1 The effect of complex dispersion and characteristic impedance on the gain of superconducting

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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
2
II. AMPLITUDE EQUATIONS WITH COMPLEX
CHARACTERISTIC IMPEDANCE AND DISPERSION
In a TL, the electric current I=I(z, t)and voltage V=
V(z, t)dynamics are given by the telegraph equations,
V
z =LI
t RI, (1a)
I
z =CV
t GV, (1b)
where zis the position along the TL and tis the time.
Here, Ris the resistance per unit length due to losses in
the conductors, Cis the capacitance per unit length due to
the close proximity between conductors, Gis the conductance
per unit length due to losses in the dielectric material between
conductors, and Lis the total self-inductance per unit length
between the conductors [22]. However, for TLs made out of
superconductors, the inductance per unit length is a function
of the current and can be modeled as
L=L01 + α
I2
I02
,(2)
where L0is the total inductance per unit length of the TL at
null electric current, αis the ratio of kinetic inductance to
total inductance, and I0
is a parameter comparable to the crit-
ical current Ic[9] (which is in the order of a few milliamperes
for realizable devices). Importantly, I=I0
/αdetermines
the strength of the nonlinear effect [9], [17].
Expression (2) comes from the fact that the current has a
cubic dependence on the velocity of Cooper pairs [23], [24]. It
is valid at temperatures TTc[24], where Tcis the critical
temperature of the superconductor, commonly of the order of
a few kelvins [11].
From (1) and (2), a nonlinear wave equation for the current
can be derived,
2
z2CL0
2
t2(CR +GL0)
t RGI
=L0
3I2
G
t +C2
t2I3.(3)
It can be compactly rewritten as
LI=NI3,(4)
where Land Nare, respectively, the differential operators
acting on the linear and nonlinear parts of the equation. The
equation LI= 0 corresponds to the well known linear wave
equation whose general solution, for waves traveling along the
TL in +zdirection, is
Ilinear =1
2X
nAnejωntγnz+c.c.,(5)
where “c.c.” stands for “complex conjugate”, ωn2πνnis
the angular frequency of the nth tone of I, and γnare the
propagation constants that fulfill the dispersion relation
γ2
n+CL0ω2
njωn(CR +GL0)RG = 0.(6)
The solution to LI= 0 can be obtained independently
at each frequency. However, if α6= 0 (i.e. N 6= 0), the
|A|ω
ω
p
ωs
p
+
pωs
pωs
p
+
pωspωs
pωs
Fig. 1. Fourier spectrum of the main signals relevant in the FWM process
[17]. This diagram shows the physically injected signals (red), and the signals
that are generated in the FWM process (blue). From the latter, the idler with
angular frequency ωi2ωpωs, and the third harmonic of the pump with
angular frequency 3ωp, are the most relevant. Many more signals appear at
higher frequencies, but they are less relevant because they have much smaller
amplitudes.
term I3allows for interaction between frequencies which,
under the correct conditions, can produce amplification of a
target frequency signal. The level of amplification depends on
a specific parameter, the pump signal, which is an injected
monochromatic wave used as source of energy in the process.
This is known as parametric amplification. Moreover, the
cubic nonlinearity means that this is a four-wave-mixing
(FWM) process. In consequence, the energy transfer occurs
fundamentally by exchange of pairs of photons, rather than
single ones [18]. Therefore, in order to transfer energy from
the pump to the signal, an additional monochromatic wave
is required, called idler, which also receives energy from the
pump. The idler is naturally generated in the FWM process
together with harmonics and sidebands [17] as depicted by
Fig. 1.
A. Multiple-scales method
We tackle the problem of solving the nonlinear current
equation (3) using a multiple-scales method by considering
the nonlinear term NI3as a perturbation to the linear wave
dynamics LI= 0 which evolves at a different rate or
scale [19], [25].
The method is applied to Eq. (4) rewritten in terms of
˜
II/I. Since |I|< Icand IIc, then |˜
I|≡|I/I|<1,
and it follows that |˜
I|3<|˜
I|. In consequence, the nonlinear
terms can be considered perturbations to the linear equation
of the current, acting at scales in zlarger than the evolution
of the linear equation. The typical spatial scale of the linear
part of (4) is the wavelength 2π/βn, where βnIm{γn}is
the wavenumber, whereas the (nonlinear) typical length scale
of the envelope Anis much larger, as it will be clear later in
subsection II-B. This information is considered by giving the
amplitudes Anin (5) a dependence on zbut only at large
scales. Then, to solve at first order, only the dominant z-
scale affecting Anis considered, giving a new wave equation,
balanced at first order. The solution to this equation must
not contain singular terms, condition that results in a set of
amplitude equations.
We focus on a zone close to the stopbands in Floquet TLs,
commonly used to implement amplifiers using this principle
[9], [17]. The set of amplitude equations display solutions that
depend on the specific TL used to transport the signals. In
particular, we obtain a new model valid in a frequency zone
摘要:

1Theeffectofcomplexdispersionandcharacteristicimpedanceonthegainofsuperconductingtraveling-wavekineticinductanceparametricampliersJavierCarrasco,DanielValenzuela,ClaudioFalc´on,RicardoFinger,andF.PatricioMenaAbstract—Superconductingtraveling-waveparametricampli-ersareapromisingamplicationtechnolo...

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