantly, the rich spatiotemporal dynamics and complex intermodal
interactions in MMFs constitute a broad avenue for controlling nonlinear wave
propagation, opening up possibilities for novel physics and new applications
such as spatiotemporal light control xiong2016spatiotemporal ;
vaughan2002automated , nonlinear frequency generation krupa2016observation ;
demas2015intermodal ; nazemosadat2016phase ; eslami2022two ; rishoj2019soliton
, nonlinear optical imaging and sensing redding2013all ; xiong2020deep ;
xu2013recent ; moussa2021spatiotemporal , optical wave turbulence
berti2022interplay ; picozzi2014optical ; garnier2019wave ;
podivilov2019hydrodynamic , and optical computing teugin2021scalable ;
rahmani2022learning .

Recently, control of the multimode spatiotemporal light fields has been
demonstrated by adjusting the input wavefront to graded-index (GRIN) MMFs for
selective mode excitation wright2015controllable ; eftekhar2017versatile ;
tzang2018adaptive ; deliancourt2019wavefront ; teugin2020controlling ;
wei2020harnessing ; bender2023spectral . These studies, by exploiting the
spatial degree of control of the input field, demonstrated the unprecedented
potential of MMFs as technological solutions for tunable high-power light
sources and spatiotemporal avenues for fundamental nonlinear optical physics.
However, the primary focus on spatial control in GRIN MMFs often subjects the
spatiotemporal pulses to limited capabilities in many applications due to two
major problems: difficulty in control and limited broadband spectral
brightness. Fiber sources with higher-dimensional tunability and broadband
spectral brightness could usher in a wave of opportunities for new types of
light sources and nonlinear phenomena renninger2013optical ;
pourbeyram2013stimulated ; lopez2016visible ; wright2016self ; liu2016kerr ;
krupa2017spatial ; ahsan2018graded ; eftekhar2019accelerated , and ignite
advances in cutting-edge applications, such as bioimaging, laser
manufacturing, chemical sensing, and optical computing. To realize this
potential, harnessing the untapped potential of temporal degrees of freedom to
control nonlinear effects in MMFs at high peak power levels presents a
compelling solution.

Here we introduce a new way of controlling multimodal nonlinear effects in
high-power regimes by exploiting not only the spatial but also the temporal
degrees of freedom. Complementary to manipulating the input pulses that seed
the nonlinear pulse propagation, we propose to modulate the multimodal
nonlinear pulse propagation processes by introducing programmable time-
dependent disorders along the fiber berti2022interplay , which is implemented
as axial-position-dependent macro-bending. This was made possible by using a
single 3D-printed modulating device dubbed fiber shaper (conceptually depicted
in Fig. 1a). By applying the fiber shaper to a standard silica step-index (SI)
fiber at high-power levels (Fig. 1b), we showed effective modulation of the
multimodal nonlinear pulse propagation and customization of the MMF output
field in the spectral, temporal, and spatial domains.

We further demonstrated the ability to generate high peak power even extending
outside the conventional soliton regime. The broadband high peak power
(approaching megawatt on average) was accomplished by combining spectral
energy reallocation (up to 166-fold) and temporal shortening (up to 4-fold).
Such capabilities are uniquely enabled by the fiber shaper, a single all-fiber
modulator without the need for external wavefront or pulse shaping. To
showcase its potential, we applied the apparatus directly to multiphoton
microscopy, an application with demanding needs for spectral and temporal
properties of pulses. Through adaptive optimization of the fiber shaper, we
achieved efficient and widely tunable two-photon and three-photon microscopy
of fluorescent beads and label-free tissues. Our proposed approach, which
presents the first method enabling simultaneous access to the spatial and
temporal degrees of control of multimodal nonlinear effects, relies on the
application of precisely controlled multi-point macro-bending to the fiber.
Recent demonstrations have shown the use of piezoelectric actuators (i.e.,
fiber pianos) to induce controllable bends in MMFs for modulating modal
dynamics in the linear regime regelman2016method ; golubchik2015controlled ;
resisi2020wavefront ; finkelstein2023spectral . Inspired by this design, we
want to develop a module that’s compatible with nonlinear regime and overcomes
the challenge of tunability and low spectral brightness in fiber sources.
Despite the promising applications in linear and quantum regimes
resisi2020wavefront ; finkelstein2023spectral ; shekel2023shaping , it is
challenging to directly adopt linear fiber pianos to the nonlinear regime due
to application-specific issues such as light loss, design flexibility, and
scalability. By employing the slip-on fiber shaper created through 3D printing
that supports smooth (minimal micro-bending) but large-curvature bending
(sufficient model coupling effects), we envision its potential as an
accessible and flexible tool for both linear and nonlinear modulations of
MMFs, with significant implications in nonlinear optics, bioimaging, and
spectroscopy.

Figure 1: Conceptual demonstration of spatiotemporal control of nonlinear
pulse propagation in MMFs by a fiber shaper. a, Illustration of the principle
of spatiotemporal control of multimodal nonlinear effects in MMFs by
introducing programmable time-dependent disorders using a fiber shaper. The
fiber shaper applies multi-point macro-bending of various bending radii to the
fiber and thus alters the multimodal spatial interactions at different time
points (e.g., $t_{1}$ to $t_{5}$) along the pulse temporal evolution. b,
Visualization of experimentally acquired output spectra of an SI MMF with the
same launching condition but different combinations of actuator positions. A
total of 3125 configurations were exhaustively searched, involving 5
actuators, each with 5 different states. Three representative spectra
corresponding to three randomly chosen configurations are highlighted in
distinct colors. Vertical solid lines mark the spectral range measured by
different spectrometers. The dash-dotted line denotes the input wavelength.

##  2 Results

###  2.1 Experimental setup

Ultrashort pulses (Light Conversion Cronus-3P) were launched into a standard
silica SI MMF with a length of 30 cm by weakly focusing (see Methods). To
explore the fiber shaper’s capability in both the normal and anomalous
dispersion regimes, we initiated the experiments with a pump wavelength of
1300 nm (46 fs). Due to its proximity to the zero-dispersion wavelength of
silica, a wide array of nonlinear effects can be effectively excited,
including self-phase modulation (SPM), cross-phase modulation (XPM), four-wave
mixing (FWM), multimode soliton formation, and dispersive wave emission
wu2023highly ; eftekhar2021general ; wright2015spatiotemporal ;
zitelli2022multimode . The fiber was mounted on a 3D-printed programmable
fiber shaper that introduces multi-point macro-bending to the MMF through five
individually controlled motorized actuators (Fig. 2). Each actuator locally
and precisely applies controlled bending on the fiber at multiple axially
dispersed positions. By producing local index perturbations, the actuators
cause energy coupling between modes snyder1972coupled ; marcuse1973coupled  at
multiple time points of the pulse evolution. When operated together, these
actuators exert the high-dimensional spatiotemporal control of the multimodal
nonlinear pulse propagation, thus creating opportunities for simultaneous
customization of the spectral, spatial, and temporal profiles of the fiber
output pulses. Besides the ease of construction and operation, the fiber
shaper maintains high light throughput with negligible transmission loss
regardless of the actuator configurations, measured at $\pm$1% at high power
levels (input pulse energy of 800 nJ, 85% coupling efficiency at initial
states).

To understand the multimodal compositions of the spatiotemporal nonlinear
effects in the SI MMF, the fiber output was spatially profiled with multiple
spectral bands of interest (see Fig. 7). The wavelength-dependent spatial
profiles of the fiber output indicate that the lower-order modes are
responsible for the formation of the most red-shifted multimode soliton
(1900$\pm$100 nm; Fig. 7b), whereas the generation of the dispersive waves
(560$\pm$5 nm, 725$\pm$25 nm; Fig. 7b) is dominated by the higher-order
radially symmetric modes, as predicted in eftekhar2021general , to satisfy the
intermodal phase-matching condition between one of the modes comprising the
dispersive waves and another lower-order mode of the multimode soliton. The
region in between (i.e., SPM, XPM, and FWM) (900$\pm$25 nm, 1200$\pm$5 nm;
Fig. 7b) exhibits a high degree of multimodal behavior, which could be
attributed to the many degrees of freedom required to satisfy the intermodal
group velocity matching and intermodal phase matching xiao2014theory ;
nazemosadat2016phase ; bendahmane2018seeded . These mechanistic studies and
observations not only shed light on the underlying mechanisms of the two-
octave spectral broadening, but also highlight the multimodal compositions of
the spectral broadening in different dispersion regimes, which lays the
foundation for the broadband tunability discussed in the following sections.

###  2.2 Effective spatiotemporal control of multimodal nonlinear effects in
SI MMF

Figure 2: Schematic of the experimental setup. The ultrabroadband light out of
the SI MMF is collimated by an off-axis parabolic mirror before directing to
the characterization apparatus and imaging system (see Methods for details).
OPA: optical parametric amplifier; M: mirror; FM: flip-mirror; BPF: bandpass
filter; BS: beam splitter; Spect: spectrometer; PMT: photomultiplier tube.

To achieve flexible tunability through spatiotemporal control of the
multimodal nonlinear effects, we designed a fiber shaper consisting of five
3D-printed linear actuators (Fig. 3a) at a total cost of 35 US dollars
including the motors and its actuators (see Methods, Section 9.2, and Figs.
10–12). The actuators can independently translate in the direction
perpendicular to the optical axis. By controlling the relative shift between
the actuators, macro-bending of various radii can be applied to the fiber
precisely and simultaneously, allowing for an exhaustive automated search and
adaptive optimization of thousands of fiber shape configurations to achieve
optimal output spectral-temporal-spatial properties. This can be readily
scaled up with more actuators to unlock the full high-dimensional
spatiotemporal control of the multimodal nonlinear optics (see Fig. 15). In
principle, these actuators alter the local refractive index profile, causing
energy coupling between modes at multiple time points during the pulse
evolution, which altogether enable the high-dimensional spatiotemporal control
of the multimodal nonlinear pulse propagation. This process can be modeled by
supplementing the generalized multimode nonlinear Schrödinger equation
(GMMNLSE) (see Methods) on the right-hand side with an additional term
representing the linear mode coupling effect mumtaz2012nonlinear ;
xiao2014theory ; wright2017multimode :

$i\sum_{n}^{N}Q_{np}{A_{n}(z,t)},$ (1)

where $Q_{np}$ denotes the linear coupling coefficient between mode-$n$ and
mode-$p$ due to a local perturbation, determined by the spatial overlap of the
three quantities (see Methods); $A_{n}(z,t)$ represents the temporal envelope
of mode-$n$, which is complex-valued to include the envelope phase due to
distinct phase velocity of each mode wright2017multimode ; horak2012multimode
; $z$ is the propagation direction. The modes can be simplified as the ideal
modes of the unperturbed waveguide using perturbative coupled-mode theory in
principle yariv1973coupled . However, in practice, the mode field can undergo
considerable deformation in curved multimode fibers, making it necessary to
consider additional variations related to mode field deformation. These
variations include changes in the propagation constant and in the nonlinear
coupling coefficient $S_{plmn}^{K}$ and $S_{plmn}^{R}$ (see Methods), which
are determined by the spatial overlap between the mode-$p$, -$l$, -$m$, and
-$n$. As a result, by introducing localized changes to $Q_{np}$ and to the
mode fields at axially dispersed positions along the fiber, the spatial and
temporal degrees of freedom of nonlinear pulse propagation can be
simultaneously controlled.

The perturbed GMMNLSE, along with the observed evolution of the spectral
broadening and spectrally distinct spatial profiles (Figs. 7 and 9), provide a
key insight into the condition of effective spatiotemporal control of
multimodal nonlinear optical processes in multimode fibers. At individual
temporal instances, the more effectively we can modulate the linear and
nonlinear mode coupling coefficients and the mode propagation constants, the
higher-dimensional control and the greater tunability the source will exhibit.
A lot of previous studies on multimodal nonlinear pulse propagation focused on
GRIN MMFs due to their unique self-imaging properties and low modal
dispersion. However, we observed that SI MMFs exhibit significantly greater
tunability and higher spectral brightness under the same experimental
conditions (Fig. 3b,g). This is likely due to their larger modal areas and the
more closely spaced propagation constants, which collectively lead to lower
peak intensity, higher sensitivity to bending, and increased susceptibility to
the mode coupling effect.

Figure 3: Mechanisms of spatiotemporal control of nonlinear effects in SI MMFs
using fiber shaper. a, Photograph of the custom-designed fiber shaper at its
initial state. b,g, Experimental results of 30-cm-long SI MMF (50/125 µm, 0.22
NA) (b) and GRIN MMF (50/125 µm, 0.2 NA) (g) with the same set of macro-
bending applied. Representative spectra corresponding to three randomly chosen
configurations out of a pool of 3125 configurations are highlighted in
distinct colors. The input pulse energy of 500 nJ is experimentally limited by
the laser-induced damage to the GRIN MMF. c,h, Normalized electric field of
the first six spatial modes in unperturbed SI MMF (c) and GRIN MMF (h), with
the core-cladding interface marked in black. d,i, Illustrative examples of the
refractive index profiles of straight ($n_{0}$) and curved ($n$) SI MMF (d)
and GRIN MMF (i). $n$ is approximately expressed by $n_{0}+n_{0}y/r_{b}$,
where $r_{b}$ represents the bend radius, which is set to 1 cm (the minimum
value the fiber shaper can introduce by design). e,j, Linear mode coupling
coefficient $Q_{np}$ resulting from the macro-bending in c,h for the first six
spatial modes in b,g in SI MMF (e) and GRIN MMF (j). f, Effective refractive
index ($n_{\text{eff}}$) for the first six spatial modes in SI and GRIN MMFs.

To gain a mechanistic understanding of this phenomenon, we simulated the
effect of bending for the two fiber types using the perturbative coupled-mode
theory, which assumes that the mode fields are not deformed by the weak
perturbation yariv1973coupled . Here we present an illustrative example
including the first six spatial modes, we show in Fig. 16 the 55-mode case.
Figure 3c–f,h–j displays the simulation results for the SI and GRIN MMFs,
showcasing the linear interactions between the first six spatial modes (Fig.
3c,h) in curved fibers (Fig. 3d,i). The resulting linear coupling coefficient
is presented in Fig. 3e,j. The phase mismatch is reflected by the effective
refractive index $n_{\text{eff}}$ in Fig. 3f. Here we chose 1 cm as the bend
radius in the simulation since it is the minimum value the fiber shaper can
introduce by design; results including three different bending radii are
displayed in Fig. 16. These simulation results show that SI MMF has a greater
$Q_{np}$ and smaller phase mismatch, both of which contribute positively to
the strength of linear coupling (see Fig. 16). This indicates richer
spatiotemporal dynamics can be introduced in SI MMFs with the application of
macro-bending. The significantly smaller $Q_{np}$ exhibited by GRIN MMFs can
be attributed to the more confined mode field which sees weaker local
perturbation at the center region (Fig. 3c,d,h,i) and hence higher resistance
to mechanical deformation such as bending olshansky1975mode ;
caravaca2017single ; flaes2018robustness ; cao2023controlling . Moreover, the
clustering nature of the propagation constants in GRIN MMFs makes the
perturbation-induced mode coupling tend to occur between the nearly degenerate
modes, commonly referred to as the degenerate-mode group
carpenter2012degenerate . Both factors make the multimodal nonlinear dynamics
in GRIN MMFs less sensitive to fiber bending compared to that in SI MMFs.

Apart from the tunability, we also observed that the GRIN MMFs exhibit a lower
threshold for laser-induced damage. Recent work has studied the effects of
damage and transmission loss induced by multiphoton absorption on GRIN and SI
MMFs ferraro2021femtosecond ; ferraro2022multiphoton . As we gradually
increased the input pulse energy for the GRIN fiber, we observed a decrease in
total output power and a reduction in the output spectral span, suggesting the
occurrence of fiber damage. This lower damage threshold is possibly due to the
parabolic index profile which leads to more severe self-focusing effects
agrawal2019invite , and the Germanium dopant in the GRIN fiber core
lancry2011dependence . These analyses and observations provide insights into
the mechanism of fiber-shaper-based spatiotemporal control and entertain the
choice of SI MMFs for a broadband high-peak-power tunable fiber source. Our
observation, although unexpected, is consistent with the recent studies that
show GRIN MMFs are not as sensitive to launching conditions or bending as SI
MMFs zitelli2022multimode ; caravaca2017single ; flaes2018robustness ;
cao2023controlling . Despite it being a drawback to applications that demand
insensitivity to bending, the usually undesirable susceptibility to bending of
SI MMFs, together with their power scalability, are essential for achieving
effective spatiotemporal control of high-power pulse propagation in multimode
fibers.

###  2.3 Enhanced tunability: customizing the fiber output in spectral,
temporal, and spatial domains

We next evaluate the effectiveness of using fiber shaper to control nonlinear
effects in SI MMFs across different dispersion regimes. This was done by
examining, in distinct spectral regions, the tunability in spectral band
energy (Fig. 4a,b), temporal duration (Fig. 4c), and spatial intensity
profiles (Fig. 4c) of the output light field while shaping the fiber in real-
time. These investigations were carried out under the same launching
conditions (800 nJ, 46 fs, and 1300 nm) using the characterization apparatus
depicted in Fig. 2. Figure 4a shows the spectral tunability factor across a
continuous two-octave bandwidth, by exhausting the combination of actuator
positions on the fiber shaper. The spectral tunability factor is defined as
the ratio of the maximally enhanced intensity to the maximally suppressed
intensity for each 20-nm-wide spectral band. We observed that the tunability
varies substantially across the spectrum, with the minimum enhancement ratio
of 3.1-fold shown in the middle of the spectral span, and significantly higher
enhancement ratios ranging from 9-fold to as much as 166-fold at the two ends
of the spectrum (below 800 nm and above 1500 nm). Such wavelength dependence
is very likely to be the product of distinct broadening mechanisms in
different dispersion regimes. To look into the versatility and effectiveness
of the fiber shaper in manipulating various nonlinear effects, we next show in
Fig. 4b,c representative examples in different spectral regions, each
featuring distinct dominant broadening mechanisms (see Fig. 7a). The highest
spectral tunability lies in the dispersive wave regime (500–900 nm; Fig. 4b)
and the soliton regime (1700–2300 nm; Fig. 4b), the 9-fold and 166-fold marked
in Fig. 4a are shown as instances in Fig. 4b. A recent study observed notable
differences in the spectra for multimode solitons with identical energy but
different initial mode components sun2022multimode , which could provide
insights into the high spectral tunability within the soliton regime.
Simultaneous dual-band spectral tuning, with an averaged enhancement ratio of
2.3-fold for the two bands, is exemplified for the nonlinear phase modulation
regime (1000–1600 nm; Fig. 4b).

Figure 4: Performance of fiber shaper: customization in spectral, temporal,
and spatial domains. a, Ultrabroadband spectral tunability, evaluated from a
set of spectra acquired from 3125 states of the fiber shaper. b,c,
Representative examples of modulations in spectral band energy (b) of selected
single bands and dual-band (denoted by vertical dotted lines), showing the
most enhanced (Enhancement) and suppressed (Suppression) cases; and in pulse
duration and the associated spatial intensity profiles (c) for selected
spectral bands, with the center wavelength and bandwidth denoted above each
panel, and the FWHM of the autocorrelation signals annotated near each trace.
For visualization purposes, each intensity profile is normalized to its own
maximum value.

Multimode solitons and their phase-matched dispersive waves consist of
multiple mode combinations, within each combination the modes are group-
velocity matched wu2023highly ; eftekhar2021general ; wright2015spatiotemporal
; zitelli2022multimode . Mechanical perturbations such as a series of macro-
bending can cause major changes in the few-mode composition and thus
significantly alter spectral profiles in order to re-match the group velocity
via mode-dependent spectral shifting. This is also manifested by dramatic
changes of the few-mode-like spatial intensity profiles of multimode solitons
(1700$\pm$42.5 nm and 1900$\pm$100 nm; Fig. 4c) and dispersive waves
(750$\pm$25 nm; Fig. 4c) across seven random states of the fiber shaper. These
changes reflect the capability of the fiber shaper to significantly alter the
mode composition, and thus the output light field in the spectral and spatial
domains. The temporal duration at the fiber output, on the other hand, shows
limited tunability as solitons and dispersive waves are intrinsically nearly
transform-limited pulses. In contrast, the regime dominated by SPM, XPM, and
FWM are known for highly nonlinear, strongly coupled, and highly multimodal
behavior. The resulting spectrum is the incoherent summation of many modes,
which makes spectral shape less sensitive to modal distribution changes and
reduces the effectiveness of tuning the spectral intensity. The speckled
spatial intensity profile (1300$\pm$25; Fig. 4c) corroborates the highly
multimode nature of the nonlinear phase modulation process, in which the many-
mode composition is needed to satisfy the intermodal phase-matching and
velocity-matching conditions. Interestingly, the control in the temporal
domain is much more significant in the nonlinear phase modulation regime than
other regimes, ranging from 1523 to 352 fs based on the full-width at half-
maximum (FWHM) of the autocorrelation signals, from 1080 to 250 fs assuming a
Gaussian pulse shape for simplicity. This observation can be attributed to the
effective reduction of modal dispersion in the multimodal output through
optimization of fiber shaper-induced linear mode coupling.

Our results demonstrate that the fiber shaper effectively manipulates the
modal compositions of the propagating pulse and the resulting spectral-
temporal-spatial properties of the output light field, exceeding what has been
shown before. The intricate nonlinear interactions emphasize the benefits of
using a fiber shaper to control the spatiotemporal dimension of the nonlinear
dynamics in order to achieve a targeted output in different dispersion
regimes. Earlier studies have successfully demonstrated broadband
supercontinuum generation using silica MMFs in longer pulse duration regimes
krupa2016spatiotemporal ; perret2019supercontinuum ; eslami2022two ;
zhou2015half ; lopez2016visible ; eftekhar2017versatile . In this work, we
demonstrate a two-octave tunable source in the femtosecond regime, which
enables applications in multiphoton microscopy due to its high-peak-power
pulses.

###  2.4 Ultrabroadband high-peak-power tunable source for multiphoton
microscopy

The results above demonstrate the potential of the fiber-shaper-controlled SI
MMF for achieving an ultrabroadband high-peak-power tunable source, which is
capable of modulating the output field in the spectral, temporal, and spatial
domains. This capacity can facilitate diverse applications in spectroscopy,
sensing, and imaging applications. As a proof-of-concept demonstration of such
potential, we directly applied the proposed source for multiphoton microscopy
(MPM) (see Methods and Fig. 17) to investigate whether it can be adaptively
optimized for nonlinear imaging which demands sources with high spectral
brightness (5–50 nJ) yildirim2019functional ; you2018intravital , short
temporal duration (10–500 fs) hoover2013advances ; huland2013three , and
confined spatial profiles dong2003characterizing .

We first characterized the performance of the proposed source with fluorescent
beads for two-photon fluorescence (2PF) and three-photon fluorescence (3PF)
imaging. For comparison, we acquired reference images with a commercial laser
source based on an optical parametric amplifier (OPA) from Light Conversion
Cronus-3P. The results are presented in Fig. 5a–h, and the experimental setup
which includes the optical path for imaging and the characterization of
spectral, temporal, and spatial profiles of the fiber source, is depicted in
Fig. 2. The raw output of the MMF was directed to a scanning MPM system after
passing through selected bandpass filters (750$\pm$25 nm for 2PF, 1300$\pm$25
nm and 1225$\pm$25 nm for 3PF). This setup can be readily improved in future
studies with advanced wavelength selection mechanisms for more systematic and
automatic multiband imaging. We observed that using the initial state of the
fiber shaper, i.e., an unoptimized form, resulted in poor image quality (e.g.
low multiphoton signals), which can be largely attributed to insufficient peak
power due to the thinning of the energy distribution that comes with spectral
broadening and/or temporal broadening caused by modal and chromatic
dispersion.

Figure 5: Multiphoton microscopy with the fiber-shaper-controlled MMF source.
a–h, Demonstration of 2PF and 3PF imaging on fluorescent beads using OPA
(a,d), initial fiber source (b,e,g), and optimized fiber source (c,f,h). The
contrasts of the insets are adjusted for visibility. Scale bars: 20 µm. i,
Pseudo-color presentation of label-free imaging of fixed tissue from the mouse
whisker pad at 1225 nm excitation using the initial and optimized fiber
source, showing THG (magenta) signals of adipocytes, 3PAF (green) signals of
muscles, and SHG (yellow) signals of collagen fibers. The contrasts of the
insets are adjusted for visibility. Scale bars: 200 µm. j–r, Characterization
of the spectral (j,m,p), temporal (k,n,q), and spatial (l,o,r) properties of
the output beam of the initial and optimized fiber source corresponding to the
images in a–h. The 50-nm spectral band energy and the FWHM of the
autocorrelation (AC) signals are indicated in the figures. The spatial
profiles are individually normalized to their maximum intensity. s, Validation
of the signal improvement from the images compared to the theoretical values
calculated from pulse measurement.

Remarkably, by adaptively and coarsely optimizing the fiber shaper using
feedback from the multiphoton signals, i.e. greedy search in our experiment,
enhancements of 15-fold, 11-fold, and 6-fold in signals were achieved for 2PF
(750 nm), 3PF (1300 nm), and 3PF (1225 nm) microscopy, respectively (Fig.
5c,f,h). To investigate the mechanisms of the improvement, we looked into the
spectral, temporal, and spatial characteristics of the output pulses before
and after optimization (Fig. 5j–r), and calculated the multiphoton signal
generation efficiency according to the method described in xu2002multiphoton
(Fig. 5s). To highlight the role of spectral and temporal tuning and for
simplicity, the high-order dispersions and multimodal spatial compositions are
neglected in the calculation and the pulse shape and spatial profile of the
fiber source are assumed to be the same as those of the laser (see Methods and
Section 9.3). The strong agreement between the measured and the calculated
improvements in signals confirms that pulse energy and pulse duration are the
dominant factors affecting multiphoton signal generation efficiency in our
experiments. Notably, the measured and calculated signal improvements for 2PF
are more closely matched than those for 3PF. This can be attributed to the
fact that the generation efficiency of 2PF is less dependent on the spatial
distribution of the beam compared to 3PF xu2002multiphoton  (see Fig. 19).

Moreover, we observed that the output field optimization for signal
enhancement of 2PF and 3PF varies as a result of different underlying
mechanisms. Specifically, the signal enhancement for 2PF was primarily driven
by a 3.8-fold increase in the spectral intensity at 750 nm, whereas the
enhancement for 3PF signals was primarily due to a 2.7-fold (at 1300 nm) and
2.1-fold (at 1225 nm) reduction in pulse duration, resulting in peak powers of
0.72, 0.12, and 0.15 MW, assuming Gaussian pulse shape. This difference is
consistent with our findings presented in Fig. 4, where the tunability in
spectral and temporal domains exhibits strong spectral dependence due to the
distinct dominant nonlinear effects: higher spectral tunability was observed
in the dispersive regime (750 nm), whereas better temporal tunability was
found in the nonlinear phase modulation regime (1300 and 1225 nm).
Additionally, we observed that the corresponding spatial profile of 3PF showed
a tendency to be less speckled and more confined when the pulse was shortened,
due to a reduced modal dispersion. The reduction in speckling in 3PF is likely
to result in less image blurring compared to 2PF (see Fig. 18).

We next examined the performance of the proposed source for label-free imaging
of freshly excised mouse tissue. We selected the 1225$\pm$25 nm excitation
band, which is not covered by most commercial high-power OPA or low-rep high-
peak-power lasers. After the same procedure of optimization (greedy search
based on the imaging signals), the optimized SI MMF allows the simultaneous
visualization of adipocytes through third-harmonic generation (THG), muscles
through three-photon autofluorescence (3PAF), and collagen fibers through
second-harmonic generation (SHG) (Fig. 5i), which demonstrated the potential
of this source as a widely and continuously tunable source for bioimaging.

###  2.5 Extension to other wavelengths

We have demonstrated above the effectiveness of using fiber shaper for
spatiotemporal control of various nonlinear effects across different spectral
regions, which leads to a high-peak-power source spanning the whole broadened
spectrum. The broadened spectral span, however, exhibits strong wavelength
dependence. For example, our results (see Fig. 13 show that pumping in the
normal dispersion regime (e.g., 800 nm) leads to limited spectral broadening
and smaller spectral tunability, while pumping deeper into the anomalous
dispersion regime (e.g., 1550 nm) results in a broad but discontinuous
spectrum. To obtain an efficient and wide continuum for sufficient broadband
spectral brightness, one often needs to pump the waveguide in the anomalous
dispersion regime close to its zero-dispersion wavelength (ZDW). For fused
silica, the ZDW is close to 1280 nm. Although the main results of this study
were generated using a pump wavelength of 1300 nm, the fiber shaper can be
easily extended to other wavelengths - with comparable broadening and
tunability - by pumping a multimode waveguide with ZDW close to the pump
wavelength.

Here, to demonstrate the extension of the fiber shaper to other more
accessible wavelengths such as 1040 nm, we leverage the blue-shifted ZDWs of
the higher-order fiber modes to match the pump wavelength (see Fig. 14. The
results are presented in Fig. 6. The upper panel shows the spectral broadening
with matched ZDW by exciting the LP${}_{\text{07}}$ mode using a spatial light
modulator (SLM) demas2015free , resulting in a spectrum spanning from 550 to
1650 nm covering 1.6 optical octaves, which is comparable to the 2-octave
spectral span we obtained in the previous sections. Inset shows the spectral
tunability between 900–1700 nm, exhibiting great capability for spectral
energy reallocation. In comparison, the reference spectrum obtained by weakly
focusing without matching the pump and zero-dispersion wavelengths is
displayed in the lower panel, limited spectral broadening was observed. Other
methods to shift the ZDW includes dispersion engineering and composing the
waveguide with non-silica materials ramsay2013generation ; petersen2014mid ;
kubat2016multimode ; eslami2022two . Through these techniques, more
wavelengths can be potentially accessed with optimized spectral span and
tunability.

Figure 6: Extension of fiber shaper to an input wavelength of 1040 nm. 220-fs
pulses were launched into the same standard silica SI MMF (50/125 µm, 0.22 NA)
with a length of 30 cm. Normalized intensity patterns in gray scale represent
the near-fields of the fiber output at low-power levels, indicating
LP${}_{\text{07}}$ and a mixture of lower-order modes were excited in the two
cases, respectively. Output energy was fixed to 600 nJ. Inset shows the
spectral tunability between 900 nm and 1700 nm, obtained with 1000 different
fiber shape configurations. Vertical solid lines mark the spectral range
measured by different spectrometers. The dash-dotted line denotes the input
wavelength.

##  3 Discussion

In summary, we have presented a new way of controlling nonlinear effects by
leveraging both the spatial and temporal degrees of freedom of the multimodal
pulse propagation through a programmable fiber shaper. This method unlocks
access to an even higher-dimensional space of spatiotemporal dynamics and two-
octave-wide high peak power in an off-the-shelf SI MMF. The major
contributions of this work include (1) opening up novel perspectives for
spatiotemporal control of nonlinear multimode pulse propagation, (2) proposing
an avenue for power scaling and field control of broadband sources from
visible to NIR-IR regime, and (3) providing accessible control designs that
can be rapidly adopted for both nonlinear and linear modulation of multimode
fibers.

Our proposed method directly modulates the modal interactions during the pulse
temporal evolution using a slip-on 3D-printed fiber shaper, without additional
non-fiber modulation layers such as free-space SLMs for wavefront shaping in
spatial and temporal domains. The all-fiber route to single-stage nonlinear
conversion inherently contributes to strong alignment robustness, great long-
term stability (see Fig. 8), and high spectral band energy. The fiber shaper
excels in modulating the multimodal pulse propagation, which SLM cannot access
easily. However, it is worth noting the SLM excels in modulating the input,
which the fiber shaper cannot access easily (e.g., the broadening shown in
Fig. 6). Because of the distinction of the modulation domains, these two
approaches are complementary and can be implemented together to maximize
control degrees of freedom.

Compared to the commonly used GRIN MMFs, the higher damage threshold and the
significantly higher tunability of SI MMFs make it an essential part of the
unprecedented spatiotemporal control of the high-power multimodal nonlinear
effects. As a result, the proposed apparatus leads to a fiber source with
great tunability in spectral (up to 166-fold reallocation), temporal (up to
4-fold shortening), and spatial domains. Furthermore, the fiber shaper enables
combined spectral and temporal tuning, leading to high peak power levels
across two-octave spectral bands. For instance, at 750$\pm$25 nm, the peak
power reached 0.72 MW and 0.15 MW at 1225$\pm$25 nm, surpassing the expected
regime of operation, which is usually the soliton regime for its inherently
high spectral density and short pulse duration.

These performances (1) overcome the bandwidth limitation of existing high-
peak-power fiber sources based on soliton formation wang2011tunable ;
wright2015controllable ; zach2019all ; rishoj2019soliton ; rishoj2020jitter
and (2) demonstrate orders of magnitude higher peak power compared to existing
tunable fiber-based broadband sources lefort2017review . These properties
could benefit the emerging but technologically demanding applications in
optical sensing, imaging, manipulation, and computing, that require light
sources with broad spectral coverage, great tunability, ultrafast pulses
(femtosecond-level pulse duration), stability, and/or high spectral density of
energy and peak power.

##  4 Methods

###  4.1 Experimental setups and measurements

Pulses with an energy up to 800 nJ and duration of 40–60 fs were launched into
30-cm-long MMFs that were mounted on a three-axis translation stage, with the
slip-on fiber shaper placed closely thereafter. This length of the fiber
allows for effective spectral tuning without significantly dispersing pulses
due to an unnecessarily long fiber. A 6-cm doublet was used to form a 16-µm
($1/\text{e}^{2}$ radius) focal spot on the input facet of the MMFs, resulting
in a coupling efficiency of 85%. Two types of MMFs of the same core sizes were
tested, namely the silica-core SI MMF (Thorlabs FG050LGA) and GRIN MMF
(Corning OM4). The pulses out of the MMFs were split twice to couple into
three different grating spectrometers after passing individual optical
diffusers. The diffusers were used to reduce the spatial variance of the
spectra out of MMFs. Each of these spectrometers covers a specific wavelength
range: 192–1020 nm (Thorlabs CCS200), 899–1702 nm (Ocean Insight
NIRQuest+1.7), and 858–2573 nm (Ocean Optics NIR256-2.5). To combine the three
spectra obtained from the individual spectrometers, we selected 950 nm and
1600 nm as the reference points for “stitching”. For each spectral region
separated by these reference points, we first subtracted the background noises
measured when the optical signals were blocked, then multiplied a scaling
factor to the spectral traces to align their overall heights with those of the
neighboring region at the stitching point. The pulse duration was measured
using an intensity autocorrelator (APE PulseScope) based on the second-
harmonic generation detection. The near-field beam profiles were acquired by
two cameras. A CMOS-based (Mako G-040B) was used for the visible and near-
infrared regions, and a thermal imaging camera based on silicon microbolometer
(DataRay WinCamD-FIR2-16-HR) was for longer wavelengths above 1700 nm. The
optical powers were measured using a thermal power sensor (Thorlabs S425C-L).
The fiber output is collimated by a parabolic mirror of 25.4-mm focal length
(Edmund Optics 36-586), and then expanded by a pair of 4-f relay systems to
slightly overfill the objective’s pupil size (15.12 mm). A dichroic mirror
(Thorlabs DMLP650L) is placed before the objective lens, which separates the
excitation signals from the emission signals in the detection channels.

###  4.2 Simulations

In all simulations, we assumed a single linear polarization for
simplification. The simulated spectra in Fig. 7b were acquired using the
GMMNLSE horak2012multimode :

$\frac{\partial A_{p}}{\partial z}=\mathcal{D}\\{A_{p}\\}+\\\
i\frac{n_{2}\omega_{0}}{c}\left(1+\frac{i}{\omega_{0}}\frac{\partial}{\partial
t}\right)\sum_{l,m,n}\Big{\\{}(1-f_{R})S_{plmn}^{K}A_{l}A_{m}A_{n}^{*}+f_{R}S_{plmn}^{R}A_{l}\big{[}h*(A_{m}A_{n}^{*})\big{]}\Big{\\}}$
(2)

implemented with a numerical solver wright2017multimode  in MATLAB. In the
equation above, $A_{p}$ is the abbreviation of $A_{p}(z,t)$ representing the
temporal envelope of the spatial mode-$p$; $\mathcal{D}$ denotes the linear
propagation including dispersion effects. $S_{plmn}^{K}$ and $S_{plmn}^{R}$
are the mode overlap factors responsible to the instantaneous Kerr effect and
Raman effect, determined by the spatial overlap of the four spatial modes
involved. We set the nonlinear index $n_{2}=2.3\times 10^{-20}$
$\text{m}^{2}/\text{W}$ and the Raman contribution $f_{R}=0.18$; we included
the self-steepening effect and up to the fourth-order linear dispersion
effects. The mode and dispersion parameters of the SI MMF were calculated
based on its pure silica core and its NA of 0.22.

The linear coupling coefficient $Q_{np}$ in equation (1) and Fig. 3e,j was
calculated as

$Q_{np}=\frac{k_{0}}{2n_{\text{eff}}}\iint\varepsilon_{0}\big{[}n^{2}(x,y)-n_{0}^{2}(x,y)\big{]}F_{n}(x,y)F_{p}^{*}(x,y),$
(3)

where the scalar function $F(x,y)$ represents the transverse fiber mode
profile (see Fig. 3c,h). The mode parameters of the GRIN MMF were calculated
based on a standard telecommunication-grade parabolic GRIN fiber used in
wright2017multimode .

###  4.3 Fiber shaper design

To generate precisely controlled macro-bending to the fiber, we designed a
device called fiber shaper. This device was fabricated using 3D printing
(Stratasys Fortus 380mc) with acrylonitrile styrene acrylate material. The
fiber shaper features a rectangular base with five slots that can hold the
translating units (referred to as actuators in Fig. 1a) and constrain their
linear motion in the desired direction. Each translating unit is powered by
its own stepper motor (ELEGOO 28BYJ-48) through a 3D-printed rack and pinion
system, allowing for individual control over their linear motion through a
microcontroller (ELEGOO Mega R3) that communicates with a computer. To mount
the fiber onto each translating unit, it is passed through a 0.5-mm-wide gap
created by two disks and secured in place with a square cap. This design
minimizes tension on the fiber during bending, and two half-disks with
securing caps at the entrance and exit of the device ensure optimal
functionality. The radius of the disk, which determines the minimum bend
radius the fiber shaper can apply, is designed to be 10 mm to minimize
bending-induced transmission loss. Different device parameters were tested to
optimize the spectral tunability of the fiber shaper (see Section 9.2 and
Figs. 10–12). As a result, the rich combinations of actuator translation
displacements create a vast array of fiber shape configurations, allowing for
effective utilization of the spatial and temporal degrees of freedom in
controlling nonlinear pulse propagation in SI MMFs with high light throughput.

###  4.4 Multiphoton microscopy

The multiphoton microscopy images were acquired using a custom-built inverted
scanning microscope. The microscope used a pair of galvanometer mirrors
(ScannerMAX Saturn-5 Galvo and Saturn-9 Galvo) to scan the beam. The beam was
then focused by a water immersion objective (Olympus XLPLN25XWMP2, 1.05 NA).
The emitted photons were collected using a photomultiplier (Thorlabs PMT2101).
During imaging, the fiber source was pumped by 800 nJ 46 fs pulses at 1300 nm
and 1 MHz repetition rate. The repetition rate of the OPA (Light Conversion
Cronus-3P) was also fixed to 1 MHz during imaging. Fluorescent carboxyl
polystyrene (Bangs Laboratories Inc FCDG009) was used as the imaging sample
for characterization, a proper emission filter (Edmund Optics 530$\pm$22.5 nm)
was chosen to match its fluorescence emission spectrum. Fixed tissue from the
mouse whisker pad was used in label-free imaging, where emission filters
(Semrock 609$\pm$28.5 nm, Edmund Optics 530$\pm$22.5 nm, Thorlabs 405$\pm$5
nm) were chosen for collecting the SHG, 3PAF, and THG signals, respectively.
Raw images were loaded onto FIJI (National Institutes of Health) to apply
pseudo-color maps. The bead sample images with 750 nm, 1225 nm, and 1300 nm
excitation were respectively acquired with pixel dwelling times of 6 µs, 6 µs,
and 10 µs, all with 150-µm field of view. The tissue sample images were
acquired with a 30-µs pixel dwelling time and a 900-µm field of view, and
4$\times$4 images were stitched to represent the mosaicked image. The greedy
search method was implemented with the imaging signals as feedback.
Specifically, the displacement of each actuator on the fiber shaper was
sequentially chosen with the imaging signals maximized.

To calculate the signals of the bead sample images, we selected a few random
regions of interest (ROI), each of which contained a spherical bead. The
signals from measurement were calculated as the sum of pixel values within the
ROIs. The theoretical value of the signals was determined by the multiphoton
fluorescence generation efficiency based solely on pulse energy and duration
as

$S_{n}=C_{n}\frac{E_{p}^{n}}{\tau_{p}^{n-1}}\quad(n=2,3),$ (4)

where $S_{n}$ refers to the $n$-photon generation efficiency, $C_{n}$ is the
corresponding constant coefficient, $E_{p}$ and $\tau_{p}$ refer to the pulse
energy and duration, respectively (see Section 9.3).

##  5 Data availability

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

##  6 Code availability

The codes that support the findings of this study are available from the
corresponding author upon reasonable request and through collaborative
investigations.

##  7 Acknowledgement

The work has been supported by MIT startup funds. H.C. acknowledges support
from the MIT Kailath Fellowship. K.L. acknowledges support from the MIT Jacobs
Presidential. We thank Peter So for sharing with us their lab space and lasers
during lab renovation, James Fujimoto for loaning the optical spectrum
analyzer, and Phillip Keathley for loaning the autocorrelator and thermal
camera. We would like to express our sincere gratitude to them and their
students Matthew Yeung, and Lu-Ting Chou for valuable insights on nonlinear
optics. We also extend our appreciation to Kristina Monakhova from our lab for
providing constructive comments on the manuscript that greatly improved its
clarity and quality. Our appreciation also goes to Miaomiao Jin from McGovern
Institute for helping provide the tissue samples for our imaging experiments.

##  8 Author contributions

T.Q., H.C., and S.Y. conceived the idea of the project. S.Y. supervised the
research and obtained the funding. T.Q., H.C., and K.L. built the optical
setup and performed the experiments and simulations. M.L., E.L., and F.W.
provided and prepared the tissue samples. T.Q, H.C., K.L., L.Y., and S.Y.
wrote the manuscript with the input from all authors.

##  9 Supplementary

###  9.1 Supplementary Note 1. Multimodal nonlinear pulse propagation and
spectral broadening in the SI MMF

To gain insights into the broadening and the tuning mechanisms, we first
investigated the evolution of the output spectra with gradually increasing
input pulse energy by using the characterization setup as depicted in Fig. 2.
The fiber shaper was set to its initial state with no bending applied to the
fiber. The results are presented in Fig. 7a. As the injected pulse energy
gradually increases, SPM, XPM, and intermodal FWM begin to dominate the
spectral broadening process. As the input energy increases further, the output
spectrum extends to the anomalous dispersion regime and multimode solitons
(i.e., solitary waves consisting of multiple spatial modes) begin to form, due
to the balancing between the nonlinear phase modulation and dispersive effects
including the intramodal group velocity dispersion and intermodal velocity
mismatch wright2015spatiotemporal . Meanwhile, excess energy from the soliton
fission process is emitted as dispersive waves through the interaction between
the nonlinear and higher-order dispersion effects. The dispersive waves are
generated at shorter wavelengths to satisfy the phase-matching condition
eftekhar2021general . With further increase in input energy, the multimode
soliton undergoes a redshift, resembling soliton self-frequency shifting
renninger2013optical , and eventually the spectrum extends beyond the near-
infrared-II region. Accordingly, the phase-matched dispersive waves shift
towards the visible region. These observations are in agreement with the
findings reported in eftekhar2021general ; zitelli2022multimode ; wu2023highly
with input wavelengths deeper into the anomalous dispersion regime.

In addition, the fiber output power at 800-nJ input pulse energy was monitored
for 1 hour to assess its stability (Fig. 8), which was measured to reflect
1–2% fluctuations at different wavelengths. Despite exceeding the critical
power for catastrophic self-focusing (about 8 MW for silica at 1300 nm
dubietis2017ultrafast ), the pronounced dispersion effects associated with
ultrashort pulses can effectively modify the dynamics of self-focusing,
thereby preventing the exponential increase in peak intensity
gaeta2000catastrophic . In this study, the fiber source was stable enough to
generate reproducible output modulation and biological images (see later
sections). Nevertheless, the fluctuation metric could potentially be improved
by using a better power meter (low dynamic range was used for the wavelength
range above 1100 nm in this study), a more environmentally controlled optical
space (temperature and humidity), and a more isolated fiber configuration (gel
or air isolation).

Figure 7: Multimodal spectral broadening in SI MMF. a, Spectral evolution as a
function of increasing input pulse energy in a 30-cm-long SI MMF, showing the
emergence of three spectral windows dominated by distinct nonlinear effects.
b, Output spectra at selected input pulse energies. The top row presents the
spectrally filtered near-field beam profiles at an input energy of 800 nJ,
numbers above each profile denote the center wavelength and bandwidth of the
bandpass filter applied. Numerical simulations are presented in dotted gray,
with the seven modes assumed to be excited with equal energy and phase. For
visualization purposes, the spectrum and beam profile at each input pulse
energy are normalized to their respective peak values at that energy.

To validate our experimental results, we performed numerical simulations (Fig.
7b) using the simplified (1+1)D model based on the generalized multimode
nonlinear Schrödinger equation (GMMNLSE) horak2012multimode ;
wright2017multimode  (see Methods) with the same fiber parameters and
launching conditions as in the experiments. Only the first seven radially
symmetric modes are included to save computation time. Overall, we observed a
good agreement in the spectral broadening between the experimental and
numerical results at all energy levels. The discrepancy in the distribution of
power spectral density is likely due to the limited number of modes in the
simulation, which may not fully capture the multimodal nonlinear broadening in
experiments.

Figure 8: Stability of the MMF source. a, Spectral stability of the SI MMF
source obtained from the output spectra recorded per second for a period of 1
hour. The solid black trace represents the mean value, and the surrounding
gray region indicates the level of variation (quantified by the standard
deviation) of the recorded spectra. b, Normalized output average power of the
SI MMF source recorded per second for a period of 1 hour. A silicon-based
photodiode sensor (Thorlabs PM16-130) was used to measure the optical power at
wavelength below 1100 nm and a thermal power sensor (Thorlabs S425C-L) was
used to measure the optical power of longer wavelengths and the whole band.
The SI MMF was pumped by 800-nJ 46-fs pulses at 1300 nm. RSD: relative
standard deviation.  Figure 9: Multimodal spectral broadening in SI MMF. a–c,
Representative results of near-field beam profiles out of a 30-cm-long SI MMF
at different input pulse energies (1300 nm input wavelength) acquired using a
CMOS camera (Mako, G-040B) with (a) and without (b) a bandpass filter; and a
thermal imaging camera based on silicon (Si) microbolometer (DataRay, WinCamD-
FIR2-16-HR) with no bandpass filter applied (c).

###  9.2 Supplementary Note 2. Optimization of the fiber shaper device

The fiber shaper enables spatiotemporal control of the nonlinear effects in SI
MMFs, underlying great tunability in the fiber source properties to adapt to
diverse applications. To maximize this tunability, we investigate several key
parameters involved in the design of the fiber shaper. These parameters
include the number of actuators, as well as the total range and resolution of
the actuator’s linear motion. The impact of these parameters on the 900–1700
nm band are shown in Figs. 10–12, where we define an average spectral tuning
ratio ($\eta$) to quantify the tunability:

$\eta=\frac{1}{N}\sum_{n=1}^{N}\frac{I^{\rm max}_{\lambda_{n}}}{I^{\rm
min}_{\lambda_{n}}}.$ (5)

$N$ denotes the number of discrete wavelengths acquired by the spectrometers,
$I^{\rm max}_{\lambda_{n}}$ and $I^{\rm min}_{\lambda_{n}}$ refer to the
maximum and minimum intensity at each wavelength. Our results indicate that
the performance is optimized when using 5 actuators, a displacement range of
20 mm, and an actuator motion resolution of 5 mm. Notably, these parameters
are specifically tailored to this device and may require adjustments for
devices with different physical dimensions.

Figure 10: Spectral tunability of the fiber shaper with different numbers of
actuators. a–d, The output spectra of the fiber-shaper-controlled SI MMF with
1 (a), 2 (b), 3 (c), and 5 (d) actuators, respectively. The input wavelength
is set at 1300 nm. Figure 11: Spectral tunability of the fiber shaper with
different displacement ranges. a–d, The output spectra of the fiber-shaper-
controlled MMF with a displacement range of 2.5 mm (a), 5 mm (b), 10 mm (c),
and 20 mm (d), respectively, all with 5 actuators and a motion resolution of 5
mm. The input wavelength is set at 1300 nm. Figure 12: Spectral tunability of
the fiber shaper with different actuator motion resolutions. a–d, The output
spectra of the fiber-shaper-controlled SI MMF with the actuator motion
resolution set at 20 mm (a), 10 mm (b), 7 mm (c), and 5 mm (d), respectively,
all with 5 actuators and a displacement range of 20 mm. The input wavelength
is set at 1300 nm.

###  9.3 Supplementary Note 3. Derivation of the multiphoton generation
efficiency using a fiber source

The derivation follows one of the foundational references of two-photon
fluorescence (2PF) imaging xu1996measurement . The motivation of this
derivation is to provide insights into how the multiphoton signal was
optimized through fiber-shaper-assisted temporal and spectral shaping of the
pulse.

Given the same sample, we consider how the properties of the excitation source
would affect the $n$-photon generation efficiency, which is described by the
time-averaged fluorescence photon fluxes $S_{n}=\langle S_{n}(t)\rangle$.
Since we assume only the excitation source varies, we can express $S_{n}(t)$
solely in terms of the intensity of the excitation $I(\bm{r},t)$, that

$S_{n}(t)\propto\int_{V}\text{d}VI^{n}(\bm{r},t),$ (6)

where $V$ is the illuminated volume. We then express $I(\bm{r},t)$ as
$I(\bm{r},t)=I_{0}(t)F(\bm{r})$ to further separate the temporal and spatial
properties of the source, where $I_{0}(t)$ is the temporal distribution of the
excitation source intensity and $F(\bm{r})$ is the normalized spatial
distribution. Substitute it back to equation (6), we have

$S_{n}\propto g^{(n)}\langle
I_{0}(t)\rangle^{n}\int_{V}\text{d}VF^{n}(\bm{r}),$ (7)

where $g^{(n)}=\langle I_{0}^{n}(t)\rangle/\langle I_{0}(t)\rangle^{n}$ is the
$n$-th order temporal coherence. In this work, the repetition rate (1 MHz) is
the same for both the laser source and the MMF source, we therefore can
further express the temporal term using the pulse energy $E_{p}$, pulse
duration $\tau_{p}$, and $g^{(n)}_{p}$ that only depends on the pulse shape:

$g^{(n)}\propto\frac{g^{(n)}_{p}}{\tau_{p}^{n-1}},\quad\langle
I_{0}(t)\rangle\propto E_{p}.$ (8)

Finally, using the normalized PSF of the source to represent $F(\bm{r})$, we
arrive at

$S_{n}\propto\frac{g_{p}^{(n)}E_{p}^{n}}{\tau_{p}^{n-1}}\int\text{d}V\text{PSF}^{n}(\bm{r}).$
(9)

Equation (9) implies that the properties of a source that would affect the
multiphoton generation efficiency are: the pulse energy $E_{p}$, the pulse
duration $\tau_{p}$, the PSF distribution $\text{PSF}(\bm{r})$, and the factor
$g^{(n)}_{p}$ related to the pulse shape. In the imaging section, we simplify
the model by assuming different sources share the same $g^{(n)}_{p}$ and
$\text{PSF}(\bm{r})$. As a result, the fiber shaper optimization (i.e.,
finding the optimal actuator configuration $s\in\mathcal{S}$) can be
formulated as

$\max_{s\in\mathcal{S}}~{}~{}\frac{E_{p}^{n}(s\mathchar
24635\relax\;\lambda)}{\tau_{p}^{n-1}(s\mathchar 24635\relax\;\lambda)},$ (10)

where $\mathcal{S}$ is the space of all configurations.

Although not accounted for in the formulation, we observed that PSF can also
be more confined with fiber shaper optimization. More advanced optimization
scheme will be useful to further improve the spectral-temporal-spatial
customization of the fiber source to push the fiber shaper to its limit.

Figure 13: Fiber-shaper-controlled SI MMF with other input wavelengths. a–e,
Spectral broadening and tunability of varied input wavelengths of 700 nm (a),
800 nm (b), 900 nm (c), 1250 nm (d), and 1550 nm (e) in the same 30-cm-long
fiber, with fixed input pulse energy of 800 nJ and the same set of macro-
bending applied. Three representative spectra corresponding to three randomly
chosen configurations are highlighted in distinct colors. Vertical solid lines
mark the spectral range measured by different spectrometers. The dash-dotted
line denotes the input wavelength. Figure 14: Shifted zero-dispersion
wavelengths of the higher-order modes.

Figure 15: Spectral tunability with respect to the number of actuators. a, The
relationship between spectral tunability and the number of actuators. The
tunability score is quantified by the mean spectral tuning range across the
entire wavelength spectrum
$\frac{1}{N}\sum_{i}\left[(I_{\lambda_{i}}^{\max}-I_{\lambda_{i}}^{\min})/\bar{I}_{\lambda_{i}}\right]$.
b, All spectral data and the relative intensity variation at each wavelength
$\left(I_{\lambda i}-\bar{I}_{\lambda_{i}}\right)/\bar{I}_{\lambda_{i}}$.

Figure 16: Simulation results of bending-induced mode coupling in step-index
(SI) and gradient-index (GRIN) multimode fibers (MMFs). The first two columns
show the effective refractive index $n_{\text{eff}}$ and the coupling
coefficients $Q_{np}$ for the first 55 spatial modes of the SI and GRIN
MMFs.The right panel shows the linear mode coupling strength
$\lvert\frac{Q_{np}}{\Delta\beta_{np}}\rvert$ ho2013mode  resulting from
macro-bending at various bending radii (1 cm, 10 cm, and 100 cm). Modes in the
GRIN MMF are clustered as the degenerate-mode group, indicated by the solid
white lines.

Figure 17: Schematic of the multiphoton imaging setup. PM: parabolic mirror;
BPF: bandpass filter; DM: dichroic mirror; LP: long pass; OBJ: objective lens;
PMT: photomultiplier tube.

Figure 18: PSF measurement of the 3PF excitation with the laser (reference)
and the 2PF and 3PF excitation with the fiber source. The size of the bead is
0.1 µm. Figure 19: Qualitative PSF comparison of the 2PF excitation (900 nm)
and the 3PF excitation (1300 nm) with the fiber source under random bending
conditions induced by the fiber shaper’s different states.The size of the bead
is 1 µm.

## References

  * (1) Wright, L. G., Renninger, W. H., Christodoulides, D. N. & Wise, F. W.  Nonlinear multimode photonics: nonlinear optics with many degrees of freedom.  _Optica_ 9, 824–841 (2022).
  * (2) Krupa, K. _et al._ Multimode nonlinear fiber optics, a spatiotemporal avenue.  _APL Photonics_ 4, 110901 (2019).
  * (3) Wright, L. G., Wu, F. O., Christodoulides, D. N. & Wise, F. W.  Physics of highly multimode nonlinear optical systems.  _Nature Physics_ 18, 1018–1030 (2022).
  * (4) Krupa, K. _et al._ Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves.  _Physical review letters_ 116, 183901 (2016).
  * (5) Krupa, K. _et al._ Spatiotemporal light-beam compression from nonlinear mode coupling.  _Physical Review A_ 97, 043836 (2018).
  * (6) Leventoux, Y. _et al._ 3d time-domain beam mapping for studying nonlinear dynamics in multimode optical fibers.  _Optics Letters_ 46, 66–69 (2021).
  * (7) Shalaby, B. _et al._ Visible supercontinuum generation by dual-wavelength pumping in multimode rectangular optical fibers.  In _2014 Conference on Lasers and Electro-Optics (CLEO)-Laser Science to Photonic Applications_ , 1–2 (IEEE, 2014).
  * (8) Zhou, R., Ren, J., Lou, S., Ju, Y. & Wang, Y.  Half mj supercontinuum generation in a telecommunication multimode fiber by a q-switched tm, ho: Yvo 4 laser.  _Journal of the Optical Society of Korea_ 19, 7–12 (2015).
  * (9) Lopez-Galmiche, G. _et al._ Visible supercontinuum generation in a graded index multimode fiber pumped at 1064 nm.  _Optics Letters_ 41, 2553–2556 (2016).
  * (10) Krupa, K. _et al._ Spatiotemporal characterization of supercontinuum extending from the visible to the mid-infrared in a multimode graded-index optical fiber.  _Optics Letters_ 41, 5785–5788 (2016).
  * (11) Eftekhar, M. A. _et al._ Versatile supercontinuum generation in parabolic multimode optical fibers.  _Optics Express_ 25, 9078–9087 (2017).
  * (12) Rishøj, L., Tai, B., Kristensen, P. & Ramachandran, S.  Soliton self-mode conversion: revisiting raman scattering of ultrashort pulses.  _Optica_ 6, 304–308 (2019).
  * (13) Wright, L. G., Christodoulides, D. N. & Wise, F. W.  Spatiotemporal mode-locking in multimode fiber lasers.  _Science_ 358, 94–97 (2017).
  * (14) Teğin, U., Kakkava, E., Rahmani, B., Psaltis, D. & Moser, C.  Spatiotemporal self-similar fiber laser.  _Optica_ 6, 1412–1415 (2019).
  * (15) Wei, X., Jing, J. C., Shen, Y. & Wang, L. V.  Harnessing a multi-dimensional fibre laser using genetic wavefront shaping.  _Light: Science & Applications_ 9, 149 (2020).
  * (16) Ding, Y. _et al._ Spatiotemporal mode-locking in lasers with large modal dispersion.  _Physical Review Letters_ 126, 093901 (2021).
  * (17) Xiong, W. _et al._ Spatiotemporal control of light transmission through a multimode fiber with strong mode coupling.  _Physical review letters_ 117, 053901 (2016).
  * (18) Vaughan, J. C., Feurer, T. & Nelson, K. A.  Automated two-dimensional femtosecond pulse shaping.  _JOSA B_ 19, 2489–2495 (2002).
  * (19) Demas, J. _et al._ Intermodal nonlinear mixing with bessel beams in optical fiber.  _Optica_ 2, 14–17 (2015).
  * (20) Nazemosadat, E., Pourbeyram, H. & Mafi, A.  Phase matching for spontaneous frequency conversion via four-wave mixing in graded-index multimode optical fibers.  _JOSA B_ 33, 144–150 (2016).
  * (21) Eslami, Z. _et al._ Two octave supercontinuum generation in a non-silica graded-index multimode fiber.  _Nature Communications_ 13, 2126 (2022).
  * (22) Redding, B., Popoff, S. M. & Cao, H.  All-fiber spectrometer based on speckle pattern reconstruction.  _Optics express_ 21, 6584–6600 (2013).
  * (23) Xiong, W. _et al._ Deep learning of ultrafast pulses with a multimode fiber.  _APL Photonics_ 5, 096106 (2020).
  * (24) Xu, C. & Wise, F.  Recent advances in fibre lasers for nonlinear microscopy.  _Nature photonics_ 7, 875–882 (2013).
  * (25) Moussa, N. O. _et al._ Spatiotemporal beam self-cleaning for high-resolution nonlinear fluorescence imaging with multimode fiber.  _Scientific Reports_ 11, 18240 (2021).
  * (26) Berti, N. _et al._ Interplay of thermalization and strong disorder: Wave turbulence theory, numerical simulations, and experiments in multimode optical fibers.  _Physical Review Letters_ 129, 063901 (2022).
  * (27) Picozzi, A. _et al._ Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics.  _Physics Reports_ 542, 1–132 (2014).
  * (28) Garnier, J. _et al._ Wave condensation with weak disorder versus beam self-cleaning in multimode fibers.  _Physical Review A_ 100, 053835 (2019).
  * (29) Podivilov, E. V. _et al._ Hydrodynamic 2d turbulence and spatial beam condensation in multimode optical fibers.  _Physical review letters_ 122, 103902 (2019).
  * (30) Teğin, U., Yıldırım, M., Oğuz, İ., Moser, C. & Psaltis, D.  Scalable optical learning operator.  _Nature Computational Science_ 1, 542–549 (2021).
  * (31) Rahmani, B. _et al._ Learning to image and compute with multimode optical fibers.  _Nanophotonics_ 11, 1071–1082 (2022).
  * (32) Wright, L. G., Christodoulides, D. N. & Wise, F. W.  Controllable spatiotemporal nonlinear effects in multimode fibres.  _Nature photonics_ 9, 306–310 (2015).
  * (33) Tzang, O., Caravaca-Aguirre, A. M., Wagner, K. & Piestun, R.  Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres.  _Nature Photonics_ 12, 368–374 (2018).
  * (34) Deliancourt, E. _et al._ Wavefront shaping for optimized many-mode kerr beam self-cleaning in graded-index multimode fiber.  _Optics Express_ 27, 17311–17321 (2019).
  * (35) Teğin, U. _et al._ Controlling spatiotemporal nonlinearities in multimode fibers with deep neural networks.  _Apl Photonics_ 5, 030804 (2020).
  * (36) Bender, N., Haig, H., Christodoulides, D. N. & Wise, F. W.  Spectral speckle customization.  _Optica_ 10, 1260–1268 (2023).
  * (37) Renninger, W. H. & Wise, F. W.  Optical solitons in graded-index multimode fibres.  _Nature communications_ 4, 1719 (2013).
  * (38) Pourbeyram, H., Agrawal, G. P. & Mafi, A.  Stimulated raman scattering cascade spanning the wavelength range of 523 to 1750 nm using a graded-index multimode optical fiber.  _Applied Physics Letters_ 102, 201107 (2013).
  * (39) Wright, L. G. _et al._ Self-organized instability in graded-index multimode fibres.  _Nature Photonics_ 10, 771–776 (2016).
  * (40) Liu, Z., Wright, L. G., Christodoulides, D. N. & Wise, F. W.  Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber.  _Optics Letters_ 41, 3675–3678 (2016).
  * (41) Krupa, K. _et al._ Spatial beam self-cleaning in multimode fibres.  _Nature Photonics_ 11, 237–241 (2017).
  * (42) Ahsan, A. S. & Agrawal, G. P.  Graded-index solitons in multimode fibers.  _Optics Letters_ 43, 3345–3348 (2018).
  * (43) Eftekhar, M. _et al._ Accelerated nonlinear interactions in graded-index multimode fibers.  _Nature communications_ 10, 1638 (2019).
  * (44) Regelman, D. V., Segev, Y. & Yosub, S.  Method for brightness enhancement and modes manipulation of a multimode optical fiber (2016).  US Patent 9,329,416.
  * (45) Golubchik, D., Segev, Y. & Regelman, D.  Controlled mode mixing in optical fibers.  In _International Conference and Exhibition on Optics and Electro-Optics (OASIS 5)_ (2015).
  * (46) Resisi, S., Viernik, Y., Popoff, S. M. & Bromberg, Y.  Wavefront shaping in multimode fibers by transmission matrix engineering.  _APL Photonics_ 5, 036103 (2020).
  * (47) Finkelstein, Z., Sulimany, K., Resisi, S. & Bromberg, Y.  Spectral shaping in a multimode fiber by all-fiber modulation.  _APL Photonics_ 8, 036110 (2023).
  * (48) Shekel, R. _et al._ Shaping single photons through multimode optical fibers using mechanical perturbations.  _arXiv preprint arXiv:2306.02288_ (2023).
  * (49) Wu, Y., Bender, N., Christodoulides, D. N. & Wise, F. W.  Highly multimode solitons in step-index optical fiber.  _APL Photonics_ 8 (2023).
  * (50) Eftekhar, M., Lopez-Aviles, H., Wise, F., Amezcua-Correa, R. & Christodoulides, D.  General theory and observation of cherenkov radiation induced by multimode solitons.  _Communications Physics_ 4, 137 (2021).
  * (51) Wright, L. G., Renninger, W. H., Christodoulides, D. N. & Wise, F. W.  Spatiotemporal dynamics of multimode optical solitons.  _Optics Express_ 23, 3492–3506 (2015).
  * (52) Zitelli, M. _et al._ Multimode solitons in step-index fibers.  _Optics Express_ 30, 6300–6310 (2022).
  * (53) Snyder, A. W.  Coupled-mode theory for optical fibers.  _JOSA_ 62, 1267–1277 (1972).
  * (54) Marcuse, D.  Coupled mode theory of round optical fibers.  _Bell System Technical Journal_ 52, 817–842 (1973).
  * (55) Xiao, Y. _et al._ Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers.  _Optics Express_ 22, 32039–32059 (2014).
  * (56) Bendahmane, A. _et al._ Seeded intermodal four-wave mixing in a highly multimode fiber.  _JOSA B_ 35, 295–301 (2018).
  * (57) Mumtaz, S., Essiambre, R.-J. & Agrawal, G. P.  Nonlinear propagation in multimode and multicore fibers: generalization of the manakov equations.  _Journal of Lightwave Technology_ 31, 398–406 (2012).
  * (58) Wright, L. G. _et al._ Multimode nonlinear fiber optics: massively parallel numerical solver, tutorial, and outlook.  _IEEE Journal of Selected Topics in Quantum Electronics_ 24, 1–16 (2017).
  * (59) Horak, P. & Poletti, F.  Multimode nonlinear fibre optics: theory and applications.  _Recent progress in optical fiber research_ 3 (2012).
  * (60) Yariv, A.  Coupled-mode theory for guided-wave optics.  _IEEE Journal of Quantum Electronics_ 9, 919–933 (1973).
  * (61) Olshansky, R.  Mode coupling effects in graded-index optical fibers.  _Applied optics_ 14, 935–945 (1975).
  * (62) Caravaca-Aguirre, A. M. & Piestun, R.  Single multimode fiber endoscope.  _Optics Express_ 25, 1656–1665 (2017).
  * (63) Flaes, D. E. B. _et al._ Robustness of light-transport processes to bending deformations in graded-index multimode waveguides.  _Physical review letters_ 120, 233901 (2018).
  * (64) Cao, H., Čižmár, T., Turtaev, S., Tyc, T. & Rotter, S.  Controlling light propagation in multimode fibers for imaging, spectroscopy, and beyond.  _Advances in Optics and Photonics_ 15, 524–612 (2023).
  * (65) Carpenter, J., Thomsen, B. C. & Wilkinson, T. D.  Degenerate mode-group division multiplexing.  _Journal of Lightwave Technology_ 30, 3946–3952 (2012).
  * (66) Ferraro, M. _et al._ Femtosecond nonlinear losses in multimode optical fibers.  _Photonics Research_ 9, 2443–2453 (2021).
  * (67) Ferraro, M. _et al._ Multiphoton ionization of standard optical fibers.  _Photonics Research_ 10, 1394–1400 (2022).
  * (68) Agrawal, G. P.  Invite paper: Self-imaging in multimode graded-index fibers and its impact on the nonlinear phenomena.  _Optical Fiber Technology_ 50, 309–316 (2019).
  * (69) Lancry, M., Poumellec, B., Chahid-Erraji, A., Beresna, M. & Kazansky, P.  Dependence of the femtosecond laser refractive index change thresholds on the chemical composition of doped-silica glasses.  _Optical Materials Express_ 1, 711–723 (2011).
  * (70) Sun, Y. _et al._ Multimode soliton collisions in graded-index optical fibers.  _Optics Express_ 30, 21710–21724 (2022).
  * (71) Perret, S. _et al._ Supercontinuum generation by intermodal four-wave mixing in a step-index few-mode fibre.  _Apl Photonics_ 4 (2019).
  * (72) Yildirim, M., Sugihara, H., So, P. T. & Sur, M.  Functional imaging of visual cortical layers and subplate in awake mice with optimized three-photon microscopy.  _Nature communications_ 10, 177 (2019).
  * (73) You, S. _et al._ Intravital imaging by simultaneous label-free autofluorescence-multiharmonic microscopy.  _Nature communications_ 9, 2125 (2018).
  * (74) Hoover, E. E. & Squier, J. A.  Advances in multiphoton microscopy technology.  _Nature photonics_ 7, 93–101 (2013).
  * (75) Huland, D. M. _et al._ Three-photon excited fluorescence imaging of unstained tissue using a grin lens endoscope.  _Biomedical Optics Express_ 4, 652–658 (2013).
  * (76) Dong, C.-Y., Koenig, K. & So, P.  Characterizing point spread functions of two-photon fluorescence microscopy in turbid medium.  _Journal of biomedical optics_ 8, 450–459 (2003).
  * (77) Xu, C. & Webb, W. W.  Multiphoton excitation of molecular fluorophores and nonlinear laser microscopy.  _Topics in Fluorescence Spectroscopy: Volume 5: Nonlinear and Two-Photon-Induced Fluorescence_ 471–540 (2002).
  * (78) Demas, J., Rishøj, L. & Ramachandran, S.  Free-space beam shaping for precise control and conversion of modes in optical fiber.  _Optics express_ 23, 28531–28545 (2015).
  * (79) Ramsay, J. _et al._ Generation of infrared supercontinuum radiation: spatial mode dispersion and higher-order mode propagation in zblan step-index fibers.  _Optics Express_ 21, 10764–10771 (2013).
  * (80) Petersen, C. R. _et al._ Mid-infrared supercontinuum covering the 1.4–13.3 $\mu$m molecular fingerprint region using ultra-high na chalcogenide step-index fibre.  _Nature Photonics_ 8, 830–834 (2014).
  * (81) Kubat, I. & Bang, O.  Multimode supercontinuum generation in chalcogenide glass fibres.  _Optics Express_ 24, 2513–2526 (2016).
  * (82) Wang, K. & Xu, C.  Tunable high-energy soliton pulse generation from a large-mode-area fiber and its application to third harmonic generation microscopy.  _Applied physics letters_ 99, 071112 (2011).
  * (83) Zach, A., Mohseni, M., Polzer, C., Nicholson, J. & Hellerer, T.  All-fiber widely tunable ultrafast laser source for multimodal imaging in nonlinear microscopy.  _Optics Letters_ 44, 5218–5221 (2019).
  * (84) Rishøj, L., Deng, F., Tai, B., Cheng, J.-X. & Ramachandran, S.  Jitter-free, dual-wavelength, ultrashort-pulse, energetic fiber sources using soliton self-mode conversion.  _Optics Express_ 28, 4333–4339 (2020).
  * (85) Lefort, C.  A review of biomedical multiphoton microscopy and its laser sources.  _Journal of physics D: Applied physics_ 50, 423001 (2017).
  * (86) Dubietis, A., Tamošauskas, G., Šuminas, R., Jukna, V. & Couairon, A.  Ultrafast supercontinuum generation in bulk condensed media (invited review).  _arXiv preprint arXiv:1706.04356_ (2017).
  * (87) Gaeta, A. L.  Catastrophic collapse of ultrashort pulses.  _Physical Review Letters_ 84, 3582 (2000).
  * (88) Xu, C. & Webb, W. W.  Measurement of two-photon excitation cross sections of molecular fluorophores with data from 690 to 1050 nm.  _JOSA B_ 13, 481–491 (1996).
  * (89) Ho, K.-P., Kahn, J. M., Kaminow, I., Li, T. & Willner, A.  Mode coupling and its impact on spatially multiplexed systems.  _Optical Fiber Telecommunications VI_ 17, 1386–1392 (2013).