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We thank D. E. Chang and H. Ritsch for insightful discussions on
dipolar interactions in a 3D lattice. We also acknowledge technical
contributions from and discussions with C. Benko, T. Bothwell,
S. L. Bromley, K. Hagen, J. L. Hall, B. Horner, H. Johnson, T. Keep,
S. Kolkowitz, J. Levine, T. H. Loftus, T. L. Nicholson, E. Oelker,
D. G. Reed, and X. Zhang. This work is supported by NIST, the
Defense Advanced Research Projects Agency, the Air Force Office
of Scientific Research Multidisciplinary University Research
Initiative, and the NSF JILA Physics Frontier Center. G.E.M. is
supported by a postdoctoral fellowship from the National Research
Council, A.G. is supported by a fellowship from the Japan Society
for the Promotion of Science, and L.S. is supported by a National
Defense Science and Engineering Graduate Fellowship.
Material and Methods
Figs. S1 to S6
Tables S1 and S2
References ( 61–63)
7 December 2016; resubmitted 4 June 2017
Accepted 24 August 2017
Spatiotemporal mode-locking in
multimode fiber lasers
Logan G. Wright,1 Demetrios N. Christodoulides,2 Frank W. Wise1
A laser is based on the electromagnetic modes of its resonator, which provides the feedback
required for oscillation. Enormous progress has been made toward controlling the interactions
of longitudinal modes in lasers with a single transverse mode. For example, the field of
ultrafast science has been built on lasers that lock many longitudinal modes together to
form ultrashort light pulses. However, coherent superposition of longitudinal and transverse
modes in a laser has received little attention. We show that modal and chromatic dispersions
in fiber lasers can be counteracted by strong spatial and spectral filtering. This allows locking
of multiple transverse and longitudinal modes to create ultrashort pulses with a variety of
spatiotemporal profiles. Multimode fiber lasers thus open new directions in studies of nonlinear
wave propagation and capabilities for applications.
The modes of a laser resonator are three- dimensional (3D) functions that vary along the axis of the resonator, as well as in the two transverse dimensions (1) (Fig. 1). Each mode has a distinct resonant frequency
(Fig. 1, B and D). In many cases of interest, the 3D
modes are separable into so-called longitudinal
and transverse modes. If the relative phases of
the modes are not controlled, the output is an
incoherent spatiotemporal field that results from
random interference. The modes can interact
with each other in the gain medium and other
components of a laser.
The situation can be simplified greatly by
restricting operation to a single transverse mode.
With the exception of high average power, the best
performance for virtually all laser parameters—
e.g., ultranarrow emission spectra, ultrashort pulse
duration, and ultralow noise—is achieved through
lasing in a single transverse mode. Operation in
multiple spatial modes, by contrast, leads to substantial complexity. Nonetheless, this technique
is widely employed in high–average-power lasers,
for applications that do not require high temporal or frequency precision or diffraction-limited
Highly refined techniques have been developed
to control the longitudinal modes of a laser. The
number of oscillating modes can range from as few
as 1 to more than 1 million perfectly synchronized
modes in a frequency comb (2). Considering the
pronounced control of the electromagnetic field
achieved in lasers with a single transverse mode,
the lack of attention paid to higher-dimensional
coherent lasing states is conspicuous. Early works
did propose locking of transverse spatial modes, as
well as transverse and longitudinal modes. Mode-locking of this kind was demonstrated with a few
modes ( 3, 4) and has since been observed with two
transverse modes of a femtosecond Ti:sapphire
laser ( 5).
Recent years have seen increased interest in
linear and nonlinear wave propagation in multiple
transverse modes, particularly in multimode (MM)
optical fibers ( 6–16) and largely in anticipation of
spatial division multiplexing ( 17) for telecommunications and as a platform for new fiber laser
sources. The presence of disorder (through random linear mode coupling) also connects MM
fibers to the field of random lasers and, more
broadly, to the optical properties of complex
media ( 18–21).
Here we employ principles of normal-dispersion
mode-locking ( 22, 23) in space and time—strong
spectral and spatial filtering in addition to the high
nonlinearity, gain, and spatiotemporal dispersion
of the fiber medium—to achieve spatiotemporal
mode-locking. The self-organized, mode-locked
pulses take a variety of spatiotemporal shapes consisting of many transverse and longitudinal modes.
Lasing modes can interact through the optical
nonlinearities of the gain, the saturable absorber,
and the fiber medium itself. These effects occur
on the time scale of a pulse and thus can couple
temporal and spatial degrees of freedom. The
slow relaxation of rare-earth gain media introduces an additional layer of temporally averaging nonlinear interactions. To realize the
simplest demonstration of highly MM spatiotemporal mode-locking (STML), we began by
constructing a cavity with a few-mode, Yb-doped
gain fiber (10-mm diameter, supporting approximately three transverse modes) spliced to a highly
MM passive graded-index (GRIN) fiber (which
supports 100 transverse modes). The use of a few-mode gain fiber provides spatial filtering and
forces the laser gain to saturate with the total
energy of the coupled modes (in general, a complex combination of the 100 transverse modes).
This therefore largely eliminates transverse gain
interactions and isolates the key nonlinear interactions involved in passive mode-locking.
GRIN fiber is used so that the modal dispersion
within the cavity is relatively small and thus of
similar magnitude to the chromatic (i.e., longitudinal mode) dispersion. As will be expanded
on later, this is an important factor for achieving
1School of Applied and Engineering Physics, Cornell University,
Ithaca, NY 14853, USA. 2Center for Research and Education
in Optics and Lasers, College of Optics and Photonics, University
of Central Florida, Orlando, FL 32816, USA.
*Corresponding author. Email: email@example.com