1
The effect of complex dispersion and characteristic
impedance on the gain of superconducting
traveling-wave kinetic inductance parametric
amplifiers
Javier Carrasco, Daniel Valenzuela, Claudio Falc´
on, Ricardo Finger, and F. Patricio Mena
Abstract—Superconducting traveling-wave parametric ampli-
fiers are a promising amplification technology suitable for appli-
cations in submillimeter astronomy. Their implementation relies
on the use of Floquet transmission lines in order to create
strong stopbands to suppress undesired harmonics. In the design
process, amplitude equations are used to predict their gain,
operation frequency, and bandwidth. However, usual amplitude
equations do not take into account the real and imaginary parts
of the dispersion and characteristic impedance that results from
the use of Floquet lines, hindering reliable design. In order
to overcome this limitation, we have used the multiple-scales
method to include those effects. We demonstrate that complex
dispersion and characteristic impedance have a stark effect on
the transmission line’s gain, even suppressing it completely in
certain cases. The equations presented here can, thus, guide to
a better design and understanding of the properties of this kind
of amplifiers.
Index Terms—parametric amplification, gain, superconductor,
nonlinear physics, four-wave-mixing.
I. INTRODUCTION
ACHIEVING larger bandwidths at the RF and IF bands,
and improving receiver sensitivity are major challenges
for future millimeter and submillimeter heterodyne observa-
tions [1]. As part of this effort, extensive work is being done in
order to improve the performance of SIS mixers [2] and HEMT
amplifiers [3], the key components of current state-of-the-art
receivers. However, on the one hand, it is not clear if HEMT
amplifiers can be further improved notwithstanding the exten-
sive work made in understanding the reasons that limit noise
temperature and operational bandwidth [4]–[6]. On the other
hand, even if SIS mixers are improved, connecting them to
HEMT amplifiers will necessarily limit their performance [7],
[8]. Recently, a promising superconducting technology that
could overcome these problems has emerged [9]. It uses the
J. Carrasco is with the Electrical Engineering Department and the Depart-
ment of Physics, Faculty of Physical and Mathematical Sciences, University
of Chile, Santiago, Chile.
D. Valenzuela is with the Electrical Engineering Department, Faculty of
Physical and Mathematical Sciences, University of Chile, Santiago, Chile.
C. Falc´
on is with the Department of Physics, Faculty of Physical and
Mathematical Sciences, University of Chile, Santiago, Chile.
R. Finger is with the Department of Astronomy, Faculty of Physical and
Mathematical Sciences, University of Chile, Santiago, Chile.
F. P. Mena is with the National Radio Astronomy Observatory, Char-
lottesville, VA, USA.
Contact e-mail: javier.carrasco@ug.uchile.cl.
kinetic inductance (KI) of superconductors [10], [11] to pro-
duce parametric amplification in a long transmission line (TL).
Devices working with this principle are dubbed Traveling-
Wave Kinetic-Inductance Parametric Amplifiers (TKIPAs) [9],
[12]–[16].
The KI, originated by the inertia of Cooper pairs [10]
in superconductors, modifies the wave-equation on the TL
by adding nonlinearities which allow the mixing of wave
amplitudes when more than one monochromatic wave are
injected [17]. Hence, it is possible to amplify the input signal
if other signals, called pumps, are simultaneously injected.
Nonetheless, more signals, including undesired harmonics,
are also generated, compromising the amplification process.
Eom et al. [9] solved this problem by implementing a suit-
able Floquet TL, also known as dispersion engineered TL,
conformed by a periodically repeating unit cell that creates
stopbands and, thus, avoids the propagation of the main
undesired harmonics of the pump signal. Such a solution,
however, translates into a TL with more intricate properties,
namely a complex dispersion and characteristic impedance,
i.e. with real and imaginary parts, that, moreover, have strong
frequency dependencies, particularly close to the stopbands.
In order to design TKIPAs, a nonlinear wave equation
must be solved. This is usually done by approximating the
process of amplitude gain as a dynamical evolution occurring
at a much larger length scale than the wavelength of the
involved signals. Within this approximation, but without taking
into account the complex nature of the Floquet TL, a set
of nonlinear amplitude equations can be obtained [18], [19].
In order to account for losses, an attempt to introduce a
complex propagation constant in this approximation has been
reported [13] but lacks justification when dealing with the
wave behavior near stopbands.
We have included the complex nature of the Floquet TL
into the amplitude equations by formally solving the nonlinear
wave equation via a multiple-scales method, widely used
in nonlinear physics and especially useful in traveling-wave
equations [19]–[21]. We demonstrate that the use of this
type of line has a profound effect on the attainable gain,
in particular when the pump signal is close to a stopband.
Depending on the specific properties of the used Floquet TL
and the amplitude and frequency of the pump signal, our
equations depart notably from the predictions given by the
traditional amplitude equations.
arXiv:2210.00626v1 [physics.app-ph] 2 Oct 2022