that the two supermodes emit with equal inten-
sity in both waveguides at either facet. Atz= 0,
the relative phaseφbetween the waveguide am-
plitudes of the supermodes is approximately
φ–(0)≈–p(p-out-of-phase) andφ+(0)≈ 0
(in-phase) forF–andF+, respectively. This is
exactly reversed at the end of the encircling
section (z=L), i.e.,φ–(L)≈0 andφ+(L)≈–p,
such that the adiabatic following along the
topological modeF–(z) continuously morphsthe transverse mode profile from beingp-out-
of-phase at one end to being in-phase at
the opposite end of the cavity [for details,
see the materials and methods ( 30 ), sec-
tions 5 and 6].886 25 FEBRUARY 2022•VOL 375 ISSUE 6583 science.orgSCIENCE
Fig. 2. Operation principle and laser structure.
(A) Parameter path encircling the EP in the plane
spanned by the normalized coupling~kand
detuning~d.(B) Illustration of the EP-encircling
laser (not to scale). In addition to the losses
caused by absorption in both waveguides, the red
waveguide experiences gain by optically pumping
the encircling section of the cavity. The separation
between the detuners and their respective main
waveguides introduces detuningd(z), whereas
the separation between the two main waveguides
generates couplingk(z). The grating reflector
on the left end of each main waveguide acts
as a wavelength filter. The steady-state topological
mode is characterized by the simultaneous
emission of the in-phase (right end) and
p-out-of-phase (left end) mode, each from
one facet. (C) SEM images (small panels)
of the laser structure demonstrating the
variation of the separations between detuners
as well as the main waveguides.
Fig. 3. Numerically simulated
transient and steady-state
behavior of the encircling
part of the cavity.(Aand
B) Numerical simulations of the
transient field evolution for
six passes in alternating
directions through the cavity in
the presence of gain saturation.
In total, 100 individual solutions
of Equations 2a and 2b based
on purely stochastic excitations
are shown as thin green
(A) and red (B) lines. The thick
gray lines show the instanta-
neous eigenstateF–(z)
without noise. (A) The relative
phase between the two
waveguides evolves continu-
ously from–pto 0 and back
within one round trip. (B) After
an initial population transfer
towardF–, the normalized
eigenvector populationpshows
that the ensuing adiabatic
following of said eigenstate leads to the emission of different supermodes from
each facet. (CandD) Evolution of the relative phase between the two cavities and
the normalized intensity difference, respectively, of the left-to-right (purple) and
right-to-left (cyan) traveling waves according to a Rigrod-type self-consistent
simulation using Equations 3a and 3b. Emissions of different supermodes from
each facet are shown. The two spatial supermodes are characterized by equal
intensity in both waveguides at the output ports and a phase difference that
evolves from–pto 0 and back.RESEARCH | REPORTS