# Curvature-induced moduli stabilization

Flavio Tonioni<EMAIL_ADDRESS>Instituut voor Theoretische Fysica,
KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

###### Abstract

We discuss FLRW-cosmologies with negatively-curved spatial slices that induce
a late-time stabilization of all the moduli of an effective field theory
through Hubble friction, independently of the moduli-space curvature. This
holds for pure moduli, which appear ubiquitously in string compactifications.
Crucially, the cosmological solutions approach a Milne universe. Hence, the
mechanism we describe in fact corresponds to late-time moduli stabilization in
Minkowski spacetime.

##  I Introduction

In string compactifications, moduli stabilization typically requires extremely
hard calculations, and it can face deep control issues. Among others, the
Dine-Seiberg problem represents a general argument as to why this is the case
[1], relating to the fact that all string-theoretic potentials are generated
dynamically, with perturbative expansions that also control the validity of
the effective field theory (EFT) itself. In fact, proposed mechanisms that are
argued to allow for lower-dimensional vacua with Minkowski or de Sitter (dS)
geometries require a delicate interplay of (non)perturbative effects such that
the moduli are stabilized somewhere in between the bulk and the asymptotic
regions [2, 3]; for recent reviews, see refs. [4, 5]. Working in intermediate
regions, however, confronts one with the issue of controlling all corrections:
see ref. [6] for a recent account. Arguments based on the completion of an EFT
into a full quantum-gravity framework, together with circumstantial evidence,
actually led some to hypothesize the absence of potentials able to stabilize
all moduli in asymptotic regions of moduli space of string compactifications
as a general fact, at least if the potential is positive [7, 8]; for a review
of the criticisms to dS constructions in string compactifications, see ref.
[9].

In this note, we describe a general dynamical mechanism by which Friedmann-
Lemaître-Robertson-Walker (FLRW) cosmologies with negatively-curved spatial
slices automatically stabilize all moduli, at late-enough times. Here, by
“moduli”, we truly mean (pseudo)scalar fields with no potential. The intuition
behind their stabilization is extremely simple. On the one hand, the curvature
term is a fluid that saturates the strong energy condition (SEC). On the other
hand, pure moduli just represent a kinating fluid, whose energy density decays
more quickly than a SEC-saturating fluid. Therefore, given enough time, only
the energy density of the curvature fluid survives. Relative to the latter,
the moduli kinetic energy must vanish. It is in this sense that in the present
work we intend the “stabilization”. The dynamical mechanism is independent of
the moduli-space curvature and of the details of the underlying string
compactification. A universe whose only component is its own negative spatial
curvature, known as the Milne universe, is actually the interior of a future
lightcone in Minkowski geometry, in suitable coordinates. Hence, what we
achieve is a dynamical method to stabilize all moduli in Minkowski spacetime.
This mechanism works exactly in the absence of a potential, thus circumventing
all arguments related to the large slope and/or convexity of scalar potentials
in string compactifications. It also evades the claim that 10d supergravity
solutions compactified to 4d Minkowski always admit a 4d massless scalar [10].

The rest of this note is organized as follows. In sec. II, we present the
details of the curvature-induced stabilization mechanism. In sec. III, we
discuss a simple example. Finally, a general discussion is in sec. IV.
Additional mathematical details are in app. A.

##  II Dynamical moduli stabilization

Throughout this note, we study a set of $n$ moduli $\varphi^{a}$ that are
associated to a moduli-space metric

$\mathrm{d}s_{n}^{2}=G_{ab}(\varphi)\,\mathrm{d}\varphi^{a}\mathrm{d}\varphi^{b},$
(1)

with a corresponding Christoffel connection ${\Gamma^{a}}_{bc}$; Einstein
summation is always implied on the field-space indices $a=1,\dots,n$. Curved
moduli spaces emerge ubiquitously in string compactifications and in
supergravity constructions. Let the non-compact $d$-dimensional spacetime be
described by the FLRW-metric

$d\tilde{s}_{1,d-1}^{2}=-\mathrm{d}t^{2}+a^{2}(t)\,\biggl{[}\dfrac{\mathrm{d}r^{2}}{1-kr^{2}}+r^{2}\mathrm{d}\Omega_{d-2}^{2}\biggr{]}.$
(2)

Here, $a$ is the scale factor, which defines the Hubble parameter
$\smash{H=\dot{a}/a}$, and $k$ is the constant curvature of the spatial slice.
Finally, we consider the presence of a set of homogeneous and isotropic fluids
of energy densities $\rho^{\alpha}$ and pressures
$p^{\alpha}=(q^{\alpha}-1)\rho^{\alpha}$, for $\alpha=1,\dots,b$. A barotropic
fluid has an equation of state $p=(q-1)\rho$, where $q$ is a constant in a
multitude of cases including curvature anisotropies, radiation, matter, and
cosmological constants, with $q=2,1/(d-1),1,0$, respectively.

In this setup, the complete cosmological equations can be written as

$\displaystyle\ddot{\varphi}^{a}+{\Gamma^{a}}_{bc}\dot{\varphi}^{b}\dot{\varphi}^{c}+(d-1)H\dot{\varphi}^{a}=0,$
(3a) $\displaystyle\dot{\rho}^{\alpha}+(d-1)H(p^{\alpha}+\rho^{\alpha})=0,$
(3b) $\displaystyle
H^{2}=\dfrac{2\kappa_{d}^{2}}{(d-1)(d-2)}\biggl{[}\dfrac{1}{2}\,G_{ab}\dot{\varphi}^{a}\dot{\varphi}^{b}+\sum_{\alpha=1}^{b}\rho^{\alpha}\biggr{]}.$
(3c)

A combination of eqs. (3a, 3b, 3c) also gives the useful relationship

$\dot{H}=-\dfrac{\kappa_{d}^{2}}{d-2}\,\biggl{[}G_{ab}\dot{\varphi}^{a}\dot{\varphi}^{b}+\sum_{\alpha=1}^{b}(p^{\alpha}+\rho^{\alpha})\biggr{]}.$
(4)

The continuity eqs. (3b) for constant values of $q^{\alpha}$ can be solved as
$\smash{\rho^{\alpha}=\rho^{\alpha}_{0}\,(a_{0}/a)^{(d-1)q^{\alpha}}}$, where
$a_{0}=a(t_{0})$ is the scale factor at an initial time $t_{0}$. Hence, the
curvature term can be formally treated as an on-shell fluid with energy
density $\rho_{k}=-[(d-1)(d-2)/(2\kappa_{d}^{2})]\,k/a^{2}$ and equation of
state given by the parameter

$q_{k}=1-\dfrac{d-3}{d-1}.$ (5)

Noticeably, this means that the negative-curvature fluid, i.e. the one with
$k=-1/\ell^{2}$, for an arbitrary length (or time) scale $\ell$, saturates the
SEC.

###  II.1 Milne universe

In the presence of a potential $V=V(\varphi)$ for the fields, eqs. (3c, 4) do
not allow one to express the kinetic energy
$\smash{T=G_{ab}\,\dot{\varphi}^{a}\dot{\varphi}^{b}/2}$ in terms of the
Hubble parameter and of the fluid energy densities. In the Friedmann eqs. (3c,
4), one generally finds the linear combinations $T+V$ and $T-V$, respectively.
However, if $V=0$, then $T$ is a variable that drops out of the equation
governing the evolution of the scale factor.

In fact, a simple manipulation leads to the equation

$\dfrac{(d-1)(d-2)}{2\kappa_{d}^{2}}H^{2}+\dfrac{(d-2)}{2\kappa_{d}^{2}}\dot{H}=\sum_{\alpha}\dfrac{2-q^{\alpha}}{2}\rho^{\alpha}.$
(6)

The $a$-dependence of the fluid energy densities is known. Hence, the above is
a second-order ordinary differential equation (ODE). If the only fluid is
negative FLRW-curvature, i.e. $k=-1/\ell^{2}$, the equation takes the form

$H^{2}+\dfrac{1}{d-1}\dot{H}=\dfrac{d-2}{d-1}\dfrac{1}{\ell^{2}a^{2}}.$ (7)

One can find the analytic solution to this equation: details are in app. A;
for the case where the only fluid is the negative FLRW-metric spatial
curvature, a previous derivation of such solution appeared in ref. [11].

Here, we can shortly illustrate a way in which such a solution can be found.
Let us reparameterize the time $t$ in terms of a time coordinate $\sigma$
defined via [11]

$\mathrm{d}t^{2}=\dfrac{\mathrm{d}\sigma^{2}}{\displaystyle\dfrac{\sigma^{2}}{\sigma_{p}^{2}}\,\biggl{(}\dfrac{\ell}{\sigma}\biggr{)}^{2(d-1)}+1},$
(8)

for a constant $\sigma_{p}\in\mathbb{R}$. Then, eq. (6) is solved by

$a(\sigma)=\dfrac{\sigma}{\ell}.$ (9)

The fact that this is the only solution is discussed in more detail in app. A.
In particular, we observe that at late-enough times it is possible to
approximate $t(\sigma\sim\infty)\simeq\sigma$.

The main observation of the present note consists in the fact that, in terms
of the cosmological time $t$, the late-time behavior of the scale factor is
such that

$a_{\infty}(t)=\dfrac{t}{\ell},$ (10)

which corresponds to a Milne universe. Therefore, by plugging this solution
into either eq. (3c) or eq. (4), one finds the requirement that

$\varphi_{\infty}^{a}(t)=\varphi^{a}_{\infty},$ (11)

where $\smash{\varphi^{a}_{\infty}\in\mathbb{R}}$ are constants. One can
immediately see that eq. (11) is a solution of eq. (3a). For generic initial
conditions, this solution is technically correct in the true asymptotic regime
$t\sim\infty$, where the scale factor takes exactly the form in eq. (10). One
might fear that, before their kinetic energy dies off, the scalars might
travel a long distance. Actually, by an asymptotic expansion of the exact
solution, one can verify that the scalar-field displacement falls off in a
power-law form, with a time scale set by $\ell$. Hence, the field displacement
is finite.

We can interpret the vanishing of the moduli kinetic energy as being due to
Hubble friction. To see this, we can observe that pure moduli represent a
perfectly stiff fluid, with a parameter

$q_{\varphi}=2.$ (12)

It is indeed trivial to verify that, in the absence of a potential, the scalar
field equation implies the continuity equation for a fluid with
$\rho_{\varphi}=p_{\varphi}=G_{ab}\,\dot{\varphi}^{a}\dot{\varphi}^{b}/2$.
Comparing with eq. (5), the energy densities associated to the negative
spatial FLRW-curvature and the moduli fall off with the scale factor as

$\displaystyle\rho_{k}$
$\displaystyle=\rho_{k0}\,\Bigl{(}\dfrac{a_{0}}{a}\Bigr{)}^{\\!2},$ (13a)
$\displaystyle\rho_{\varphi}$ $\displaystyle=\rho_{\varphi
0}\,\Bigl{(}\dfrac{a_{0}}{a}\Bigr{)}^{\\!2(d-1)}.$ (13b)

Hence, by a reasoning that is typical in cosmology, eventually the fluid with
with the lowest $q$-parameter dominates over the others. One can also check
that the scale factor in eq. (10) determines a late-time Hubble parameter
$H_{\infty}(t)=1/t$. Hence, eq. (3c) requires
$\smash{\dot{\varphi}_{\infty}^{a}=0}$. We stress that none of these
conclusions depends on the curvature of the moduli space.

If there are multiple fluids, the individual energy densities fall off more
quickly for larger $q$-parameters. Hence, after long enough, only the fluid
with the smallest value of $q$ survives. Therefore, the mechanism we presented
keeps working even if we include any number of SEC-fulfilling fluids, such as
radiation, matter and curvature anisotropies.

###  II.2 Minkowski spacetime

As is well known, the Milne universe coincides with the interior of a future
lightcone in Minkowski spacetime [12]. To see this, we can define the radial
coordinate

$\chi(r)=\mathrm{artanh}\,\biggl{[}\dfrac{r}{\sqrt{r^{2}+\ell^{2}}}\biggr{]},$
(14)

which is such that
$\smash{\mathrm{d}\chi^{2}=\mathrm{d}r^{2}/(r^{2}+\ell^{2})}$. With this
coordinate choice, if the scale factor is $a(t)=t/\ell$, the metric in eq. (2)
reads

$d\tilde{s}_{1,d-1}^{2}=-\mathrm{d}t^{2}+t^{2}\Bigl{[}\mathrm{d}\chi^{2}+\mathrm{sinh}^{2}\,\chi\,\mathrm{d}\Omega_{d-2}^{2}\Bigr{]}.$
(15)

Now, we can write the Minkowski metric as

$ds_{\mathbb{M}^{1,d-1}}^{2}=-\mathrm{d}\tau^{2}+\mathrm{d}\rho^{2}+\rho^{2}\,\mathrm{d}\Omega_{d-2}^{2}.$
(16)

By the coordinate transformation

$\displaystyle\tau$ $\displaystyle=t\,\mathrm{cosh}\,\chi,$ (17a)
$\displaystyle\rho$ $\displaystyle=t\,\mathrm{sinh}\,\chi,$ (17b)

we recover exactly the negatively-curved FLRW-metric with $a(t)=t/\ell$.
Crucially, the late-time regime $t\sim\infty$ in the Milne universe is a truly
late-time regime in Minkowski time $\tau\sim\infty$, at any radial distance
$\chi$ from the coordinate origin.

Solutions where the late-time scale factor is linear in cosmological time may
be expected in FLRW-universes with negative spatial curvature, as the latter
saturates the SEC. However, such solutions are not necessarily Milne
universes, and hence not Minkowski spacetimes after a coordinate redefinition.
To see this, let

$a(t)=\alpha\,\dfrac{t}{\ell},$ (18)

where $k=-1/\ell^{2}$. Then, the Ricci scalar for the metric in eq. (2) reads

$\tilde{R}=\dfrac{(d-1)(d-2)}{t^{2}}\,\biggl{(}1-\dfrac{1}{\alpha^{2}}\biggr{)},$
(19)

which is clearly non-zero for any value except $\alpha=1$. It is this specific
value that corresponds to a Milne universe. Scalar fields tend to have a
“tracker behavior”: their $q$-parameter is not constant and in many known
examples it tracks down the $q$-value of other fluids in the theory, at late
times [13]. For instance, a single scalar $\phi$ with a potential
$V=\Lambda\,\mathrm{e}^{-\kappa_{d}\gamma\phi}$, for a constant $\gamma$,
admits a solution $a(t)=(1/\sqrt{1-4/[(d-2)\gamma^{2}]})\,t/\ell$ [14, 15]. It
is only in the limit $\gamma\to\infty$ that one recovers a Milne universe,
which indeed corresponds to $V\to 0$. This can be understood by noticing that
it is the scalar potential that induces the tracker behavior: in the absence
of a potential, a set of scalars is instead a purely kinating fluid, with
constant parameter $q_{\varphi}=2$.

###  II.3 Cosmic deceleration

One might wonder whether the late-time Minkowski geometry can be attained
along a trajectory undergoing accelerated expansion, in the FLRW-formulation.
Unfortunately, here we show that this is not the case.

Given an FLRW-metric as in eq. (2), cosmic acceleration is defined as the set
of conditions $\dot{a}>0$ and $\ddot{a}>0$. In particular, the condition on
the second time derivative is equivalent to the condition that
$\epsilon=-\dot{H}/H^{2}<1$. In terms of the time coordinate $\sigma$ defined
in eq. (8), we can express the $\epsilon$-parameter as

$\epsilon(t)=1+\dfrac{d-2}{1+\Bigl{(}\dfrac{\sigma(t)}{\ell}\Bigr{)}^{\\!2(d-2)}}.$
(20)

This means that we have $\epsilon(t)>1$ at all times $t$, and in particular
that $\smash{\lim_{t\to\infty}\,\epsilon(t)=1^{+}}$. Hence, in the FLRW-
formulation, the expansion is always decelerated. This does not necessarily
represent a problem for exploiting the stabilization mechanism into string
embeddings of cosmic inflation, since the moduli might only need to be
stabilized after the inflationary phase. On the other hand, it does not
provide a viable way for getting a transient quasi-dS expansion at late times.

##  III Example: GKP compactifications

A protypical example of a string compactifcation down to a Minkowski geometry
with a set of unstabilized moduli is the class of Giddings-Kachru-Polchinski
(GKP) backgrounds [16]. These are type-IIB compactifications on a Calabi-Yau
orientifold $\mathrm{CY}^{\prime}{\\!\\!}_{3}$ with RR-3- and NSNS-3-form
fluxes in which the Kähler sector happens to be flat in all directions. Here
we illustrate how the dynamical stabilization of the latter can work.

As a compactification Ansatz, the 10d Einstein-frame line element
$ds_{10}^{2}=\hat{g}_{MN}\,\mathrm{d}x^{M}\mathrm{d}x^{N}$ is taken to be

$ds_{1,9}^{2}=\mathrm{e}^{2A(y)}\,\tilde{ds}_{1,3}^{2}(x)+\mathrm{e}^{-2A(y)}\,\breve{ds}_{6}^{2}(y),$
(21)

where the function $\mathrm{e}^{2A}$ is the so-called warp factor. The blocks
$\tilde{g}_{\mu\nu}(x)$ and $\breve{g}_{mn}(y)$ are a 4d metric depending only
on the 4d coordinates $x^{\mu}$ and the metric of a 6d Calabi-Yau orientifold
depending on the 6d internal coordinates $y^{m}$, respectively. Due to Lorentz
invariance, the background axio-dilaton $\tau$ varies only over the internal
manifold, i.e. $\smash{\tau=\tau(y)}$. Similarly, the complex 3-form flux
$G_{3}=F_{3}-\tau H_{3}$ only has internal indices, i.e.
$\smash{G_{3}=G_{mnl}\,\mathrm{d}y^{m}\wedge\mathrm{d}y^{n}\wedge\mathrm{d}y^{l}/3!}$,
and the self-dual 5-form flux $\tilde{F}_{5}$ is of the form
$\smash{\tilde{F}_{5}=(1+\hat{*}_{10})\,\mathrm{d}\alpha\wedge\tilde{\mathrm{vol}}_{1,3}}$,
for a real scalar $\alpha=\alpha(y)$. Finally, the theory includes a set of
localized sources such as D- and anti-D-branes as well as O- and anti-O-
planes, compatibly with the Ansatz metric.

The 4d components of Einstein’s equations and the 5-form flux field equation
can be combined in an enlightening way, giving [17]

$\begin{split}\breve{\nabla}^{2}\Phi_{-}=\tilde{R}_{4}&+\dfrac{(\Phi_{+}+\Phi_{-})^{2}}{4\,\mathrm{Im}\,\tau}\,G{}_{3}^{-}\,\breve{\cdot}\,\overline{G}{}_{3}^{-}\\\
&+\dfrac{2}{\Phi_{+}+\Phi_{-}}\,\partial\Phi_{-}\breve{\cdot}\,\partial\Phi_{-}+T_{4,6},\end{split}$
(22)

where $\smash{G_{3}^{-}=[(\breve{*}_{6}G_{3})-\mathrm{i}G_{3}]/2}$,
$\smash{\Phi_{\pm}=\mathrm{e}^{4A}\pm\alpha}$, and $\smash{T_{4,6}}$ is a
combination of the energy-momentum tensor of the localized sources, which
vanishes e.g. for appropriate D3-brane/O3-plane configurations; we use the
notation
$\smash{(A_{q}\breve{\cdot}B_{q})\,\breve{\mathrm{vol}}_{n}=A_{q}\wedge\breve{*}_{n}B_{q}}$,
on an $n$-dimensional space. The left handside integrates to zero over a
compact manifold, so the same must happen with the right-handside. In a 4d
Minkowski spacetime, with $\smash{\tilde{R}_{4}=0}$, each term in the right
handside must integrate to zero, too, which implies the conditions

$\displaystyle\breve{*}_{6}G_{3}$ $\displaystyle=\mathrm{i}G_{3},$ (23a)
$\displaystyle\mathrm{e}^{4A}$ $\displaystyle=\alpha.$ (23b)

Such conditions can also solve the remaining field equations. In terms of a 4d
effective theory, such fluxes stabilize the axio-dilaton and the complex-
structure moduli. However, the Kähler moduli are purely flat directions. For
instance, the overall volume is controlled by the imaginary part of the chiral
supermultiplet $\rho=\theta+\mathrm{i}\,\xi$, whose field-space metric block
can be read off the Kähler potential
$\kappa_{4}^{2}K=-3\,\mathrm{ln}\,[-\mathrm{i}(\rho-\overline{\rho})]$. The
mechanism we presented above stabilize both fields $\theta$ and $\xi$ to
finite values, while solving the 10d equations of motion.

##  IV Discussion

In this note, we have described a generic framework in which all the moduli of
a given EFT are dynamically stabilized into constant finite values in a
Minkowski geometry. The process takes place independently of the moduli-space
curvature and of any possible compactification details, such as the internal
geometry. Strictly speaking, the stabilization mechanism takes place in the
asymptotic future. However, throughout the evolution, the kinetic energy of
the fields becomes parametrically smaller and smaller compared to the energy
set by the Hubble scale, in an associated formulation in terms of a FLRW-
cosmology with negative spatial curvature. The Hubble-sized energy is stored
in the spatial FLRW-curvature.

It is of course not surprising that fluids (help) freeze the scalar kinetic
energy, relative to the Hubble scale. For instance, this is an integral
element in the recent studies of kination phases in early moduli-driven
stringy cosmologies, where radiation terminates the kination epoch [18, 19].
Hubble damping generally helps overcome the overshoot problem [20]. In fact,
such mechanisms have been exploited extensively in the literature: see e.g.
refs. [21, 22, 23, 24]. What is special about negative spatial FLRW-curvature,
however, is the fact that asymptotically it gives a perfect Milne universe,
which is actually the Minkowski spacetime. Hence, not only does the scalar
kinetic energy vanish, compared to the overall energy scale of the problem
(the Hubble scale, in a cosmological setting), but it does so into a Minkowski
vacuum (and not necessarily in a cosmological setting).

The setup for the result is that of classical field theory. The final state
that one finds is a Minkowski vacuum, with all the scalars fixed at a finite
value. However, such scalars have no Lagrangian mass. Hence, one should expect
that their fluctuations mediate long-range forces, and possibly the time
dependence of fundamental constants; see e.g. ref. [25] for an account.
Looking at the problem from a different angle, the stabilization mechanism
that we have presented is generally unstable under the presence of scalar
potentials. At some point, as the Hubble energy asymptotes to zero, any scalar
potentials (that perhaps one may have consistently neglected at higher
energies) might become relevant, thus changing the dynamics of the problem.
Nonetheless, if such potentials are irrelevant at least initially, the
dynamics that we described might at least serve as an explanation as to why
the field kinetic energy is small, and thus also circumvent the discussion on
long-range forces. This might be of use in constructions that require some
degree of fine tuning of the initial conditions to accommodate for
realizations of dark energy, such as in refs. [26, 27]. We leave studies of
other phenomenological questions such as the cosmological moduli problem [28,
29] for future work.

Although with a number of caveats, the instance of a dynamical moduli
stabilization into a Minkowski vacuum presented in this note is applicable to
string compactifications in generality. Recently, Minkowski vacua with no
running moduli have also been obtained in non-geometric backgrounds that are
mirror duals of rigid Calabi-Yau manifolds [30, 31].

###### Acknowledgements.

### Acknowledgments

I am grateful to Thomas Van Riet and Timm Wrase for inspirational
conversations and for their precious feedback on the manuscript. My work is
supported by the FWO Odysseus grant GCD-D0989-G0F9516N.

##  Appendix A Exact solution

A solution $a=a(t)$ to eq. (7) can be found analytically; the overall sign is
arbitrary, but we focus on solutions with a positive scale factor $a>0$. It is
convenient to express it as the inverse of a function $b$, i.e.

$a(t)=b^{-1}(t),$ (24)

which reads

$b(t)=\ell
t\,{}_{2}F_{1}\biggl{[}\dfrac{1}{2},-\dfrac{1}{2}\dfrac{1}{d-2},\dfrac{1}{2}\dfrac{2d-5}{d-2},\dfrac{p}{t^{2(d-2)}}\biggr{]},$
(25)

where $\smash{{}_{2}F_{1}={}_{2}F_{1}(a,b,c;z)}$ is the ordinary
hypergeometric function and $p\in\mathbb{R}$ is an integration constant. An
initial time gauge has been chosen for brevity. Although this expression is
complicated, it admits an extremely simple late-time behavior. For simplicity,
let $\smash{p=s\,t_{p}^{2(d-2)}}$, with $s=\pm 1$ and $t_{p}>0$. An asymptotic
expansion then reveals that

$\displaystyle b(t)=\ell
t\biggl{[}1-\dfrac{1}{2}\dfrac{s}{2d-5}\,\biggl{(}\dfrac{t_{p}}{t}\biggr{)}^{\\!\\!2(d-1)}+O\biggl{(}\dfrac{t_{p}}{t}\biggr{)}^{\\!\\!4(d-1)}\biggr{]}.$

At leading order in $t_{p}/t$, we may approximate the function $b$ with the
function $b_{\infty}(t)=\ell t$, which leads immediately to the late-time
scale factor in eq. (10). Working at next-to-leading order is harder, but the
expression of $b(t)$ motivates the expansion

$a(t)=\dfrac{t}{\ell}\biggl{[}1-q\,\biggl{(}\dfrac{t_{p}}{t}\biggr{)}^{\\!\\!m}+O\biggl{(}\dfrac{t_{p}}{t}\biggr{)}^{\\!\\!m_{1}}\biggr{]},$
(26)

for a constant $q\in\mathbb{R}$ that is yet undetermined and two positive
constants $m>m_{1}>0$. Plugging this back into eq. (7) indicates that one must
fix $m=2(d-2)$. Plugging this solution into eq. (3c), one finds the late-time
behavior

$\displaystyle
T(t)=\dfrac{(d-2)q_{d}}{\kappa_{d}^{2}t_{p}^{2}}\,\biggl{(}\dfrac{t_{p}}{t}\biggr{)}^{\\!\\!2(d-1)}\,\biggl{[}1+O\biggl{(}\dfrac{t_{p}}{t}\biggr{)}^{\\!\\!m_{2}}\biggr{]},$

for some positive constant $m_{2}>0$, and where we defined
$q_{d}=(d-1)(2d-5)\,q$ for convenience. This means that one must have $q>0$;
the same expression is confirmed by plugging the scale factor into eq. (4).
One can also check that we may write
$\smash{T=T_{0}\,(a_{0}/a)^{2(d-1)}\,\bigl{[}1+O(a_{0}/a)^{m_{0}}\bigr{]}}$,
for some constant $m_{0}>0$ and some reference values $a_{0}$ and $T_{0}$, in
agreement with the fact that the moduli represent a stiff fluid (see eq.
(13b)). For instance, in the case of a single canonical field, where
$T=\dot{\varphi}^{2}/2$, one finds

$\varphi(t)=\varphi_{\infty}-\dfrac{\sqrt{2q_{d}}}{\kappa_{d}\sqrt{d-2}}\,\biggl{(}\dfrac{t_{p}}{t}\biggr{)}^{\\!\\!d-2}\,\biggl{[}1+O\biggl{(}\dfrac{t_{p}}{t}\biggr{)}^{\\!\\!m_{3}}\biggr{]},$
(27)

up to a sign, where $\varphi_{\infty}$ is an integration constant and
$m_{3}>0$ is another positive constant. The integrated scalar field excursion
from an initial time $t_{0}$ until infinity, defined as
$\smash{\delta=\kappa_{d}\int_{t_{0}}^{\infty}\mathrm{d}t\;\varphi(t)}$, is

$\displaystyle\delta=\dfrac{t_{0}}{d-3}\,\dfrac{\sqrt{q_{d}}}{\sqrt{d-2}}\,\biggl{(}\dfrac{t_{p}}{t_{0}}\biggr{)}^{\\!\\!d-2}\biggl{[}1+O\biggl{(}\dfrac{t_{p}}{t_{0}}\biggr{)}^{\\!\\!m_{3}}\biggr{]},$

if $d>3$. One has $\delta=\infty$ if $d=3$.

The solution in eqs. (24, 25) was first derived in ref. [11]. Here we revisit
this derivation to provide further understanding of the results in the main
text. Let the spacetime metric be parameterized as

$d\tilde{s}_{1,d-1}^{2}=-f^{2}(\sigma)\,\mathrm{d}\sigma^{2}+a^{2}(\sigma)\,h_{ij}(x)\,\mathrm{d}x^{i}\mathrm{d}x^{j},$
(28)

where $h_{ij}(x)$ is the metric over a maximally-symmetric $(d-1)$-dimensional
space $\mathbb{X}^{d-1}$ with constant curvature $k$, for $i,j=1,\dots,d-1$.
Let $\varphi^{a}=\varphi^{a}(\sigma)$ be a set of scalars. In terms of the
time variable $\omega$, defined through

$\dfrac{\mathrm{d}\omega}{\mathrm{d}\sigma}(\sigma)=\dfrac{f(\sigma)}{a^{d-1}(\sigma)},$
(29)

the action for the scalars takes the form

$S[\varphi]=\dfrac{1}{2}\,\int\mathrm{d}\omega\int_{\mathbb{X}^{d-1}}\mathrm{d}^{d-1}x\;G_{ab}(\varphi)\,\dfrac{\mathrm{d}\varphi^{a}}{\mathrm{d}\omega}\dfrac{\mathrm{d}\varphi^{b}}{\mathrm{d}\omega},$
(30)

which is formally independent of the metric. Hence, the scalars have trivial
dynamics, as they just trace out geodesics in moduli space. The Einstein
equations for the function $a(\sigma)$ then read

$\biggl{(}\dfrac{\mathrm{d}a}{\mathrm{d}\sigma}\biggr{)}^{2}=\biggl{[}\dfrac{\kappa_{d}^{2}\,p^{2}}{(d-1)(d-2)}\,\dfrac{1}{a^{2(d-2)}}-k\biggr{]}\,f^{2},$
(31)

where
$p^{2}=G_{ab}(\varphi)\,(\mathrm{d}\varphi^{a}/\mathrm{d}\omega)\,(\mathrm{d}\varphi^{b}/\mathrm{d}\omega)$
is a constant. This explains the disappearing of the scalar dynamics from the
equations governing the metric evolution that we observe in eq. (6). A general
solution takes the form

$a^{2}(\sigma)=\Bigl{(}\dfrac{\sigma}{\ell}\Bigr{)}^{\\!2},$ (32)

where

$f^{2}(\sigma)=\dfrac{1}{\ell^{2}}\,\dfrac{1}{\displaystyle\dfrac{\kappa_{d}^{2}\,p^{2}}{(d-1)(d-2)}\,\biggl{(}\dfrac{\ell}{\sigma}\biggr{)}^{2(d-2)}-k}.$
(33)

Let us reparameterize time as in eq. (2), namely through the definition
$\mathrm{d}t=f(\sigma)\,\mathrm{d}\sigma$. In case $k=-1/\ell^{2}$, we have
$a(\sigma(t))=a(t)$, with $a=a(t)$ in eqs. (24, 25), confirming the result. In
terms of the time coordinate $\sigma$, it is already clear that the late-time
behavior of the solution is a Milne universe. Indeed, we see the behavior
$\smash{f^{2}(\sigma\sim\infty)\simeq 1}$, which trivially reduces the metric
to the form in eq. (15). The late-time behavior is independent of the initial
conditions.

## References

  * Dine and Seiberg [1985] M. Dine and N. Seiberg, Phys. Lett. B 162, 299 (1985).
  * Kachru  _et al._ [2003] S. Kachru, R. Kallosh, A. D. Linde, and S. P. Trivedi, Phys. Rev. D68, 046005 (2003), arXiv:hep-th/0301240 [hep-th] .
  * Balasubramanian  _et al._ [2005] V. Balasubramanian, P. Berglund, J. P. Conlon, and F. Quevedo, JHEP 03, 007 (2005), arXiv:hep-th/0502058 [hep-th] .
  * Cicoli  _et al._ [2024] M. Cicoli, J. P. Conlon, A. Maharana, S. Parameswaran, F. Quevedo, and I. Zavala, Phys. Rept. 1059, 1 (2024), arXiv:2303.04819 [hep-th] .
  * McAllister and Quevedo [2023] L. McAllister and F. Quevedo, (2023), arXiv:2310.20559 [hep-th] .
  * McAllister  _et al._ [2024] L. McAllister, J. Moritz, R. Nally, and A. Schachner, (2024), arXiv:2406.13751 [hep-th] .
  * Obied  _et al._ [2018] G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa, (2018), arXiv:1806.08362 [hep-th] .
  * Ooguri  _et al._ [2019] H. Ooguri, E. Palti, G. Shiu, and C. Vafa, Phys. Lett. B 788, 180 (2019), arXiv:1810.05506 [hep-th] .
  * Danielsson and Van Riet [2018] U. H. Danielsson and T. Van Riet, Int. J. Mod. Phys. D 27, 1830007 (2018), arXiv:1804.01120 [hep-th] .
  * Andriot  _et al._ [2022] D. Andriot, L. Horer, and P. Marconnet, JHEP 08, 109 (2022), [Erratum: JHEP 09, 184 (2022)], arXiv:2204.05327 [hep-th] .
  * Bergshoeff  _et al._ [2009] E. Bergshoeff, W. Chemissany, A. Ploegh, M. Trigiante, and T. Van Riet, Nucl. Phys. B 812, 343 (2009), arXiv:0806.2310 [hep-th] .
  * Mukhanov [2005] V. Mukhanov,  _Physical Foundations of Cosmology_ (Cambridge University Press, Oxford, 2005).
  * Copeland  _et al._ [1998] E. J. Copeland, A. R. Liddle, and D. Wands, Phys. Rev. D 57, 4686 (1998), arXiv:gr-qc/9711068 .
  * Marconnet and Tsimpis [2023] P. Marconnet and D. Tsimpis, JHEP 01, 033 (2023), arXiv:2210.10813 [hep-th] .
  * Andriot  _et al._ [2023] D. Andriot, D. Tsimpis, and T. Wrase, Phys. Rev. D 108, 123515 (2023), arXiv:2309.03938 [hep-th] .
  * Giddings  _et al._ [2002] S. B. Giddings, S. Kachru, and J. Polchinski, Phys. Rev. D66, 106006 (2002), arXiv:hep-th/0105097 [hep-th] .
  * Baumann  _et al._ [2010] D. Baumann, A. Dymarsky, S. Kachru, I. R. Klebanov, and L. McAllister, JHEP 06, 072 (2010), arXiv:1001.5028 [hep-th] .
  * Conlon and Revello [2022] J. P. Conlon and F. Revello, JHEP 11, 155 (2022), arXiv:2207.00567 [hep-th] .
  * Apers  _et al._ [2024] F. Apers, J. P. Conlon, E. J. Copeland, M. Mosny, and F. Revello, (2024), arXiv:2401.04064 [hep-th] .
  * Brustein and Steinhardt [1993] R. Brustein and P. J. Steinhardt, Phys. Lett. B 302, 196 (1993), arXiv:hep-th/9212049 .
  * Barreiro  _et al._ [1998] T. Barreiro, B. de Carlos, and E. J. Copeland, Phys. Rev. D 58, 083513 (1998), arXiv:hep-th/9805005 .
  * Huey  _et al._ [2000] G. Huey, P. J. Steinhardt, B. A. Ovrut, and D. Waldram, Phys. Lett. B 476, 379 (2000), arXiv:hep-th/0001112 .
  * Brustein  _et al._ [2004] R. Brustein, S. P. de Alwis, and P. Martens, Phys. Rev. D 70, 126012 (2004), arXiv:hep-th/0408160 .
  * Conlon  _et al._ [2008] J. P. Conlon, R. Kallosh, A. D. Linde, and F. Quevedo, JCAP 09, 011 (2008), arXiv:0806.0809 [hep-th] .
  * Cicoli  _et al._ [2019] M. Cicoli, S. De Alwis, A. Maharana, F. Muia, and F. Quevedo, Fortsch. Phys. 67, 1800079 (2019), arXiv:1808.08967 [hep-th] .
  * Gomes  _et al._ [2024] J. M. Gomes, E. Hardy, and S. Parameswaran, Phys. Rev. D 110, 023533 (2024), arXiv:2311.08888 [hep-ph] .
  * Casas  _et al._ [2024] G. F. Casas, M. Montero, and I. Ruiz, (2024), arXiv:2406.07614 [hep-th] .
  * Banks  _et al._ [1994] T. Banks, D. B. Kaplan, and A. E. Nelson, Phys. Rev. D 49, 779 (1994), arXiv:hep-ph/9308292 .
  * de Carlos  _et al._ [1993] B. de Carlos, J. A. Casas, F. Quevedo, and E. Roulet, Phys. Lett. B 318, 447 (1993), arXiv:hep-ph/9308325 .
  * Rajaguru  _et al._ [2024] M. Rajaguru, A. Sengupta, and T. Wrase, (2024), arXiv:2407.16756 [hep-th] .
  * Becker  _et al._ [2024] K. Becker, N. Brady, M. Graña, M. Morros, A. Sengupta, and Q. You, (2024), arXiv:2407.16758 [hep-th] .


# Planarized THz quantum cascade lasers for broadband coherent photonics

Urban Senica1,∗ Andres Forrer1 Tudor Olariu1 Paolo Micheletti1 Sara Cibella2
Guido Torrioli2 Mattias Beck1 Jérôme Faist1 Giacomo Scalari1,∗

###### Abstract

Recently, there has been a growing interest in integrated THz photonics for
various applications in communications, spectroscopy and sensing. We present a
new integrated photonic platform based on active and passive elements
integrated in a double-metal, high confinement waveguide layout planarized
with a low-loss polymer. An extended top metallization results in low
waveguide losses and improved dispersion, thermal and RF properties, as it
enables to decouple the design of THz and microwave cavities. Free-running on-
chip quantum cascade laser combs spanning 800 GHz, harmonic states over 1.1
THz and RF-injected broadband incoherent states spanning over nearly 1.6 THz
are observed. With a strong external RF drive, actively mode-locked pulses as
short as 4.4 ps can be produced, as measured by SWIFTS. We demonstrate as well
passive waveguides with low insertion loss, enabling the tuning of the laser
cavity boundary conditions and the co-integration of active and passive
components on the same THz photonic chip.

Quantum Optoelectronics Group, Institute of Quantum Electronics, ETH Zürich,
8093 Zürich, Switzerland

Istituto di Fotonica e Nanotecnologie, CNR, Via del Fosso del Cavaliere 100,
00133 Rome, Italy

## Introduction

Integrated photonics [1] makes extensive use of on-chip optical elements such
as sources, splitters, modulators, and high-confinement waveguides embedded in
a planar platform to efficiently process and route optical signals. There is a
growing interest in integrated Mid-IR and THz photonics for telecommunications
and sensing [2, 3]. In the THz frequency range, a prominent candidate for
source integration is the THz quantum cascade laser [4]. Recent advances in
the high temperature operation of these devices, [5, 6] combined with their
frequency agility [7] and the possibility to operate as frequency combs [8, 9]
as well as very fast detectors [10] make them extremely appealing as key
building blocks for THz photonics.

Crucial features for laser integration [11] in more complex photonic systems
[12] are the reduction of the electrical consumption, and consequently of the
injected current, and the efficient coupling to low-loss passive waveguides.
Here, we propose a new platform for integrated THz photonics that allows
signal propagation with passive elements and coherent source integration for
applications such as broadband sensing [13] and coherent telecommunications
[14]. In this first demonstration, we leverage the presence of a common
metallic ground plane to demonstrate the integration of several active and
passive THz photonic components onto the same semiconductor platform, allowing
for efficient signal processing at THz and RF frequencies.

## Planarized waveguide platform

The basic building block is a high-performance planarized double-metal
waveguide with an extended top metallization, as shown in Fig. 1 on the left.
A similar kind of waveguide has already proven to be very effcient both for
THz and microwave applications [15]. Following a standard double metal
waveguide fabrication process [16] with dry-etched active region waveguides, a
microelectronic-grade low-loss polymer Benzocyclobutene (BCB) is spin-coated
and baked as the surrounding material (see Methods for details). The latter is
widely used in microelectronics and has already been successfully employed in
several THz applications [17, 18]. Specifically, we use Cyclotene 3022-57
(BCB), with a refractive index of 1.57 and relatively low losses (3 cm-1 = 1.3
dB/mm) at 3 THz [19], making it an ideal planarization material. To ensure a
flat and smooth profile, the BCB spin and bake step is repeated five times,
and the top surface is subsequently etched with RIE to the same height as the
active region waveguide. An extended top metallization with a typical width of
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is deposited over the active
region and the BCB-covered area on the sides, which offers several advantages,
as discussed below. From the active region standpoint, for these first
demonstrations we used a strongly diagonal, low-threshold broadband
GaAs/AlGaAs heterostructure, fully described in Ref. [20].

Figure 1: Our new platform for broadband coherent THz photonics is based on
planarized active and passive waveguides embedded in BCB, a low-loss polymer.
I. The planarized active waveguide consists of a standard double metal
waveguide encompassed in BCB and an extended top contact metallization. With
bonding wires placed over the passive section, improved waveguide losses, RF
and thermal properties are observed (see text for details). II. Passive
waveguides can also be fabricated, consisting of metallic stripes on top of
BCB, which provide confinement and guide the optical mode. These can be used
to co-integrate active and passive elements on the same chip.

We start by presenting results on the integration of on-chip THz frequency
combs making use of planarized waveguides and optimized RF input and output
coupling. In the context of THz frequency combs, we have shown that the
control over transverse modes is essential in order to obtain a regular and
flat-top comb spectrum [21]. The introduction of side absorbers mitigates
lasing in higher-order transverse modes due to increased waveguide losses. A
similar result can be achieved by reducing the transverse dimension of the
laser ridge to $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and below. However,
for conventional double metal waveguides, the width cannot be arbitrarily
small since the waveguides are usually contacted by wire bonding on the top
metallic cladding. This inherently limits the effective ridge width $w$ to the
dimensions of the bonding wire patch, making reliable devices with ridges of
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ or below challenging to contact and
prone to failures. Bonding directly on the active region can moreover
introduce defects, increasing the waveguide losses and non-intentionally
selecting specific modes, potentially compromising the long-term performance
of the device.

With our planarized platform, all of these issues can be solved. Placing the
bonding wires on top of the extended top metallization over the passive, BCB-
covered area, prevents formation of any defects or local hotspots on top of
the active region, and enables the fabrication of very narrow waveguides, well
below the bonding wire size. The narrow waveguide width can be employed as an
efficient selection mechanism for the fundamental transversal lasing mode, and
is also beneficial for heat dissipation and high temperature continuous wave
(CW) operation. With reduced waveguide widths we enter the regime of “wire”
lasers [22], which have a very favourable figure of merit for their surface-
to-volume ratio as it scales as the inverse of the width S/V$\propto$1/w. The
extended contact eases also the heat extraction as in a radiator scheme.

## Simulations

With the extended top metallization, the active region is sandwiched in a
symmetric structure. As a consequence, the propagating optical mode does not
feature any field spikes on the corners and at the edges of the ridge, as is
the case for a standard double metal waveguide, as illustrated in Fig. 2(a).
One major advantage of using double metal waveguides is the large overlap
factor of the propagating mode, reaching nearly unity. It can, however, reduce
with the waveguide width. In Fig. 2(b), we show that the computed overlap
factor remains above 90% for ridge widths of above
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for both the standard and
planarized waveguides at a frequency of 3 THz. Since we are interested in
frequency combs, another important figure of merit is the group velocity
dispersion of the waveguide. In Fig. 2(c) we display results of COMSOL 2D
eigenmode simulations, showing that the dispersion of the planarized
waveguides is significantly reduced with respect to the standard ones,
especially at low ($<$3 THz) frequencies. This is again due to the absence of
any field spikes which introduce dispersion, as the optical mode starts to
leak to the sides in standard double metal waveguides. The computation
includes waveguide and material (GaAs) dispersion for a
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide ridge waveguide.

Heat dissipation properties are improved as well, as lateral heat transport
takes place through the extended top metallization and the BCB polymer. COMSOL
2D thermal simulations show that for a $40\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ wide waveguide, the maximum temperature inside the active region is
reduced by around 7 K in a planarized waveguide at maximum bias conditions (11
V, 400 mA/cm2) and a heat sink temperature of 100 K. A measurement study of
the threshold current density as a function of increasing heat sink
temperature [23] and a comparison with data from our previous work [24] also
show a factor of 1.6 higher thermal conductivity of 130 W/Kcm2. More detailed
results of these simulation and characterization studies can be found in the
Supplementary Material.

The lateral heat transport could be enhanced by functionalizing BCB with
nanoparticles, as demonstrated in [25], without affecting the waveguide’s
optical losses. Moreover, as the planarized platform allows for the
fabrication of even narrower waveguides, heating effects can be mitigated
further. Recent results also display the positive impact of a thinner active
region in obtaining high temperature CW operation [26]. This approach can be
as well combined with our planarized geometry to push the high temperature
operation even higher.

Next, we study the radio-frequency (RF) properties of the planarized
waveguides. These are crucial both for the extraction and measurement of
frequency comb beatnotes as well as for an efficient injection of RF signals
which can affect and control laser operation. The relevant RF frequency range
is close to the cavity repetition rate, defined as
$f_{\mathrm{rep}}=\frac{c}{2\ n_{\mathrm{g}}\ L}$, where $c$ is the speed of
light in vacuum, $n_{\mathrm{g}}$ the mode group index, and $L$ the cavity
length. For typical waveguide lengths between 2-4 mm, the corresponding
$f_{\mathrm{rep}}$ lies roughly between 10-20 GHz, although harmonic comb
states can generate RF beatnotes well above 50 GHz.

To fully capitalize on the improved RF properties, a dedicated RF PCB was
developed and used for the laser mounting on copper submounts. As illustrated
in Fig. 2(d), the PCB features independent DC bias and RF readout/injection
lines. The RF line is a straight 50 $\Omega$ matched coplanar waveguide which
enables the placing of several short bonding wires at the backside of the
laser waveguide, while short ground wires on each side close the loop to
minimize RF losses. The positioning of all the RF signal wires on only one end
of the waveguide is crucial in order to maximize the RF readout/injection
efficiency, since the RF field produces a standing wave across the whole
cavity with a node in the center and maxima at each end,[27] as shown in the
simulation in Fig. 2(e). A separate dedicated, non-matched contact pad
provides the DC bias to the device, with bonding wires distributed along the
whole length of the laser waveguide for a homogeneous current injection.

2D and 3D eigenmode numerical simulations of the RF field of the planarized
waveguide suggest that the whole extended top metallization encompasses a
cavity with the ground plane, with the electromagnetic field oscillating
across the whole patch, as shown in Fig. 2(e) for a frequency of 20 GHz.
Compared to standard waveguides, the RF properties of the planarized
waveguides are modified in several aspects. First, due to the large modal
overlap with BCB (low refractive index of 1.57), the effective index of the
microwave mode $n_{\mathrm{eff}}^{\mathrm{GHz}}$ is reduced, which results in
an increase of the fundamental microwave resonance frequency, detuning it from
the THz mode spacing frequency $f_{\mathrm{rep}}$. Second, this also reduces
the waveguide impedance at RF frequencies, as shown in Fig. 2(f) for a
frequency of 20 GHz and a varying width of the active waveguide. Here, we
performed 2D COMSOL simulations of the waveguide cross section, where the
refractive index of the active region was fixed to 3.6, while the metals were
approximated as a perfect electric conductor (PEC). The lower impedance
results in higher Q-factors of the microwave modes in the planarized
waveguide, and in reduced radiative RF losses. Since the impedance of the
planarized waveguide depends mostly on the width of the extended top contact,
this allows for an independent design of the RF properties (e.g., by
changing/modulating the width of the extended contact), while keeping the
waveguide properties at THz frequencies unchanged. A more detailed analysis of
RF properties with 3D numerical simulation results can be found in the
Supplementary Material.

Figure 2: (a) COMSOL 2D eigenmode simulation of a
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide waveguide at 3 THz. While
there are field spikes present in the standard waveguide at the corners, for
planarized waveguides the field distribution is smooth also at the environment
boundary. (b) Computed overlap factors with the active region are comparable
for the standard and planarized waveguides. The dashed line indicates the
width where the overlap factor drops below 90%, for a frequency of 3 THz. (c)
Computed chromatic dispersion for a waveguide width of
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$. At low frequencies (3 THz and
below), the planarized waveguide has a significantly lower GVD than a standard
waveguide, where there are dispersive field spikes as the mode starts to leak
out at lower frequencies. (d) Illustration of the sample mounting with a
custom RF-optimized PCB. Several short signal wires are connected from a 50
$\Omega$ matched coplanar waveguide to one end of the waveguide, with short
ground wires for minimal RF injection and readout losses (inset, top left).
The DC laser bias is provided through a separate contact pad (inset, top
right). (e) 2D and 3D eigenmode numerical simulations of the RF field at 20
GHz show that the whole extended top metallization encompasses a cavity with
the ground plane. (f) Computed impedance from 2D COMSOL simulations for a
frequency of 20 GHz and a varying active waveguide width. Typical waveguide
widths are marked with circles.

## Experimental results

We present now experimental measurements, first investigating the performance
of simple ridge devices. These have a typical waveguide width of
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$, narrow enough for fundamental
transversal mode selection, and wide enough for a large overlap factor and low
propagation losses. Similar as in the case of standard double metal
waveguides, a Fabry-Pérot cavity is formed by mechanical cleaving, which forms
atomically flat end facets. In Fig. 3(a) we show an SEM image of the front
part of the ridge waveguide with a cleaved facet and DC bonding wires on the
extended top metallization. 3D electromagnetic numerical simulations show the
formation of standing waves inside the cavity due to the finite end facet
reflectivities (in this case in the order of 60% at 3 THz, very similar as for
standard double metal waveguides).

Light-current-voltage (LIV) characteristics, shown in Fig. 3(b), display a
very low threshold current density in the order of 140 A/cm2 at a heat sink
temperature of 40 K, which is due to the low-loss Cu-Cu planarized waveguide
and the low-dissipation superdiagonal active region [20]. These devices
typically operate up to around 115 K in continuous wave (CW). A
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide and 2.7 mm long ridge
waveguide with cleaved facets reaches output powers in the order of 3.0 mW at
40 K and 2.4 mW at 80 K in CW at rollover. The measured powers are from a
single facet and uncorrected for any absorption in the cryostat window.

In recent years, the possibility to obtain frequency combs from compact, on-
chip sources[28] has opened several possibilities from spectroscopy to LIDAR,
remote sensing and coherent communications [29]. Quantum cascade lasers in the
Mid-IR and THz proved to be excellent candidates for integrated,
semiconductor-based comb sources [30]. We examine now the comb properties of
our lasers based on planarized waveguides. Free-running devices can generate
frequency comb states, where the THz modes are exactly equidistantly spaced,
have a fixed phase relationship and produce a single RF signal (beatnote) at
the mode spacing frequency [31]. A typical measurement result is shown in Fig.
3(c), where the THz spectrum spans around 800 GHz and there is a single stable
RF beatnote at the cavity repetition rate $f_{\mathrm{rep}}$. The measured
free-running RF beatnotes in planarized waveguides can reach relatively high
powers of -60 to -55 dBm at the spectrum analyzer readout, confirming improved
RF properties (as described in more detail in the previous section). Free-
running beatnote maps of several devices can be found in the Supplementary
Material. Moreover, self-starting pure harmonic states [32, 33] (Turing
patterns) can be observed at specific bias points, where the mode spacing is
an integer multiple of the fundamental $f_{\mathrm{rep}}$ [34]. A typical
harmonic state spectrum spanning over 1.1 THz is shown in Fig. 3(d), in this
case corresponding to the third harmonic.

By injecting an external RF signal to the laser cavity, it is possible to
strongly modify the lasing operation both of a comb state [35, 36] or also of
a high phase noise state[20, 37]. In Fig. 3(e) we show an example where
injecting a strong RF signal (+32 dBm at source) close to the natural cavity
mode spacing ($f_{\mathrm{rep}}$ $\pm\leq$200 MHz) can broaden the THz
emission spectrum to over 1.5 THz. These are typically not frequency comb
states due to the limiting chromatic dispersion over such a wide bandwidth (as
evident already from the asymmetric interferograms in standard FTIR spectrum
measurements [34]), but still very useful as sources of broadband THz
radiation.

Figure 3:  (a) SEM image of a planarized,
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide ridge waveguide device,
showing the cleaved front facet. The bonding wires are placed on the extended
top metallization over the BCB-covered area. (b) LIV curves of a ridge laser
sample measured in pulsed mode (500 ns pulses, 10% duty cycle) and in CW at a
heatsink temperature of 40 K. (c) A free-running frequency comb spanning
around 800 GHz, with the measured single strong RF beatnote (-55 dBm) shown in
the inset. (d) A broadband free-running third harmonic state, covering a
bandwidth of 1.1 THz. (e) With strong RF injection (+32 dBm at source) close
to the free-running mode spacing, the emission spectrum can be broadened,
spanning around 1.5 THz.

In order to assess the comb coherence and retrieve the phase between adjacent
modes for reconstructing the time-domain emission profile, we used Shifted
Wave Interference Fourier Transform Spectroscopy (SWIFTS) [8, 38]. This is a
coherent beatnote technique that requires a fast detector combined with an
FTIR, as illustrated in the schematic in Fig. 4(a). A hot electron bolometer
(HEB) [39] was used as the fast detector. It consists of a thin film (6 nm)
NbN detecting element [40], with a metallic (Ti/Au) log spiral antenna for an
efficient in-coupling of the incoming THz radiation. Mounting and antenna-
coupling have been optimized to enhance RF performance. During operation, it
is cooled down to the superconducting state (below 10 K) and features
ultrafast rise times in the order of $\sim$40 ps [41, 39]. When illuminated
with a THz QCL frequency comb, the optical beatnote generated between adjacent
THz modes can be measured directly on the bias line of the detector and fed
into a spectrum analyzer with an IQ demodulator for SWIFTS [42].

Here, we present two measurement examples in two different regimes. First, we
analyze the comb state of a ridge waveguide device, injection-locked to the
natural $f_{\mathrm{rep}}$ = 14.540 GHz with a relatively weak reference RF
signal (+5 dBm at the source) in order to stabilize the repetition. The
measured spectra and relative phases are shown in Fig. 4(b), where the THz
spectra measured with a slow DTGS detector (blue) and the HEB (orange) have a
good overlap and comparable signal-to-noise ratio. This is attributed to the
high coherence of the comb state. The relative phases are scattered across
different values (with some mode groups sharing the same phase), which also
depends on the amplitude of the injected RF signal. The reconstructed time
profile produces a quasi-continuous periodic output with some amplitude
modulation, as shown in Fig. 4(c).

The second example is the same ridge waveguide device operating just above
lasing threshold and driven by a strong RF injection (+32 dBm at the source)
at $f_{\mathrm{inj}}$ = 14.538 GHz, close to the natural repetition rate
$f_{\mathrm{rep}}$ = 14.540 GHz. The measurement results in Fig. 4(d) show
that we can reach an active mode locking regime, where all the measurable
modes have a flat relative phase profile, and the reconstructed time profile
in Fig. 4(e) is a train of nearly Fourier-limited pulses as short as 4.4 ps.
In this case, the measurement with the HEB (orange) suffers from a worse
signal-to-noise ratio as compared to the DTGS (green). This is due to the
limited dynamic range of the HEB (short pulses have higher peak intensities)
and RF pickup problems due to the strong RF driving signal. Since only the
central high-intensity modes could be measured with the HEB detector, the
actual emitted THz pulses are very likely even shorter (assuming all the modes
have the same phase).

Figure 4: (a) Schematic of the SWIFTS setup, featuring an FTIR and a hot
electron bolometer (HEB) as a fast detector. (b, c) Measurements of a weakly-
injected free-running ridge device show scattered phase differences, and the
reconstructed time profile has a quasi-continuous periodic output intensity.
(d, e) Strong RF injection (+32 dBm at the source) close to the repetition
rate frequency on a ridge device close to lasing threshold results in active
mode locking, producing pulses as short as 4.4 ps (close to the Fourier limit,
dotted line). Here, the signal-to-noise ratio of the HEB measurement is
reduced due to RF pickup problems with stronger RF injection.

We should also emphasize that for all the frequency comb results presented in
this paper (including SWIFTS), the lasing modes throughout the full measured
THz emission spectrum share both the same repetition frequency
$f_{\mathrm{rep}}$ and the same offset frequency $f_{\mathrm{ceo}}$, and are
thus part of the same comb. In the case of matching $f_{\mathrm{rep}}$, but
several different values of $f_{\mathrm{ceo}}$, the spectrum consists of
several (individual) sub-combs, in which case the SWIFTS time-domain
reconstruction could not have been computed for the complete emission spectrum
at once. Instead, it would be sliced into the contributions of each individual
sub-comb. These aspects together with a thorough description of the FM to AM
transition under RF injection will be discussed elsewhere [42].

## Passive waveguide components

Another important advantage of our planarized platform is the possibility to
co-integrate active and passive elements. While we have already demonstrated
planar reflective and outcoupling passive antenna structures [43, 44], here we
designed passive waveguides for on-chip signal routing between various
elements. Integration of passive waveguides is critical in the development of
an integrated photonic platform. Near-infrared photonic circuits are nowadays
a reality, and a similar approach can be envisioned for THz frequencies. An
optical microscope image of a fabricated device is shown in Fig. 5(a), where
an active ridge is connected to straight and bent passive waveguides on each
side. The passive waveguide is a metallized stripe on top of BCB which
continues beyond the active region waveguide. Full-wave 3D numerical
simulations show that the optical mode is guided below the metal stripe,
following also non-straight paths (see insets in Fig. 5(a)). The reflectivity
(coupling efficiency) at the active/passive waveguide interface can be tuned
by the shape of the active and passive waveguides. In the simplest case of a
flat active waveguide facet, a reflectivity of $R=\lvert
S_{11}\rvert^{2}=22.5\%$ into the fundamental active waveguide mode, and a
transmission of $T=\lvert S_{21}\rvert^{2}=57.0\%$ into the fundamental
passive waveguide mode is obtained from a 3D numerical numerical simulation at
a frequency of 3 THz. This gives an insertion loss of
$L_{\mathrm{ins}}=1-\frac{T}{1-R}=26.5\%=1.34\ \mathrm{dB}$.

We performed 2D COMSOL simulations to evaluate the passive waveguide
propagation losses. For a straight waveguide (field profile in the top panel
of Fig. 5(b)), a propagation loss of around $\alpha_{\mathrm{pass}}=$ 9.5 cm-1
= 4.1 dB/mm at 3 THz is obtained. This originates mainly from the non-
optimized optical mode overlap with the top and bottom metal layers (a
copper/gold stack), as BCB has a relatively low loss of 3 cm-1 = 1.3 dB/mm at
around 3 THz [19]. If we consider a waveguide bend with a radius Rbend, the
electric field intensity is shifted towards the outer part of the passive
waveguide, as shown in the bottom panel in Fig. 5(b). Below a certain critical
radius Rbend, which depends on the frequency and passive waveguide width, the
propagation losses start to increase exponentially due to bending losses. In
Fig. 5(c), we plot the propagation losses of a passive waveguide with a top
metallic stripe width of $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ at 3 THz.
The radiative losses are negligible for a bend radius above around
$230\text{\,}\mathrm{\SIUnitSymbolMicro m}$, slightly above two times the
free-space wavelength. This value however does not impose a definitive
limitation for the integration density of passive components, as it is
possible to implement, for example, low-loss sharp waveguide bends with
inverse-designed optimized components [45].

Figure 5: (a) Optical microscope image of a ridge device co-integrated with
straight and curved passive waveguide elements, consisting of a metallized
stripe on top of BCB. Insets show the computed THz wave propagation with
minimal scattering and bending losses at 3 THz. (b) Simulated electric field
distribution in the passive waveguide cross-section at 3 THz. In the case of a
straight waveguide, the intensity is concentrated symmetrically below the
metal stripe, while it shifts to the outer side in the case of a bent path.
(c) False color SEM images of the cleaved passive waveguide with a flat BCB
end facet. (d) Computed passive waveguide propagation losses in dB/mm for a
varying bend radius, obtained by COMSOL 2D eigenmode simulations at a
frequency of 3 THz. The total losses are split into contributions from
absorption loss (overlap with lossy metals and BCB) and radiative loss
(bending loss), which is negligible for bend radii above
$230\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (e) The LIVs measured in pulsed
(500 ns pulses, 10% duty cycle, 20 K) show an increased threshold current
density and a higher slope efficiency, both consistent with the lower end
facet reflectivity. (f) Spectrum of a free-running device (CW, T = 20 K) in
the comb regime with a bandwidth over 1 THz and a single RF beatnote above -60
dBm, indicating that the reduced cavity feedback can be beneficial for comb
formation.

To characterize the passive waveguides experimentally, we performed a
mechanical cleaving around 1 mm away from the active-passive waveguide
interfaces on both sides and mounted the sample on a custom holder to measure
the laser output from both sides of the device (straight and bent). The active
region waveguide has a width of $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
a length of 2 mm. Compared to cleaved active ridge facets, due to refractive
index matching and the continued top metallized stripe, the end facet
reflectivity reduces from around 60% to around 22.5%, as predicted by 3D
numerical simulations. A false-color SEM image of the cleaved passive
waveguide can be seen in Fig. 5(c). The measured LIV curves in pulsed (500 ns
pulses, 10% duty cycle, 20 K) are shown in Fig. 5(e). The peak output powers
from the straight (1 mm long) and bent side (around 1.2 mm long) of the device
are comparable, indicating low propagation and bending losses. The threshold
current density increased to around 160 A/cm2 due to higher mirror losses.
Using the mirror loss formula
$\alpha_{\mathrm{mirr}}=-\frac{1}{2L}\ln(R_{1}R_{2})$ and comparing a 2 mm
long waveguide with cleaved facets ($R_{1}=R_{2}=60\%$) versus coupling on
both sides to a passive waveguide ($R_{1}=R_{2}=22.5\%$), the mirror losses
increase from 2.6 cm-1 to 7.5 cm-1. The slope efficiency of the measured
samples is increased from 11.9 mW/A to 13.6 mW/A. This is not as large as
anticipated from the reduced facet reflectivity, however we need to take into
account that the tested sample is coupled to a relatively long (around 1 mm)
passive waveguide, which induces propagation losses. A simple back-propagation
calculation gives us an upper estimate of the power emitted from the active
facet (without absorption losses in the passive section): $P^{{}^{\prime}}=P\
e^{(\alpha_{\mathrm{pass}}\,L)}=7.1$ mW. Here, we used the numerical
simulation result of $\alpha_{\mathrm{pass}}=9.5$ cm-1 and $L=$ 1 mm. This
would yield an increased slope efficiency of 37.6 mW/A. Although the actual
measured power is lower, it could be reached with a shorter passive waveguide,
and possibly increased further by using a passive outcoupling structure as in
Ref. [43].

The frequency comb performance does not deteriorate, but is actually improved.
In Fig. 5(f) we showcase the largest measured comb bandwidth, broader than 1
THz (wider than any other measured sample), and a single RF beatnote stronger
than -60 dBm (without using the custom RF PCB from Fig. 2(d)). As anticipated
in the Mid-IR by numerical results in Ref.[46] and theoretical models in Ref.
[47], the product of the facet reflectivities of a QCL ridge laser affects the
maximum bandwidth obtainable by a given gain medium, and a lower reflectivity
should enhance the comb bandwidth. In particular, the maximum bandwidth is
achieved for a product $R_{1}R_{2}\simeq 0.2$. In our case, the lower
reflectivity at the ends of the double metal cavity due to the coupling to the
passive section brings the product $R_{1}R_{2}|_{active}=0.36$ to
$R_{1}R_{2}|_{act+pass}=0.062$.

By using different combinations of cleaved and passive waveguide facets, the
planarized geometry offers the possibility to change the cavity reflectivity,
adding an important element to the comb engineering toolbox. This could be
expanded further by fabricating more complex dry-etched waveguide facet
geometries, for example an adiabatic taper coupled to a passive waveguide, to
make the reflectivity even lower. Finally, coupling broadband frequency combs
to passive waveguides on the same chip, as already demonstrated in the Mid-
IR[48], is an important milestone towards fully integrated active and passive
THz photonic circuits and spectrometers.

## Conclusions

In conclusion, we have presented a novel platform for broadband coherent THz
photonics based on high-confinement active and passive planarized double metal
waveguides. The extended top contact metallization enables bonding wires to be
placed over the BCB-covered area, which results in lower waveguide losses,
improved RF and thermal dissipation properties, and allows for an independent
design of THz and RF waveguide properties. The fabrication of narrow
waveguides acts as a fundamental mode selection mechanism and further improves
heat dissipation. Free-running broadband frequency combs over 800 GHz and
harmonic states over 1.1 THz are observed. Driven with an additional external
RF signal, broadband THz emission spectra over nearly 1.6 THz, and actively
mode-locked pulses as short as 4.4 ps can be generated on demand. The co-
integration of passive elements with custom on-chip guiding, reflection and
outcoupling properties complete the diverse integrated THz photonics toolbox
enabled by our planarized platform.

## Methods

###  0.1 Planarized waveguide fabrication

From MBE-grown wafers, samples with a typical size of (9x10) mm2 were cleaved.
A metallic stack of Ta(5nm)/Cu(250nm)/Ti(50nm)/Au(500nm) was deposited on the
sample and on a carrier n+ GaAs substrate by electron beam evaporation. These
were bonded using thermocompression wafer bonding at a temperature of 320°C
and a pressure of 5 MPa for 15 minutes in vacuum. After a mechanical thinning
and several wet etch steps to expose the active region, the waveguides were
dry-etched in a Cl2/H2 inductively-coupled plasma (ICP) using SiNx as a
hardmask. Subsequently, AP3000 was spun on the sample as an adhesion layer for
BCB (3000 RPM for 30 s). Cyclotene 3022-57 (BCB) was both spin coated (5000
RPM for 50 s) and baked sequentially five times (4x softbake at 210°C and 1x
hardbake at 250°C for 2h). This resulted in a total BCB thickness of around
$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ to ensure a smooth profile across
the sample. Using Reactive Ion Etching (RIE), the BCB was etched to match the
height of the active region waveguide
($10.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$). Finally, the extended top
metallization (same metallic stack as the wafer-bonded side) with a typical
width of $300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ was defined with
photolithography and lift-off, spanning over the active waveguides and the
BCB-covered area on the sides. The passive waveguides were fabricated over
BCB-covered areas in this final step as well.

###  0.2 RF mounting

The custom two-layer RF PCB was fabricated on a Rogers 4350 substrate with a
thickness of 0.8 mm. The coplanar waveguide has a width of
$800\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and is separated from the side
ground planes by $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. A series of vias
connecting to the ground plane improves the RF performance and the isolation
between RF and DC ports. The PCB is screwed to a copper laser submount and
mounted on a cryostat cold finger. It is connected with a standard RF
connector (HK-LR-SR2(12)) to low-loss semi-rigid RF cables (SUCOFORM 86 FEP),
accessible from the outside of the cryostat.

The authors gratefully acknowledge funding from the ERC Grant CHIC (No.
724344) and in part by the NATO Science for Peace and Security Programme under
grant G5721 THESEUs. We also thank Tabea Bühler, Sebastian Gloor and Johannes
Hillbrand for technical help.

The authors declare that they have no competing financial interests.

U.S. and G.S. conceived the idea. U.S. designed and fabricated the devices,
carried out all the measurements, analysed experimental data and performed
numerical simulations under the supervision of G.S. and J.F.. A. F. designed
the RF PCB board and built the SWIFTS setup. U.S. and T.O. developed the
planarized waveguide fabrication process. P.M. performed some of the device
characterizations. S.C. and G.T. provided the HEB detectors, A.F. optimized
the HEB RF coupling. M.B. performed the epitaxial growth. U.S. and G.S. wrote
the manuscript. All authors discussed the results and commented on the
manuscript.

*Correspondence should be addressed to U. Senica (email<EMAIL_ADDRESS>and G. Scalari (email: scalari@phys.ethz.ch).

## References

## References

  * [1] Thomson, D. _et al._ Roadmap on silicon photonics.  _Journal of Optics_ 18 (2016).
  * [2] Smit, M., Williams, K. & van der Tol, J.  Past, present, and future of InP-based photonic integration.  _APL Photonics_ 4, 050901 (2019).
  * [3] Sengupta, K., Nagatsuma, T. & Mittleman, D. M.  Terahertz integrated electronic and hybrid electronic-photonic systems.  _Nature Electronics_ 1, 622–635 (2018).
  * [4] Köhler, R. _et al._ Terahertz semiconductor-heterostructure laser.  _Nature_ 417, 156–159 (2002).
  * [5] Bosco, L. _et al._ Thermoelectrically cooled THz quantum cascade laser operating up to 210 K.  _Applied Physics Letters_ 115, 010601 (2019).
  * [6] Khalatpour, A., Paulsen, A. K., Deimert, C., Wasilewski, Z. R. & Hu, Q.  High-power portable terahertz laser systems.  _Nature Photonics_ 15, 16–20 (2021).
  * [7] Curwen, C. A., Reno, J. L. & Williams, B. S.  Broadband continuous single-mode tuning of a short-cavity quantum-cascade VECSEL.  _Nature Photonics_ 13, 855–859 (2019).
  * [8] Burghoff, D. _et al._ Terahertz laser frequency combs.  _Nature Photonics_ 8, 462–467 (2014).
  * [9] Rösch, M., Scalari, G., Beck, M. & Faist, J.  Octave-spanning semiconductor laser.  _Nature Photonics_ 9, 42–47 (2015).
  * [10] Micheletti, P. _et al._ Regenerative terahertz quantum detectors.  _APL Photonics_ 6, 106102 (2021).
  * [11] Roelkens, G. _et al._ III-V/silicon photonics for on-chip and inter-chip optical interconnects.  _Laser & Photonics Reviews_ 4, 751–779 (2010).
  * [12] Piccardo, M. _et al._ Roadmap on multimode light shaping.  _Journal of Optics_ 24, 013001 (2022).
  * [13] Consolino, L. _et al._ Quantum cascade laser based hybrid dual comb spectrometer.  _Communications Physics_ 3, 69 (2020).
  * [14] Koenig, S. _et al._ Wireless sub-THz communication system with high data rate.  _Nature Photonics_ 7, 977–981 (2013).
  * [15] Maineult, W. _et al._ Microwave modulation of terahertz quantum cascade lasers: a transmission-line approach.  _Applied Physics Letters_ 96, 021108 (2010).
  * [16] Williams, B. S.  Terahertz quantum-cascade lasers.  _Nature Photonics_ 1, 517–525 (2007).
  * [17] Perret, E., Zerounian, N., David, S. & Aniel, F.  Complex permittivity characterization of benzocyclobutene for terahertz applications.  _Microelectronic Engineering_ 85, 2276–2281 (2008).
  * [18] Bosco, L. _et al._ A patch-array antenna single-mode low electrical dissipation continuous wave terahertz quantum cascade laser.  _Applied Physics Letters_ 109, 201103 (2016).
  * [19] Bonzon, C. B.  _Phase and mode control of structured semiconductor lasers_.  Ph.D. thesis, ETH Zurich (2016).
  * [20] Forrer, A. _et al._ Photon-Driven Broadband Emission and Frequency Comb RF Injection Locking in THz Quantum Cascade Lasers.  _ACS Photonics_ 7, 784–791 (2020).
  * [21] Bachmann, D. _et al._ Short pulse generation and mode control of broadband terahertz quantum cascade lasers.  _Optica_ 3, 1087–1094 (2016).
  * [22] Amanti, M. I., Scalari, G., Castellano, F., Beck, M. & Faist, J.  Low divergence Terahertz photonic-wire laser.  _Optics Express_ 18, 6390–6395 (2010).
  * [23] Faist, J.  _Quantum cascade lasers_ (Oxford University Press, Oxford, United Kingdom, 2013), first edition edn.
  * [24] Forrer, A., Bosco, L., Beck, M., Faist, J. & Scalari, G.  RF Injection of THz QCL Combs at 80 K Emitting over 700 GHz Spectral Bandwidth.  _Photonics_ 7, 9 (2020).
  * [25] Xu, X. & Wang, Z.  Thermal conductivity enhancement of benzocyclobutene with carbon nanotubes for adhesive bonding in 3-d integration.  _IEEE Transactions on Components, Packaging and Manufacturing Technology_ 2, 286–293 (2012).
  * [26] Curwen, C. A. _et al._ Thin THz QCL active regions for improved continuous-wave operating temperature.  _AIP Advances_ 11, 125018 (2021).
  * [27] Piccardo, M. _et al._ Time-dependent population inversion gratings in laser frequency combs.  _Optica_ 5, 475–478 (2018).
  * [28] Chang, L., Liu, S. & Bowers, J.  Integrated optical frequency comb technologies.  _Nature Photonics_ 16, 95 (2022).
  * [29] Marin-Palomo, P. _et al._ Microresonator-based solitons for massively parallel coherent optical communications.  _Nature_ 546, 274–279 (2017).
  * [30] Faist, J. _et al._ Quantum Cascade Laser Frequency Combs.  _Nanophotonics_ 5, 272–291 (2016).
  * [31] Hugi, A., Villares, G., Blaser, S., Liu, H. C. & Faist, J.  Mid-infrared frequency comb based on a quantum cascade laser.  _Nature_ 492, 229–233 (2012).
  * [32] Kazakov, D. _et al._ Self-starting harmonic frequency comb generation in a quantum cascade laser.  _Nature Photonics_ 11, 789 (2017).
  * [33] Burghoff, D.  Unraveling the origin of frequency modulated combs using active cavity mean-field theory.  _Optica_ 7, 1781–1787 (2020).
  * [34] Forrer, A. _et al._ Self-starting harmonic comb emission in thz quantum cascade lasers.  _Applied Physics Letters_ 118, 131112 (2021).
  * [35] Schneider, B. _et al._ Controlling Quantum Cascade Laser Optical Frequency Combs through Microwave Injection.  _Laser & Photonics Reviews_ 15 (2021).
  * [36] Hillbrand, J., Andrews, A. M., Detz, H., Strasser, G. & Schwarz, B.  Coherent injection locking of quantum cascade laser frequency combs.  _Nature Photonics_ 13, 101–104 (2019).
  * [37] Gellie, P. _et al._ Injection-locking of terahertz quantum cascade lasers up to 35GHz using RF amplitude modulation.  _Optics Express_ 18, 20799–20816 (2010).
  * [38] Burghoff, D., Ren, D. & Han, Z.  Sensitivity of SWIFT spectroscopy.  _Optics Express_ 28, 6002–6017 (2020).
  * [39] Semenov, A., Gol’tsman, G. & Sobolewski, R.  Hot-electron effect in superconductors and its applications for radiation sensors.  _Superconductor Science and Technology_ 15, R1–R16 (2002).
  * [40] Martini, F., Cibella, S., Gaggero, A., Mattioli, F. & Leoni, R.  Waveguide integrated hot electron bolometer for classical and quantum photonics.  _Optics Express_ 29, 7956–7965 (2021).
  * [41] Bründermann, E., Hübers, H.-W. & Kimmitt, M. F.  _Terahertz Techniques_ , vol. 151 of _Springer Series in Optical Sciences_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012).
  * [42] Forrer, A. _et al._ Coherence of RF Injected THz Quantum Cascade Laser Combs.  _in preparation_ (2022).
  * [43] Bosco, L. _et al._ A patch-array antenna single-mode low electrical dissipation continuous wave terahertz quantum cascade laser.  _Appl. Phys. Lett_ 109, 201103 (2016).
  * [44] Senica, U. _et al._ An antipodal Vivaldi antenna for improved far-field properties and polarization manipulation of broadband terahertz quantum cascade lasers.  _Applied Physics Letters_ 116, 161105 (2020).
  * [45] Liu, Y. _et al._ Very sharp adiabatic bends based on an inverse design.  _Opt. Lett._ 43, 2482–2485 (2018).
  * [46] Beiser, M., Opačak, N., Hillbrand, J., Strasser, G. & Schwarz, B.  Engineering the spectral bandwidth of quantum cascade laser frequency combs.  _Opt. Lett._ 46, 3416–3419 (2021).
  * [47] Humbard, L. & Burghoff, D.  Analytical theory of frequency-modulated combs: generalized mean-field theory, complex cavities, and harmonic states.  _Opt. Express_ 30, 5376–5401 (2022).
  * [48] Wang, R. _et al._ Monolithic integration of mid-infrared quantum cascade lasers and frequency combs with passive waveguides.  _ACS Photonics_ 9, 426–431 (2022).