1]Time and Frequency Division, National Institute of Standards and Technology,
325 Broadway MS 688, Boulder, 80305, Colorado, USA

2]Department of Physics, University of Colorado, 390 UCB, Boulder, 80305,
Colorado, USA

3]Laboratoire ICB, UMR 6303 CNRS-Université de Bourgogne, 21078 Dijon, France

4]Octave Photonics, 325 W South Boulder Rd Suite B-1, Louisville, 80027,
Colorado, USA

# The bandgap-detuned excitation regime in photonic-crystal resonators

Yan Jin<EMAIL_ADDRESS>Erwan Lucas  Jizhao Zang  Travis Briles  Ivan
Dickson  David Carlson  and Scott B. Papp  [  [  [  [

###### Abstract

Control of nonlinear interactions in microresonators enhances access to
classical and quantum field states across nearly limitless bandwidth. A recent
innovation has been to leverage coherent scattering of the intraresonator pump
as a control of group-velocity dispersion and nonlinear frequency shifts,
which are precursors for the dynamical evolution of new field states. A
uniform periodicity nanostructure addresses backscattering with one resonator
mode, and pumping that mode enables universal phase-matching for four-wave
mixing with control by the bandgap. Yet, since nonlinear-resonator phenomena
are intrinsically multimode and exhibit complex modelocking, here we
demonstrate a new approach to controlling nonlinear interactions by creating
bandgap modes completely separate from the pump laser. We explore this
bandgap-detuned excitation regime through generation of benchmark optical
parametric oscillators (OPOs) and soliton microcombs. Indeed, we show that
mode-locked states are phase matched more effectively in the bandgap-detuned
regime in which we directly control the modal Kerr shift with the bandgaps
without perturbing the pump field. In particular, bandgap-detuned excitation
enables an arbitrary control of backscattering as a versatile tool for mode-
locked state engineering. Our experiments leverage nanophotonic resonators for
phase matching of OPOs and solitons, leading to control over threshold power,
conversion efficiency, and emission direction that enable application advances
in high-capacity signaling and computing, signal generation, and quantum
sensing.

Lasers are versatile and precise sources with diverse applications across a
range of industries and scientific disciplines, contributing to advanced
optical communication [1], atomic and molecular spectroscopy [2], and quantum
technologies [3]. The capability to generate laser sources with a nearly
arbitrary spectrum and quantum-limited noise properties is an enduring
challenge that stimulates exploration of new light-matter interactions.
Integrated nonlinear photonics plays a key role in enabling the spectrum of
light to be dramatically transformed from the input to the output of a device.
In particular, nonlinear waveguides transform a mode-locked laser into
supercontinuum outputs with exceptionally broad bandwidth at low operating
power [4, 5, 6] and with complex mixtures of nonlinear interactions [7]. Kerr
microresonators enable the generation of vastly broadband optical parametric
oscillators (OPOs) and soliton microcombs through the conversion of a
continuous-wave (CW) pump laser, according to a complex mixture of resonance
conditions, group-velocity dispersion of the resonator modes, and the
intraresonator pump field [8, 9, 10, 11, 12, 13, 14, 15, 16].

Control of nonlinear interactions in microresonators offers the opportunity to
engineer new laser sources, especially through foundational behaviors of the
intraresonator field [17]. Group-velocity dispersion (GVD, hereafter
dispersion) engineering has been an important advance, enabling direct
spectrum control in microcombs [18, 19]. Adjusting the pump laser power and
detuning also directly controls a soliton microcomb [20]. Given that solitons
are isolated excitations of the microresonator, interacting soliton ensembles
offer unique properties [21]. Moreover, ensembles of coupled microresonators
enable fundamentally new interactions, including solitons that coexist in
several devices [22]. Direct modification of the intensity-dependent
refractive index is also a natural tool for controlling solitons [23, 24].
Among the methods that have been explored to manipulate microcombs, a recent
innovation has been photonic-crystal ring resonators (PhCRs), which employ a
sub-wavelength nanostructure to couple forward and backward propagation of
light and induce mode-frequency shifts [25, 18]. Indeed, PhCRs provide a rich
capability for mode-by-mode dispersion control and to rebalance Kerr frequency
shifts.

PhCRs offer direct control of the phase matching and nonlinear dynamics of
microresonators. Indeed, access to mode-by-mode frequency shifts represents a
fundamentally new degree of freedom for the constituent Lugiato-Lefever
equation (LLE) description of a microcomb. In particular, pump-laser
excitation of the split mode in PhCRs allows universal phase matching for
four-wave mixing in both the normal and anomalous dispersion regimes, enabling
OPOs [26, 27], bright solitons in anomalous dispersion [25], access to the
dark-to-bright soliton continuum in normal dispersion [28], and use in
nonlinear resonator circuits to optimize microcomb performance metrics [9].
Recently, PhCR bandgaps for unpumped modes have been used to define the output
wavelength in OPOs [29] and induce exotic microcomb dynamics [18, 30].
Although PhCRs intrinsically program mode-by-mode dispersion, since the
nanostructured waveguide defines the optical mode, other approaches, including
coupled resonators, enable similar controls [31, 32]. Still, the use of
coherent backscattering in PhCRs opens excess optical loss channels,
particularly since practical devices operate in the large bandgap limit where
forward and backward propagation are strongly coupled. These devices include
specific limitations, such as increased threshold power, reduced conversion
efficiency, and predominant emission of newly generated OPO and soliton light
in the backward direction, i.e. emitted toward the pump laser.

Here, we explore the bandgap-detuned excitation regime of PhCRs in which we
open optical bandgaps mode-detuned from the pump laser. Operating PhCRs in
this regime avoids splitting the pump amongst the forward and backward
directions, yet we preserve the capability for universal phase matching.
Moreover, by designing bandgaps to interact directly with mode-locked
nonlinear states of the resonator, we exploit the inherent multimode
composition of target states in their construction. The ratio of forward and
backward coupling, and hence the effective excess loss toward the backward
direction, is tunable in the bandgap-detuned regime. Through this effect, we
introduce a new framework for nonlinear-state engineering based on an
effective integrated dispersion parameter, which directly characterizes phase
matching of resonator-field states. We explore benchmark OPOs and soliton
microcombs in bandgap-detuned PhCRs, demonstrating that these states are more
effectively phase-matched and can be created with reduced threshold power,
higher efficiency, and the control to realize predominantly forward emission.
Furthermore, we demonstrate a design process for OPO lasers and solitons
microcombs that could be used to create sources for applications.

Figure 1:  (a) Conventional regime where the red shifted mode is pumped. (b)
Bandgap-detuned regime. Both (a) and (b) present the ring structure,
$D_{\mathrm{int}}$ normalized to $\kappa/2$, mode structure, the forward and
backward spectra, and the pulse intensity normalized to $F^{2}$. (c)
Integrated dispersion for $\mu_{s}=10$ with low pump power (blue) and high
pump power (orange). (d) Setup. EDFA: Erbium-doped fiber amplifier. OSA:
optical spectrum analyzer. ESA: electrical spectrum analyzer. PD: photodiode.
(e) SEM images for PhCR. (f) Forward and backward spectra for $\mu_{s}=10$ in
simulation (orange dashed lines) and experiment (solid lines).

A PhCR is a three-port nonlinear system with an input port receiving the
pumping field $F$, and two output ports with modal fields $A_{\mu}$ and
$B_{\mu}$ propagating respectively in the forward direction (i.e., the same as
the pump) and the backward direction, where $\mu$ is the mode number with
respect to the pumped mode. Intraresonator coherent backscattering caused by
the periodic sub-wavelength pattern of the PhCR couples the two propagating
directions at a specific Bragg criterion and hence at a specific longitudinal
mode. At a designated mode $\mu_{s}$, this coupling lifts the initial
degeneracy of $A_{\mu_{s}}$ and $B_{\mu_{s}}$, leading to a red-shifted (lower
frequency) and a blue-shifted (higher frequency) resonance, separated by a
bandgap $\Gamma_{\mu_{s}}$. The dispersion of the resonance frequencies is
perturbed and follows the integrated dispersion relation
$D_{\mathrm{int}}(\mu)\equiv\omega_{\mu}-\omega_{0}-\mu
D_{1}=D_{2}\mu^{2}/2\pm\Gamma_{\mu}/2$, where $\omega_{\mu}$ is the resonance
frequencies of the ring resonator, $\omega_{0}$ is the pump mode frequency,
$D_{1}/2\pi$ is the free spectral range, $D_{2}$ is the second-order
dispersion coefficient, and $\Gamma_{\mu}=0$ if $\mu\neq\mu_{s}$. To describe
the dynamics of the PhCR, we introduce the coupled-mode LLEs, written in the
spectral domain [33]:

$\displaystyle\dot{A}_{\mu}$
$\displaystyle=-(1+i\alpha)A_{\mu}-i\frac{d_{2}}{2}\mu^{2}A_{\mu}+i\beta_{\mu}B_{\mu}+i\mathcal{F}(|A|^{2}A)_{\mu}+i2A_{\mu}P_{b}+F\delta_{\mu,0}$
(1) $\displaystyle\dot{B}_{\mu}$
$\displaystyle=-(1+i\alpha)B_{\mu}-i\frac{d_{2}}{2}\mu^{2}B_{\mu}+i\beta_{\mu}^{*}A_{\mu}+i\mathcal{F}(|B|^{2}B)_{\mu}+i2B_{\mu}P_{a},$
(2)

where $F$ refers to the normalized pump field, $\alpha$ is the detuning,
$\delta_{\mu,0}$ is the Kronecker symbol, $\kappa$ is the linewidth,
$\beta_{\mu}=\Gamma_{\mu}/\kappa$ is the normalized half-bandgap in mode
$\mu$, $d_{2}=2D_{2}/\kappa$ is the normalized second-order dispersion
coefficient, $P_{a}=\sum_{\eta}|A_{\eta}|^{2}$,
$P_{b}=\sum_{\eta}|B_{\eta}|^{2}$, and
$\dot{A}_{\mu}=\mathrm{d}A_{\mu}/\mathrm{d}\tau$ where $\tau=t\kappa/2$ is the
normalized time. $A(\theta,\tau)=\sum A_{\mu}(\tau)e^{i\mu\theta}$,
$B(\theta,\tau)=\sum B_{\mu}(\tau)e^{i\mu\theta}$ are the normalized
electrical fields in the ring coordinates, and $\mathcal{F}(\cdot)_{\mu}$
represents the Fourier operator.

Our framework for nonlinear-state engineering in PhCRs is based on an
effective integrated dispersion $\tilde{D}_{\mathrm{int}}(\mu)$, which
characterizes the frequency mismatch of the field states,

$\tilde{D}_{\mathrm{int}}(\mu)\equiv\operatorname{Re}\left(i\frac{dA_{\mu}/dt}{A_{\mu}}\right)=\frac{\kappa}{2}\operatorname{Re}\left(i\frac{\dot{A}_{\mu}}{A_{\mu}}\right)=\alpha\dfrac{\kappa}{2}+\dfrac{D_{2}}{2}\mu^{2}+\epsilon_{\mu}-\Delta_{\mu}-P_{b}\kappa-\delta_{\mu,0}\operatorname{Im}\left(\dfrac{F}{A_{0}}\right)\dfrac{\kappa}{2}$
(3)

where
$\Delta_{\mu}\equiv\operatorname{Re}(\mathcal{F}(|A|^{2}A)_{\mu}/A_{\mu})\kappa/2$
is the modal Kerr shift [25], and
$\epsilon_{\mu}\equiv\operatorname{Re}(-\Gamma_{\mu}\frac{B_{\mu}}{A_{\mu}})/2$
is the bandgap-induced frequency shift. The three main contributors in
Equation 3 are the background dispersion $D_{2}\mu^{2}/2$, the Kerr
nonlinearity $\Delta_{\mu}$, and the mode shift $\epsilon_{\mu}$ due to
coupling between the couterpropagating fields. For $F=0$, $\Delta_{\mu}=0$,
$\epsilon_{\mu}=\pm\Gamma_{\mu}/2$ indicates two components of the split mode,
one blue shifted mode ($\epsilon_{\mu}>0$) and the other red shifted mode
($\epsilon_{\mu}<0$) , and $\tilde{D}_{\mathrm{int}}(\mu)$ reduces to the
dispersion of the resonator modes $D_{2}\mu^{2}/2\pm\Gamma_{\mu}/2$. In mode-
locked states, $\tilde{D}_{\mathrm{int}}(\mu)$ is linear with $\mu$, denoting
a stable and stationary field configuration. Therefore, we express
$\tilde{D}_{\mathrm{int}}(\mu)=\mu\omega_{\mathrm{rep}}$, where
$\delta\omega_{\mathrm{rep}}/2\pi=\mathrm{f}_{\mathrm{rep}}-\mathrm{FSR}$ is
the difference between the repetition rate of the comb (or line spacing) and
the free spectral range. In mode-locked states, where
$\tilde{D}_{\mathrm{int}}(\mu)$ and $D_{2}\mu^{2}/2$ remain constant, the
relative field amplitudes $A_{\mu}$ and $B_{\mu}$ are influenced by the
interplay between the bidirectional coupling strength $\epsilon_{\mu}$ and the
Kerr nonlinearity $\Delta_{\mu}$.

In the bandgap-detuned regime, we implement nonlinear state engineering by
introducing bandgaps in unpumped modes. First, we consider the case
$\Gamma_{\mu}=\Gamma\delta{\mu,\mu_{s}}$, where $\mu_{s}\neq 0$ is the split
mode. Due to the constraint of $\tilde{D}_{\mathrm{int}}$ for a mode-locked
state, we can control each of the field amplitudes $A_{\mu}$ or $B_{\mu}$ with
each bandgap $\Gamma_{\mu}$, which sets the bandgap-induced frequency shift
$\epsilon_{\mu}$. This powerful control parameter also sets the excess power
“lost” to backscattering in the PhCR. Furthermore, in the bandgap-detuned
regime, mode-locked states are excited in either the red mode
($\epsilon_{\mu}<0$) or the blue mode ($\epsilon_{\mu}>0$), allowing another
new facet to control such states. In contrast, in the conventional regime
($\Gamma_{\mu}=\Gamma\delta_{\mu,0}$) where the red-shifted resonance of the
split mode is pumped to generate OPOs or solitons [25], only $\epsilon_{0}<0$
is allowed.

To understand the nonlinear state excitation in this new scenario, we present
benchmark simulations of OPO states in the conventional (Figure 1(a)) and
bandgap-detuned (Figure 1(b)) regimes. These simulations compare the spectra
of forward and backward emission and the intraresonator pulse profiles. In
conventional PhCRs, the generated OPO light propagates in the backward
direction ($\sum_{\mu\neq 0}|A_{\mu}|^{2}<\sum_{\mu\neq 0}|B_{\mu}|^{2}$),
while in the bandgap-detuned regime it propagates predominantly in the forward
direction. This difference emerges because at the parametric threshold the
backward gain exceeds the forward gain in the conventional regime, while the
bandgap-detuned regime is characterized by the dominance of the forward gain.
Furthermore, employing split-mode pumping, as in the conventional regime,
results in a notably higher threshold power, nearly twice that of the bandgap-
detuned scenario. By comparing the sideband power between the two regimes in
simulation, we find that the bandgap-detuned regime also makes efficient use
of the pump energy.

Since the constraint on $\tilde{D}_{\mathrm{int}}(\mu)$ is the basis for
nonlinear-state engineering, we show in Figure 1(c) that this quantity
explains the formation and advantages of OPOs in the bandgap-detuned regime.
For $F=0$, we plot the cold cavity effective dispersion
$\tilde{D}_{\mathrm{int}}(\mu)=D_{2}\mu^{2}/2\pm\Gamma_{\mu}/2$ in blue, and
the pump mode is indicated by the red circle. Since self-phase modulation is
twice smaller than cross-phase modulation in the homogeneous (flat) state, the
modal Kerr shift at $0,\pm\mu_{s}$ obeys
$\Delta_{0}<\Delta_{\mu_{s}},\Delta_{-\mu_{s}}$. Thus, the blue mode at
$\mu_{s}$ is excited so that $\epsilon_{\mu_{s}}>0$, and
$\tilde{D}_{\mathrm{int}}(\mu)$ at $\mu=-\mu_{s},0,\mu_{s}$ connect directly
in a straight-line fashion, as shown by the orange dashed line. This
characteristic stands in contrast to the conventional regime, where the red
mode of the pumped split mode is excited ($\epsilon_{0}<0$). The split mode
$\mu_{s}$ targets the frequency of the signal or idler, and
$\epsilon_{\mu_{s}}>0$ increases $\Delta_{\mu_{s}}$. The increased Kerr shift
$\Delta_{\mu_{s}}$ implies that higher power accumulates in the target mode
$\mu_{s}$ and the symmetric mode $-\mu_{s}$, which explains the enhanced OPOs
conversion in the bandgap-detuned regime. The nonzero slope of
$\tilde{D}_{\mathrm{int}}$ indicates a mismatch between the OPO line spacing
and the free spectral range, which we will analyze in Figure 2.

We explore the generation of OPOs and soliton microcombs throughout this paper
with the experimental setup in Figure 1(d). We generate the pump field with a
continuous-wave (CW) laser, amplified by an erbium-doped fiber amplifier
(EDFA), and we couple this light into a tantala
($\mathrm{Ta}_{2}\mathrm{O}_{5}$) PhCR [34, 35]. An optical circulator
separates the forward propagating pump and the backward propagating field from
the resonator. We measure both the forward- and backward-propagating fields
with an optical spectrum analyzer (OSA), and we use an electrical spectrum
analyzer (ESA) to measure the relative intensity noise of the forward spectra.

We show the physical structure of the PhCR with scanning electron microscope
(SEM) images in Figure 1(e). The outer sidewall of the ring is completely
circular, while we create a spatial modulation on the inner wall of the ring
by means of a periodic nanostructure. We denote the average ring radius as
$\mathrm{RR}$, the average width as RW, and the inner corrugated wall is
parametrized as
$\rho_{\mathrm{in}}(\theta)=\mathrm{RR}-\mathrm{RW}/2+\rho^{\textsc{phc}}\,\sin(2m\theta)$,
where $(\rho,\theta)$ are the polar coordinates, $\rho^{\textsc{phc}}$ is the
amplitude of the photonic crystal and $m$ is the longitudinal order (number of
wavelength) of the targeted split mode. In this paper,
$\mathrm{RR}=\qty{54.4}{\micro}$ for all devices so that the FSR is
approximately 400. $\mathrm{RW}$ varies from [range-units = single]22.1 so
that the dispersion coefficient $D_{2}/2\pi\simeq-10$ MHz,
$\rho^{\textsc{phc}}$ is around 5, and $m$ varies to split the modes at
different wavelengths. The ring resonator design shown in this figure is
exaggerated so that the SEM images are more clear.

To generate bandgap-detuned OPOs, we tune the pump laser frequency to a mode
of the PhCR and monitor the device output. Our framework provides a specific
expectation for the resulting state. Figure 1(f) shows the simulated and
experimental spectra for case $\mu_{s}=10$, on a linear power scale,
highlighting the close correspondence of the traces. In the forward direction,
both the simulation and the experimental data corroborate that the pump is
almost fully depleted, with its power reduced below that of the signal and the
idler. This observation underscores good use of the pump power and the high
efficiency of OPO formation in the bandgap-detuned regime. In our simulation,
only the comb mode at $\mu=\mu_{s}$ is nonzero in the backward direction, but
in the experiment the backward comb lines at $\mu=0,-\mu_{s}$ also appear due
to the reflection of the angled waveguide facets at the edge of the chip.

Figure 2: (a) Existence map with $F=2$, $d_{2}=-0.37$. The colors indicate the
values of $\delta\omega_{\mathrm{rep}}$ as the signal and idler are excited,
the brown empty circles marked with (i)(ii)(iii) are explained in (b) and (c),
and the green dots refer to the efficiency data points in (d). (b) Phase
matching diagrams with (i) $\mu_{s}=1$ and (ii) $\mu_{s}=-1$. (c) Simulated
(orange) and measured (blue) spectra with (i) $\mu_{s}=1$, (ii) $\mu_{s}=-1$
and (iii) $\mu_{s}=15$. Red circle: pump, green triangle: split mode at
$\mu_{s}$, purple square: mode at $-\mu_{s}$. (d) Forward efficiency of two
devices (blue and green) as we pump different modes. The solid lines are the
simulated efficiency, and the dots are the efficiency measured in experiments.

The central characteristic that controls the generation of bandgap-detuned
OPOs is the position of the bandgap relative to the pump $\mu_{s}$. In
particular, we explore phase matching with specific parameters ($F$, $d_{2}$,
$\mu_{s}$, and normalized half-bandgap $\beta_{\mu}\equiv\Gamma_{\mu}/\kappa$)
that enable OPO generation. Figure 2(a) presents the OPO existence map with
respect to the split-mode number $\mu_{s}$ and the bandgap strength
$\beta=\beta_{\mu_{s}}$ for given driving $F=2$ and dispersion $d_{2}=-0.37$.
The gray region indicates the area in which no OPO is generated. The colored
region gives the values of $\delta\omega_{\mathrm{rep}}$ that characterize the
difference between $f_{rep}$ and $\mathrm{FSR}$. Note that this map was not
computed via LLE simulations, but by exploring the steady state of the pump
mode ($a_{0}$) and evaluating the dependence of the parametric gain at the
modes $A_{\mu_{s}}$, $A_{-\mu_{s}}$ and $B_{\mu_{s}}$ as a function of
$\mu_{s}$ and $\beta$. The brown circles (marked with (i)(ii)(iii)) and green
dots are experimentally measured data points for devices with different
$\mu_{s}$ and $\beta$, which represents our observation points in Figure
2(b)(c)(d).

In the bandgap-detuned mode-locked regime, the unpumped split modes make the
integrated dispersion asymmetric, leading to a non-zero slope coefficient
$\delta\omega_{\mathrm{rep}}$ in $\tilde{D}_{\mathrm{int}}(\mu)$. The mode
detunings and the strength of the bandgaps determine the sign of
$\delta\omega_{\mathrm{rep}}$. For OPOs, if $\mu_{s}>0$, then
$\delta\omega_{\mathrm{rep}}>0$ and vice versa. This is illustrated by the
phase matching diagrams in Figure 2(b); for simplicity, we take (i)
$\mu_{s}=1$ and (ii) $\mu_{s}=-1$. For either case, the blue mode at
$\mu=\mu_{s}$ is excited. When the Kerr shift compensates for the original
dispersion, if $\mu_{s}=1$, then $\epsilon_{1}>0$, and the slope of
$\tilde{D}_{\mathrm{int}}(\mu)$ is greater than 0; while if $\mu_{s}=-1$, then
$\epsilon_{-1}>0$ and the slope is negative. To verify this, we select a
device with FSR = 398.078 and pump it into the modes adjacent to the split
mode so that $\mu_{s}$ can be $1$ or $-1$. The cold cavity dispersion
$D_{\mathrm{int}}$ displayed in Figure 2 (b) corresponds to the cases
$\mu_{s}=\pm 1$ for this device, and the first two panels in Figure 2(c) show
the corresponding spectra when we pump these different split modes. The green
triangles point to the split mode $\mu_{s}$, the red circles indicate the pump
mode, and the purple squares indicate the mode at $-\mu_{s}$. We then measure
the repetition rates (line spacing) and find
$f_{\mathrm{rep}}=\qty{398.488}{}$ for $\mu_{s}=1$, and
$f_{\mathrm{rep}}=\qty{397.569}{}$ for $\mu_{s}=-1$. This corresponds to
$\delta\omega_{\mathrm{rep}}/2\pi=\qty{410}{}$ ($\mu_{s}=1$), and
$\delta\omega_{\mathrm{rep}}/2\pi=\qty{-509}{}$ ($\mu_{s}=-1$). The difference
in the values of $\delta\omega_{\mathrm{rep}}$ is due to different linewidths
of the pump mode for $\mu_{s}=1$ or $-1$. In our simulation, we found
$\delta\omega_{\mathrm{rep}}/2\pi=\qty{392}{}$, and -459 respectively, which
is close to our experimental results.

In addition to nonlinear-state engineering, which involves controlling the
relative amplitude of $A_{\mu}$ and $B_{\mu}$, the bandgap-detuned regime
provides the capability to efficiently generate forward-propagating OPOs with
varying bandwidth, while minimizing excess loss resulting from backward
reflection. We select a device with a split mode wavelength around 1550 and
$\mu_{s}\in\\{\pm 1,\pm 2,\pm 3,\pm 4\\}$, and another device with a split
mode around 1521 and $\mu_{s}\in[6,15]$. Their locations on the existence map
are marked with green dots in Figure 2(a). In particular, the data point with
$\mu_{s}=15$ lies at the boundary of the existence map and is marked with a
brown circle, showing the prediction ability of the existence map. We plot its
forward spectra in panel (iii) of Figure 2(c). The pump mode is well depleted
and has a lower power than the signal and idler, indicating good efficiency
performance. Figure 2(d) shows the efficiency of the two devices in the
forward direction as a function of $\mu_{s}$, obtained by pumping different
modes. The simulations (lines) match the experiments (dots), indicating a high
forward efficiency, consistently above 20%. In the conventional regime, the
best total efficiency can reach 41% in a complex design that includes a
waveguide reflector to recycle the pump field [27], while in the bandgap-
detuned regime, forward efficiency alone can reach 48%.

Figure 3:  (a) Conventional regime of generating combs in a photonic-crystal
ring resonator. The red mode of the pump mode is on resonance and the backward
combs dominate. (b) Bandgap-detuned excitation regime. Modes $\mu_{s}=\pm 1$
are split, and the simulated spectra and pulse indicate the forward-
propagating combs. In (a)(b), the blue traces indicate the forward spectra or
pulse, and the cyan traces indicate the backward spectra or pulse. (c) Phase-
matching diagram for the bandgap-detuned regime in the cold and hot cavities.
(d) SEM images. (e) Experimental (solid) and simulated (orange dashed) spectra
in the forward ($|A_{\mu}|^{2}$) and backward ($|B_{\mu}|^{2}$) direction for
the bandgap-detuned regime. (f) Measured dispersion and simulated pulse on the
ring resonator. (g) Simulated spectra with respect to $\alpha$. The unit of
the map is dB.

We now turn to the generation of soliton microcombs in normal dispersion with
the bandgap-detuned scheme. By designing PhCRs with multiple unpumped
bandgaps, we open up mode-locked-state engineering of solitons, following the
same analysis based on the frequency mismatch operator
$\tilde{D}_{\mathrm{int}}$ of the mode fields in the device. We begin by
comparing soliton generation in PhCRs with conventional (Figure 3(a)) and
bandgap-detuned (Figure 3(b)) excitation. With the same pump power setting
$F$, we calculate the evolution of the forward (blue) and backward (cyan)
propagating comb fields as the pump tunes into the mode $\mu=0$, into the
lower frequency resonance in the conventional case ($\Gamma_{0}>0$) or to a
bandgap-detuned scenario with equal symmetric bandgaps $\Gamma_{\pm 1}$ at
$\mu_{s}=\pm 1$. We obtain the comb spectra $|A_{\mu}|^{2}$ and
$|B_{\mu}|^{2}$, and the corresponding intraresonator temporal profiles in the
forward (blue) and backward (cyan) directions. In the conventional PhCR
scheme, the intracavity field evolves from primary OPO sidebands to a soliton
microcomb in the backward direction. In the case of bandgap-detuned PhCR,
parametric gain arises in the split modes $\mu_{s}$, resulting in the direct
generation of sidebands into these modes. This initial modulation further
evolves into a deterministic, forward-propagating dark soliton, made possible
by the non-split nature of the pumped mode, ensuring that most of its power
propagates forward. Conversely, the backward-coupled comb modes at
$\mu_{s}=\pm 1$ give rise to a weaker 2-FSR spaced comb in this direction.

To understand phase matching and the formation of bandgap-detuned soliton
microcombs, we plot $\tilde{D}_{\mathrm{int}}(\mu)$ in Figure 3(c).
Considering an example where the bandgaps at $\mu_{s}=\pm 1$ are not equal
($\Gamma_{1}>\Gamma_{-1}$) highlights an important practical element of
bandgap-detuned microcombs [30]. In this case, $D_{\mathrm{int}}(\mu)$ is
plotted in blue, and the red circle indicates the pump mode. At threshold, the
modal Kerr shift obeys $\Delta_{1}\simeq\Delta_{-1}>\Delta_{0}$ because cross-
phase modulation is larger than self-phase modulation, which is shown by the
black arrow. As we increase the pump power, the modal Kerr shift
$\Delta_{\mu}$ compensates for the dispersion $D_{\mathrm{int}}$ with the help
of the mode splittings at $\mu_{s}=\pm 1$, thus the phase matching is achieved
and $\tilde{D}_{\mathrm{int}}(\mu)$ evolves into a straight line (orange). The
mode in $\mu=1$ is more blue-shifted than in $\mu=-1$, therefore,
$\epsilon_{1}>\epsilon_{-1}$, thus $\tilde{D}_{\mathrm{int}}(\mu)$ exhibits a
positive slope in the mode-locked state, increasing the repetition rate of the
comb ($\delta\omega_{\mathrm{rep}}>0$).

To experimentally explore the generation of bandgap-detuned dark-soliton, we
study the PhCR device in Figure 3(d) with bandgap modes $\mu_{s}=\pm 1$. We
design this resonator by superposing two grating patterns in the inner wall of
the ring that are written as a parametric curve [36, 18]:
$\rho_{\mathrm{in}}(\theta)=\mathrm{RR-
RW/2}+\sum_{\mu\in\mu_{s}}\rho^{\textsc{phc}}_{\mu}\,\sin(2(m_{0}+\mu)\theta)$.
We visualize the fabricated pattern by imaging a device with an exaggerated
amplitude $\rho$, using a high-resolution SEM. We characterize
$D_{\mathrm{int}}$ of the fabricated device by a calibrated frequency scan of
our tunable laser across several resonator modes; see Figure 3(f). In
particular, we resolve the resonator dispersion and the bandgaps
$\Gamma_{1}/2\pi=\qty{1.04}{}$, $\Gamma_{-1}/2\pi=\qty{0.94}{}$,
$D_{1}/2\pi=\qty{397.848}{}$, and $D_{2}/2\pi=\qty{-12.3}{}$. On the basis of
this information, we simulate the intraresonator intensity pattern of the
bandgap-detuned soliton. Experimentally, we pump this PhCR with a power
corresponding to $F=2.5$ and slowly tune the pump laser frequency on
resonance, starting from the blue-detuned side. This procedure yields a
soliton microcomb propagating in the forward direction, which we characterize
by measuring the optical spectrum of the forward (solid blue line) and
backward (solid teal line) outputs from the device. Comparisons to our
simulation (orange line in Figure 1e) show a good match between the predicted
forward comb and the experiment, confirming the close connection between the
designed bandgaps, our mode-locked state engineering with
$\tilde{D}_{\mathrm{int}}(\mu)$, and the operational generation of solitons in
such devices. As we predict from $\tilde{D}_{\mathrm{int}}(\mu)$ in Figure
3(c), the larger bandgap at $\mu=1$ than $\mu=-1$ leads to
$\delta\omega_{\mathrm{rep}}>0$, so there are more modes with positive
$\Delta_{\mu}$ for $\mu<0$ than $\mu>0$, leading to an asymmetric comb with a
flatter comb profile for $\mu<0$, as shown in the forward combs in Figure
3(e). Indeed, we measured $\delta\omega_{\mathrm{rep}}/2\pi$ to be 21 MHz,
consistent with our prediction that $\delta\omega_{\mathrm{rep}}>0$ in this
device. Regarding the measured spectra and simulation in the backward
direction, chip facet reflections obscure the observation of the entire
backward spectrum. Nonetheless, characteristic spectral peaks in the modes
$\pm 1$ are prominently evident. We obtain a conversion efficiency from pump
to comb of 37 in the forward direction and a 15 conversion efficiency in the
backward direction, which underscores the fundamental benefits of bandgap-
detuned excitation. This fraction of backscattering enables mode-locked-state
engineering with the designed bandgaps, creating the forward propagating dark
soliton. Figure 3(g) shows the simulated spectra, focusing on the first few
lines, upon scanning the detuning $\alpha$. It verifies our prediction that
the comb lines at $\mu_{s}=\pm 1$ are generated first and then trigger the
development of the entire comb, which is different from traditional soliton
generation where the primary combs are formed away from the pump.

Figure 4: Photonic-crystal ring resonator with (a) $\mu_{s}=\pm 3$. The OPOs
and soliton crystals generated with this design are shown in (c). (b)
$\mu_{s}=\pm 1,2,...,10$. The bright soliton generated with this design is
shown in (d). From left to right: Phase matching diagram; pulse shape; SEM
images; measured $D_{\mathrm{int}}$. (c) Spectra and pulses for design in (a)
with (i) low and (ii) high detuning. (d) Spectra and pulses for design in (b).
For (c) and (d), forward ($|A_{\mu}|^{2}$) and backward ($|B_{\mu}|^{2}$)
spectra are plotted in solid lines, and orange dashed lines are simulation
results. The forward and backward pulses are plotted in the same color as the
spectra in the same direction.

Expanding on this framework for mode-locked-state engineering of dark
solitons, we explore other states by way of design with
$\tilde{D}_{\mathrm{int}}$. Important examples include soliton crystals [21]
and bright pulses in normal dispersion [28]. To create a soliton crystal in
the bandgap-detuned regime, we place bandgaps at $\mu_{s}=\pm 3$. Figure 4(a)
shows the phase-matching diagram with the dispersion of the designed resonator
and the simulated $\tilde{D}_{\mathrm{int}}$ when soliton crystals are
generated. We show an exaggerated PhCR nanostructure with high-resolution SEM,
and we characterize $D_{\rm{int}}$ with a calibrated laser frequency scan. To
generate soliton crystal microcombs, we pump the device at the level of
$F=2.4$ by scanning the laser frequency into the mode $\mu=0$. Figure 4(b)
shows spectra and soliton-crystal formation at an initial detuning setting (i)
and then at a different value of $\alpha$ (ii) where the laser frequency was
slightly reduced. Because phase matching is designed to occur on modes
$\mu_{s}=\pm 3$, the initial modulation pattern (i) is composed of three
periods in the resonator, leading to the formation of a soliton crystal with
three pulses of equal temporal space in the resonator [21]. When the laser
frequency is decreased, the pulse spacing is perturbed, leading to the
appearance of natively spaced lines in the spectra (see panel (ii)), as
observed in soliton crystal combs with a defect [21]. We attribute this
phenomenon to the interaction between the oscillating tails of the switching
waves that form dark pulses [37].

We can form bright pulses in the bandgap-detuned regime by splitting many
modes of the PhCR except for the pump mode; as shown in Figure 4(c). These
split modes individually control the relative amplitude of $A_{\mu}$ and
$B_{\mu}$, creating the dispersion profile needed for bright solitons and
controlling the excess loss to coherent backscattering. We study an example in
which twenty modes $|\mu_{s}|\leq 10$ except the pump are split by an equal
amount to create a so-called meta-dispersion in the mode spectrum [18].
Despite the large number of split modes, the principle for phase matching
remains the same, and more phase matching pathways are enabled. At the
parametric oscillation threshold, the entire higher frequency branch of the
$D_{\rm{int}}$ spectrum is shifted to lower frequency due to the Kerr effect.
Therefore, a soliton forms as the pump mode and the blue-shifted branch align,
creating a $\tilde{D}_{\mathrm{int}}$ profile that is a straight line. The
associated photonic crystals form a more intricate corrugated pattern in the
ring resonator as shown in the SEM image in Figure 4(c). The measured
dispersion verifies that the targeted mode structure is implemented correctly
and that the pump mode is not split. Interestingly, this split mode
distribution consistently creates a single pulse in the ring resonator. Figure
4(d) presents the spectra from simulation and experiments, which allow us to
characterize the generated state with respect to the
$\tilde{D}_{\mathrm{int}}$ design. From the spectral data, we find that the
forward efficiency is 24 and the backward efficiency is 14, resulting from the
more equal ratio of forward and backward emission of this design. Despite an
increased fraction of the comb power emitted in the backward direction because
multiple modes are split, more power accumulates in the modes $|\mu|\leq 10$
and create a bright pulse within this PhCR. The comb also features a
relatively flat spectral intensity profile and the distribution of several
split modes about the pump mitigates imbalance in the bandgaps of fabricated
devices, creating a symmetrical soliton microcomb profile.

In summary, we have explored the bandgap-detuned excitation regime of PhCRs,
demonstrating modelocked state engineering by design of modal Kerr shifts. The
bandgap-detuned regime supports a rich state space of unique OPOs and soliton
microcombs that offer low threshold power, high efficiency, and control over
forward and backward emission. In exchange for controlled loss directed
backward, the bandgaps on the side modes facilitate effective phase matching
in the forward direction, optimizing efficiency. Notably, the forward
efficiency of OPOs reaches 48, while that of the dark soliton achieves 37.
Although bright solitons exhibit a slightly lower forward efficiency (24) due
to the splitting of more modes, this regime unveils new possibilities for comb
states. The bandgap-detuned excitation regime exhibits significant promise in
realizing the application potential of OPOs and microcombs, spanning areas
such as spectroscopy, telecommunications, computing, and quantum sensing.

## Acknowledgements

We thank Grant Brodnik and Lindell Williams for reviewing the manuscript. This
work is funded by the DARPA PIPES program HR0011-19-2-0016, the DARPA QuICC
program FA8750-23-C-0539, the DARPA NaPSAC program, AFOSR FA9550-20-1-0004
Project Number 19RT1019, NSF Quantum Leap Challenge Institute Award OMA -
2016244, and NIST. This work is a contribution of NIST and not subject to US
copyright. Mention of specific companies or trade names is for scientific
communication only and does not constitute an endorsement by NIST.

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