[1]Thomas R. Jones

1]Elmore Family School of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN, USA

2]Department of Physics, Technion-Israel Institute of Technology,
Haifa, Israel

# Article Title

<EMAIL_ADDRESS>Alexander V. Kildishev<EMAIL_ADDRESS>Mordechai
Segev<EMAIL_ADDRESS>Dimitrios Peroulis<EMAIL_ADDRESS>[  [

# Time-Reflection of Microwaves by a Fast Optically-Controlled Time-Boundary

<EMAIL_ADDRESS>Alexander V. Kildishev<EMAIL_ADDRESS>Mordechai
Segev<EMAIL_ADDRESS>Dimitrios Peroulis<EMAIL_ADDRESS>[  [

###### Abstract

When an electromagnetic (EM) wave is propagating in a medium whose properties
are varied abruptly in time, the wave experiences refractions and reflections
known as “time-refractions” and “time-reflections”, both manifesting spectral
translation as a consequence of the abrupt change of the medium and the
conservation of momentum. However, while the time-refracted wave continues to
propagate with the same wave-vector, the time-reflected wave is propagating
backward with a conjugate phase, despite the lack of any spatial interface.
Importantly, while time-refraction is always significant, observing time-
reflection poses a major challenge – because it requires a large change in the
medium occurring within a single cycle. For that reason, time-reflection of EM
waves was observed only recently. Here, we present the observation of
microwave pulses at the highest frequency ever observed (0.59 GHz), and the
experimental evidence of the phase-conjugation nature of time-reflected waves.
Our experiments are carried out in a periodically-loaded microstrip line with
optically-controlled picosecond-switchable photodiodes. Our system paves the
way to the experimental realization of Photonic Time-Crystals at GHz
frequencies.

###### keywords:

time-reflection, photonic time-crystal, PIN photodiode, time-boundary
reflection, time-interface

##  1 Introduction

When the refractive index is changed abruptly in time, a wave propagating in
the medium experiences time-refraction and time-reflection, similar to the
refraction and reflection at dielectric interfaces [1, 3, 2]. However, despite
the similarity, time-reflection and refraction are fundamentally different
than their spatial counterparts. Namely, whereas energy (frequency) is
conserved at spatial interfaces between dielectric media, a time-interface
necessarily changes the frequency, because time-translation symmetry is broken
by the temporal modulation. On the other hand, when the abrupt temporal change
occurs in a homogeneous medium - momentum (wave-vector $k$) is conserved,
which implies that both the time-refraction and the time-reflection experience
translation of their spectrum [5, 4]. Also, causality implies that the time-
reflection cannot go back in time (unfortunately) but instead is back-
reflected in space. That is, while the time-refracted wave continues to
propagate with the same wave-vector, the time-reflected wave is propagating
backwards with a conjugate phase [7, 6]. In this way, modulating the
refractive index strongly and periodically in time gives rise to multiple
time-reflections and time-refractions, which interfere and yield dispersion
bands separated by band-gaps in the momentum $k$ [9, 10, 8]. This temporal
structure is known as Photonic Time-Crystals (PTCs), and their most intriguing
property is that states residing in the momentum gap can have exponentially
increasing amplitudes, which draw energy from the modulation [11, 13, 12].
Importantly, PTCs offer a plethora of new possibilities of new physics and
applications, ranging from the generation of pairs of entangled photons and
below-threshold Cherenkov radiation to new widely-tunable laser sources at THz
frequencies [14, 15, 16, 17, 18, 19, 20, 21]. However, all of these rely on
time-reflections: without significant time-reflections there would be no PTCs.
This understanding poses a major challenge: for the time-reflection to be
significant, the refractive index change has to be large (order of unity) and
abrupt (occurring within 1-2 cycles of the time-reflected waves), otherwise
the time-reflection is extremely weak, and PTCs become impossible [23, 22].
This tough requirement to have a strong and abrupt change in the refractive
index is the reason why the universal phenomenon of time-reflected waves has
been observed only recently. Its properties were thus far studied with water
waves [24] and with EM waves at radio frequencies [25], and only very recently
with microwaves [26].

Here, we study the propagation of microwaves in a periodically-loaded
microstrip line with optically-controlled picosecond-switchable photodiodes,
and present time-reflection at the highest frequency ever observed (0.59 GHz)
along with direct experimental evidence for its phase-conjugate nature. Our
experiments demonstrate the time-resolved reflection of an ultra-high
frequency pulse propagating along a periodically-loaded microstrip
transmission line, occurring due to a time-modulated distributed impedance.

For a strong time-reflection at microwave frequencies, the change in the
characteristic impedance due to the temporal event must be fast – within 1-2
cycles of the time-reflected waves. Consequently, we use high-speed
photodiodes with switching times in the picosecond regime, to control the
capacitive loading and achieve the desired bandwidth. Moreover, the entire
experimental setup is judiciously designed to maximize the available bandwidth
in both states, such that we can study the propagation of very short pulses in
the microstrip line. Also, the electric length of the microstrip line is
optimized to be as short as possible, so that the pulse always remains
completely within the line during the temporal event. Such a unique
experimental system allows us to explore the evolution of ultra high frequency
signals, and observe time-reflections at 0.59 GHz occurring due to an almost-
instantaneous time-interface. Subsequently, we study the propagation of a
hybrid two-peak asymmetric pulse, and demonstrate the time-reversal nature of
the time-reflected signal, along with its frequency translation arising from
conservation of momentum. We also study the time reflection of a chirped pulse
with phase variation, providing experimental evidence of the phase-conjugation
nature of time-reflected waves.

Apart from the exciting fundamental aspects of this study, our prototype
system has important practical value. It employs realistic microwave circuit
designs with surface-mount high-speed photodiodes, and demonstrates the viable
potential of ultra-fast time-modulated microstrip circuits for nascent real-
world applications in novel microwave pre- and post-processing systems, AI-
driven navigation, and satellite communication platforms.

The paper is organized as follows. First, Section 2 describes the general
concept of our time-boundary experiment and provides the theoretical
background. Section 3 outlines the design of the time-reconfigurable
microstrip-line and switchable capacitive loading using high-speed
photodiodes. The experimental setup and results are presented in Section 4.
Here the frequency response of the time-reconfigurable microstrip is reported,
along with the time-boundary experiments using single, two-peak asymmetric,
and chirped pulses demonstrating time-reflection and its time-reversal and
phase-conjugate nature, along with spectral translation due to conservation of
momentum. The conclusion provides a short summary and discussion on future
ideas.

##  2 Principle of Operation

Figure 1:  A conceptual 3D illustration of a time boundary’s impact on a two-
peak asymmetric pulse. An input two-peak assymetric pulse (yellow ”input
pulse”) with the smaller amplitude peak leading the larger, propagates along a
periodically-loaded microstrip transmission line. The effective impedance of
the microstrip is modulated in time by high-speed photodiodes. The diodes are
synchronized by simultaneous optical triggers (green ”optical trigger”)
delivered through the optical fibers from a single laser diode (not shown).
Upon activation of the time interface, the input pulse is scattered into a
transmitted pulse (black ”transmitted pulse”) and reflected pulse (red
”reflected pulse”). The transmitted pulse continues to travel in the same
direction as the input with the smaller amplitude peak still leading the
larger, with reduced magnitude. The reflected pulse travels in the opposite
direction as the input – the larger amplitude peak now leads the lower one,
demonstrating the time-reversal property of PTCs. The pulses are shown in the
spatial domain propagating along the x-axis.

In this work, a reconfigurable periodically-loaded microstrip with rapidly
switchable effective characteristic impedance models an abrupt change in the
effective permittivity of the medium. A time interface is generated while a
microwave pulse propagates within the microstrip, and the corresponding
reflected and transmitted signals are measured at the input and output,
respectively.

Fig. 1 illustrates a 3D concept of our reconfigurable distributed microwave
transmission line (DMTL) used to experimentally demonstrate time-boundary
reflection and transmission of a microwave pulse. The DMTL is designed to be
periodically loaded with two parallel shunt capacitors, $C_{\mathrm{a}}$ and
$C_{\mathrm{b}}$. The second capacitor $\left(C_{\mathrm{b}}\right)$ is
connected in series to a switch controlled by a high-speed photodiode. Both
capacitors are terminated with conductive vias to ground. Upon excitation by
light (green ”optical trigger” pulse), the high-speed photodiodes switch from
an isolating OFF-state to a conductive ON-state, connecting the additional
shunt capacitor $C_{\mathrm{b}}$ to the circuit. This reconfiguration changes
the effective impedance of the microstrip from State 1 to a lower impedance
value in State 2, generating a time-boundary reflection (red ”reflected
pulse”) and transmission (black ”transmitted pulse”) of the input (yellow
”input pulse”) wave travelling within the medium.

With picosecond rise times of high-speed photodiodes, the time interface is
much faster than the operating bandwidth of the microwave pulse, enabling a
near-instantaneous time-boundary akin to spatial reflections with an abrupt
spatial-boundary and allowing the measurement of an ultra high frequency time-
reflected pulse of 0.59 GHz. In addition to fast rise times, the use of
photodiodes enables almost perfect synchronization of each switch periodically
loading the microstrip, in which the optical trigger is produced from a single
laser diode source, generating a truly homogeneous time-switched medium.

Two unique properties distinguish time-reflected waves from spatially-
reflected waves [1]. The first is time reversal, due to the homogeneous change
in space of the properties of the medium. This instant change leads to a
broadband time reversal of the reflected wave. Fig. 1 illustrates this effect
in the spatial domain with a two-peak asymmetric pulse scattered at a time
interface. Observing the input pulse (yellow trace), we see that the higher
amplitude peak lags the lower one as the pulse travels towards the top right
along the x-axis in the positive x-direction. Upon activation of the time
boundary, the order of the peaks is now reversed – the low-amplitude pulse now
lags behind the higher one, as the direction of travel changes towards the
bottom left or negative x-direction. This effect is completely different with
a spatial boundary – for a wave bouncing off a spatial discontinuity (spatial
wall) the leading edge reaching the wall is the first one to get reflected and
received back at the source. With a time boundary, the entire pulse inside the
”slab” gets reflected at the same time. This distributed reflection
effectively reverses the pulse in time, its trailing edge becomes the leading
edge, and the leading edge becomes the trailing one.

The second unique property of time-reflected waves is their broadband
frequency translation due to spatial translational symmetry, since the wave’s
momentum is conserved across the time interface. Following [26], we derive the
frequency translation due to a scattering event at a time interface based on a
distributed circuit model of a microwave transmission line. For details of
this derivation, see Supplementary Section 1. The frequency of the input wave
$(\omega_{1})$ is transformed to that of the reflected and transmitted waves
$(\omega_{2})$ as,

$\omega_{2}=\sqrt{\frac{C_{1}}{C_{2}}}\omega_{1}=\frac{Z_{2}}{Z_{1}}\omega_{1}$
(1)

where $C_{1}$ and $Z_{1}$ are the loading capacitance and effective
characteristic impedance of the DMTL in State 1, respectively, while $C_{2}$
and $Z_{2}$ are the loading capacitance and effective characteristic impedance
of the DMTL in State 2, respectively. This straightforward model immediately
leads to the time-boundary reflection and transmission coefficients:

$R=\frac{1}{2}\left[\frac{C_{1}}{C_{2}}-\sqrt{\frac{C_{1}}{C_{2}}}\right]=\frac{Z_{2}(Z_{2}-Z_{1})}{2Z_{1}^{2}}$
(2)

$T=\frac{1}{2}\left[\frac{C_{1}}{C_{2}}+\sqrt{\frac{C_{1}}{C_{2}}}\right]=\frac{Z_{2}(Z_{2}+Z_{1})}{2Z_{1}^{2}}$
(3)

The above properties and theoretical derivations of time-boundaries will now
be validated through experiments in the following sections.

##  3 Design

This section now outlines the design of a time-reconfigurable periodically-
loaded microstrip line. First, the detailed design steps of a reconfigurable
periodically-loaded microstrip line are presented, leading to a realizable
prototype suitable for our time-boundary experiment. The details of the time-
switched capacitor loading element using high-speed photodiodes are then
outlined, with considerations towards the photodiodes measured dc
characteristics, synchronization, and switching speed.

###  3.1 Time-Reconfigurable Periodically-Loaded Microstrip Line

The design methodology of the time-reconfigurable periodically-loaded
microstrip transmission line follows the procedure for reconfigurable DMTLs in
[27, 28]. Fig. 2 shows the equivalent circuit diagram where each loading
element consists of two shunt capacitors $C_{\mathrm{a}}$ and
$C_{\mathrm{b}}$, along with a high-speed photodiode configured to operate as
a switch. The photodiode is connected in series with $C_{\mathrm{b}}$, and is
reversed biased by a dc voltage applied at the input of the transmission line
(not shown). With no light incident on the reversed biased photodiode, the
switch is in the OFF-state – the time-switched capacitor is in State 1, with a
total loading capacitance of $C_{1}=C_{\mathrm{a}}$. When light with optical
power $P$ is incident on the photodiode, the switch moves into the ON-state
(State 2), and $C_{\mathrm{b}}$ becomes connected in parallel to
$C_{\mathrm{a}}$. The total loading capacitance in State 2 becomes
$C_{2}=C_{\mathrm{a}}+C_{\mathrm{b}}$. Thus, by controlling all photodiodes
with synchronized light pulses, the total loading capacitance, or
characteristic impedance, of the entire DMTL is reconfigured in time.

Figure 2:  The equivalent circuit of a time-reconfigurable periodically-loaded
microstrip line. a, Circuit diagram of a time-switched capacitor, where a
high-speed photodiode triggered by an optical pulse changes the impedance of
the load from State 1 with $C_{1}=C_{\mathrm{a}}$ to State 2 with
$C_{2}=C_{\mathrm{a}}+C_{\mathrm{b}}$. Above the equivalent circuit diagram,
the effect of time scattering on a two-peak asymmetric pulse is shown in the
time domain, compared to the spatial domain shown in Fig. 1. Here, the input
pulse (yellow trace) is measured at voltage reference $V_{\mathrm{in}}$ versus
time, with the lower amplitude peak measured before the higher peak. Upon
activation of the time boundary, the transmitted pulse (black trace) measured
at voltage reference $V_{\mathrm{out}}$ maintains the same pulse profile as
the input. However, the reflected pulse (red trace) measured at
$V_{\mathrm{in}}$ becomes reversed in time compared to the input, with the
higher amplitude peak measured before the lower one.

For a DMTL, the characteristic impedances of the loaded line in State 1(2) are
calculated as [27, 28],

$\displaystyle Z_{1(2)}$ $\displaystyle=\frac{Z_{0}}{\sqrt{K_{1(2)}}}$ (4)
$\displaystyle K_{1(2)}$
$\displaystyle=1+\frac{C_{1(2)}}{d}\frac{c_{0}Z_{0}}{\sqrt{\varepsilon_{\mathrm{reff}}}}$
(5)

where $c_{0}$ is the free-space velocity of light in vacuum, $K_{1(2)}$ is the
scaling factor due to the periodic loading in each state, and $C_{1(2)}$ is
the capacitance of each loading element, either in State 1
($C_{1}=C_{\mathrm{a}}$) or State 2 ($C_{2}=C_{\mathrm{a}}+C_{\mathrm{b}}$),
$d$ is the separation distance between loading elements. $Z_{0}$ and
$\varepsilon_{\mathrm{reff}}$ are the characteristic impedance of the unloaded
microstrip and the effective dielectric constant of the unloaded transmission
line [29]. The effective dielectric constant of the loaded transmission line
in either State 1 or State 2 can then be calculated as,

$\varepsilon_{\mathrm{reff-1(2)}}^{\mathrm{DMTL}}=\varepsilon_{\mathrm{reff}}K_{1(2)}$
(6)

As the guided wavelength within a periodically-loaded structure approaches the
separation distance between loadings, a cutoff frequency called the Bragg
frequency $(f_{\mathrm{Bragg-1(2)}})$ for State 1(2),

$f_{\mathrm{Bragg-1(2)}}=\frac{c_{0}Z_{1(2)}}{\pi
dZ_{0}\sqrt{\varepsilon_{\mathrm{reff}}}}$ (7)

Here, due to the smaller impedance of $Z_{2}$ compared to $Z_{1}$, the lower
Bragg frequency in State 2 limits the overall bandwidth of the DMTL.

These constraints must be carefully accounted for to maximize the available
bandwidth of the line. Further, the loading elements have self-resonance
frequencies which can contribute to operating bandwidth limitations; however,
these can be also designed to resonate at frequencies above the Bragg
frequency, and are therefore less significant.

Within the linear operating region of the microstrip, the phase velocity
${v_{\mathrm{p-1(2)}}=c_{0}(\varepsilon_{\mathrm{reff-1(2)}}^{\mathrm{DMTL}})^{-1/2}}$
is equivalent to the group velocity, and is used to determine the time-frame
when the pulse is within the transmission line.

In our study, the effective impedance of the microstrip is designed to change
from 60 $\Omega$ to 40 $\Omega$ upon light activation of the photodiode
switch. The reason for taking this impedance range is two-fold. First, as the
magnitude of the time reflection is proportional to the change in impedance
between the two states, we found a 20 $\Omega$ change in impedance would
generate a reasonable time reflection, strong enough to be detectable. Second,
a State 1 impedance of 60 $\Omega$ minimizes the impedance mismatch of the
spatially reflected pulse back into the 50 $\Omega$ measurement equipment.
Likewise, a State 2 impedance of 40 $\Omega$ after the time-boundary minimizes
the spatial reflections of the reflected and transmitted pulses exiting the
DMTL.

Considering the spatial distribution of a 0.5 GHz Gaussian pulse within the
DMTL, a total of 16 loading elements with a spacing distance of $d$ = 18.3 mm
are chosen to accept the entire pulse completely within the microstrip during
the time-boundary event. The final DMTL design parameters are given in Fig. 2.

Since the experimental results in Section 4 below were taken with an
oscilloscope, an example of time-reversal of a two-peak asymmetric pulse
reflected and transmitted at a time-boundary is shown in the time-domain at
the top of Fig. 2, referenced at the input voltage $V_{\mathrm{in}}$ and
output voltage $V_{\mathrm{out}}$. We see within the input pulse (yellow
trace), the low-amplitude peak leads the larger peak. After the time-boundary
at time $t=t_{\mathrm{kick}}$, the large amplitude peak leads the smaller one
within the reflected wave (red trace) due to time-reversal. This will help
with understanding the measured results to follow.

###  3.2 A Switchable Capacitive Load with High-Speed Photodiodes

A critical design consideration for our time-boundary experiment is the method
of modulating the effective homogeneous medium. The speed at which the time
boundary occurs must be within the order of the period of the signal to be
scattered [8]. Hence, a Gaussian pulse with a carrier frequency of 0.5 GHz,
requires a modulation speed on the order of 1 ns or less. Furthermore, the
modulation of each loading element of the periodically-loaded microstrip line
must be synchronized to occur at precisely the same time to achieve a truly
homogeneous time-switched medium. Finally, the switching element must be
readily available and simple to integrate within the circuit.

To match these requirements, commercially available high-speed PIN photodiodes
are used to switch between the two different capacitive states of the
microstrip line. Photodiodes offer $ps$-scale rise/fall times, low junction
capacitances (${\sim}2~{}\mathrm{pF}$ or less), and are available in surface
mount and fiber-coupled packaging for easy assembly. To activate each
photodiode at the same time, a single laser source is triggered, and its
output is evenly split to each photodiode using optical fibers for near-
perfect (within a few 10s of picoseconds) synchronization.

The equivalent circuit of a high-speed photodiode switch is shown in Fig. 3
(a). Upon excitation by light, a photocurrent
$(I_{\mathrm{ph}}=R_{\mathrm{d}}\,P)$ is generated within the photodiode,
where $R_{\mathrm{d}}$ is the photodiode responsivity with units A/W, and $P$
is the applied optical power. Application of this photocurrent transitions the
diode from a reverse bias OFF-state (State 1) to a conductive ON-state (State
2). The photodiode operation within a linear regime is simulated using the
small-signal approximation [30], with the total photodiode current
$I_{\mathrm{d}}$ computed as follows,

$I_{\mathrm{d}}=-I_{\mathrm{sat}}(\mathrm{e}^{qV_{\mathrm{d}}/nk_{\mathrm{B}}T}-1)+I_{\mathrm{ph}}$
(8)

where $q$ is a unit charge, $n$ is the diode ideality factor, $k_{\mathrm{B}}$
is the Boltzmann constanst, and $T$ is the temperature.

Figure 3:  Equivalent circuit and dc I-V characteristics of a high-speed
photodiode operating as a switch. a, Equivalent circuit diagram of a high-
speed photodiode configured to operate as a switch for input signal
$V_{\mathrm{RF}}$. b, the measured dc I-V characteristics of three different
photodiodes taken from the same lot, demonstrating the change from the
isolating OFF-state ($P$ = 0 mW) to the conductive ON-state (fourth quadrant)
at different applied optical powers $P$. The difference between photodiodes
for both the OFF-state and ON-state is minimal. For the dc I-V curve
measurements, $R_{\mathrm{L}}=3\,\mathrm{k}\Omega$.

Sixteen Fermionics FD50S8-F8-FC/APC InGaAs PIN photodiodes are chosen for our
design. The datasheet specifications of the photodiodes at $V_{\mathrm{bias}}$
= 5 V are: rise/fall times of 200 ps max, saturation currents of
$I_{\mathrm{sat}}$ = 0.05 $\pm$ 0.03 nA, junction capacitances of $C_{j}$ =
0.33 $\pm$ 0.05 pF, and responsivities of $R_{\mathrm{d}}$ = 0.94 $\pm$ 0.04
A/W at 1300 nm. Fig. 3 (b) compares the dc operation of the select group of
three photodiodes, depicting their measured dc I-V curves at different optical
powers. With increased illuminating power, the magnitude of the diode current
in the reverse bias state increases, pushing the operating point of the diodes
into the conductive region within the fourth quadrant, showing insignificant
deviation in the ON-state regimes of the given photodiodes.

To verify the picosecond rise time of the high-speed photodiodes, Fig. 4 shows
the measured response of both a laser diode pulser and a single photodiode in
series. A Highland Technology T165-12 Picosecond Laser Diode Pulser drives a
laser diode providing a picosecond optical pulse to the photodiode. In Fig. 4,
the measured voltage from an optical waveform monitor pin available on the
laser pulser shows a 10-90% rise time of 283 ps, corresponding to a 10-90%
rise time of a 100 MHz sine wave through the photodiode of 311 ps. The
noticable ringing in the laser pulser response is due to EMI artifacts at the
monitor pin, which are not present at the laser diode anode.

Figure 4:  Measured picosecond rise time of a high-speed photodiode. Response
of the laser diode pulser measured at its monitor pin along with the measured
transmission response of the series connected photodiode. A 100 MHz sine wave
is sent into the photodiode circuit from a RIGOL DG4102 function generator,
and the transmitted signal is captured using a high-speed oscilloscope. The
measured 10-90% rise time (OFF-to-ON) of the 100 MHz signal through the
photodiode circuit is 311 ps.

##  4 Experimental Results

Fig. 5 shows the block diagram of the time reflection experiment along with a
picture of the experimental setup, while Fig. 6 depicts a close up of the
device under test (DUT) along with a picture of a single loading element. The
most critical aspect of this experiment was to synchronize the timing of the
high-speed photodiode activation while the RF pulse was within the microstrip
TL. To achieve this, a Tektronix AWG700002A Arbitrary Waveform Generator with
a sampling rate of 25 GSa/s was used to generate both the optical trigger and
RF pulse. This enabled synchronization between the two signals, which could be
tuned in 40 ps steps.

A block diagram showing the procedure we used to generate the time reflection
is shown in Fig. 5 (a). First, a control signal was sent from an arbitrary
waveform generator (AWG) to a Highland Technology T165-12 Picosecond Laser
Diode Pulser, which drove an optical pulse with a picosecond rise time from an
attached laser diode coupled into an optical fiber. A Thorlabs FPL1053S Fabry-
Perot Laser Diode in a butterfly package operating at a wavelength of 1550 nm
provides the optical source. This optical pulse is then split evenly into 16
different optical fibers using a Thorlabs TDS1315HA 1x16 beam splitter, with
each fiber terminating into a high-speed photodiode periodically loading the
microstrip. Considering a tolerance of $\pm$ 5 mm for the length of an
individual fiber and $n\approx~{}1.97$ for its refractive index, leads to a
$\pm$ 33 ps tolerance in synchronized timing between each switch, which is
significantly shorter than the switching time and the carrier period of the
microwave pulse ($2\,\mathrm{ns}$). Hence, the optical pulses reach all 16
photodiodes at essentially the same time, generating a homogeneous and near-
instantaneous change in the microstrip’s effective impedance.

Simultaneous to the optical pulse, a microwave pulse is sent into the
microstrip from the same AWG carefully synchronized and slightly delayed from
the optical control signal such that the microwave pulse is entirely within
the microstrip when the optical pulse reaches the photodiodes and activates
the time reflection. The microwave pulse then interacts with a time-boundary,
with both the transmitted and reflected waves measured with a high-speed
oscilloscope.

A Picosecond Pulse Labs 5330A-140 6-dB power splitter at the input side is
used to enable the measurement of the input signal and the time-reflected
signal at the scope. The measured insertion loss of the splitter is less than
0.1 dB with a return loss greater than 27 dB up to 1 GHz. It is important to
note that both the spatial-reflected and time-reflected signals pass through
the power splitter twice before they are recorded by the scope.

An amplifier is used to boost the signal from the AWG due to its maximum
output swing of 500 mVpp. A Picosecond Pulse Labs 5840A Series amplifier is
used, with 41 ps rise time, 80 kHz to 9.3 GHz of bandwidth, 22 dB of gain, +12
dBm saturation, and 5.8 dB noise figure. A specific gain is dialed in to
achieve a measured input pulse amplitude of 0.5 V at the scope.

A Tektronix DPO 71604C Digital Phosphor Oscilloscope with 16 GHz bandwidth and
100 GSa/s is used to measure both the input and output signals, the trigger to
the laser diode driver, and the monitor pin of the laser diode driver, with an
input impedance of 50 $\Omega$ for each channel. The oscilloscope is set to
average mode collecting 100 samples for each measurement.

Figure 5:  Experimental setup to demonstrate time reflection. a, Block
diagram. b, Picture of the experimental setup. Figure 6:  Fabricated and
assembled prototype. A picture of the fabricated and assembled device under
test (DUT) along with a close up of a single loading element with a surface
mount high-speed photodiode. Figure 7:  Measured and simulated frequency
response of the periodically loaded microstrip transmission line. a,
Scattering parameters of both State 1 (photodiode switch OFF) and State 2
(photodiode switch ON). b, Corresponding dispersion diagram.

###  4.1 Frequency Response

Figure 8:  Signal flow graph for the experiment with no time boundary. The
signal enters from the source (AWG), is divided by the 6-dB power splitter
(PS), and is then scattered at the first spatial boundary (SB1). The
transmitted wave ($T_{\mathrm{SB1}}$) travels through the periodically-loaded
microstrip line (device under test - DUT), and is scattered at the second
spatial boundary (SB2) before exiting the DUT ($T_{\mathrm{SB2}}$) and
measured at the output $V_{\mathrm{out}}$ (channel 2 of the scope). The
reflected signal at SB1 ($\Gamma_{\mathrm{SB1}}$) is again split by the PS and
measured at the input $V_{\mathrm{in}}$ (channel 1 of the scope), with the
other half absorbed by the source. The reflected signal at SB2
($\Gamma_{\mathrm{SB2}}$) travels back down the DUT, and is again scattered by
SB1. The transmitted signal ($T_{\mathrm{SB1}}$) is split by the PS and
measured at $V_{\mathrm{in}}$. The characteristic impedance $Z_{0}$ of the
system outside the DUT is 50-$\Omega$.

Fig. 7 (a) compares the measured and simulated scattering parameters of the
periodically loaded microstrip line for the two states, with good agreement.
While initially designed for operation up to 1 GHz by limitation of the Bragg
frequency cutoff, the measured bandwidth has been limited by the insertion
loss, most significantly in State 2. This is due to the $R_{\mathrm{on}}$ of
the photodiode, estimated to be approximately 8 $\Omega$ using the method in
[30]. Losses and parasitics from the discrete components also contribute to
the reduced bandwidth. A more optimized photodiode package and periodic
loading design using integrated elements could improve the overall bandwidth
of the device beyond 1 GHz.

Fig. 7 (b) shows the measured vs. simulated dispersion for the two states,
again with good agreement. We use the experimental data to predict the
measured frequency translation for a pulse centered at 0.6 GHz. From the
conservation of momentum described above in Section 2, the momentum in State 1
and State 2 are equivalent across the time-boundary, i.e., $k_{1}$ = $k_{2}$.
Thus, translating the frequency of 0.6 GHz in State 1 to State 2, gives a new
frequency of 0.51 GHz, yielding a measured (simulated) frequency translation
ratio of $\omega_{2}/\omega_{1}$ = 0.85 (0.75), compared to the theoretical
frequency translation of 0.67 for an ideal impedance transformation
$Z_{1}\rightarrow Z_{2}$ of 60 $\Omega$ $\>\rightarrow\>$ 40 $\Omega$. The
total length of the periodically-loaded microstrip line is 332.7 mm.

###  4.2 Time Reflection of a Single Pulse

This section presents our simulated and measured results of both the reflected
and transmitted signals of a single Gaussian pulse with a full-width half-
maximum (FWHM) bandwidth of 0.35 GHz scattered at a time interface. Due to the
presence of both spatial boundaries and time boundaries in the experiment,
signal flow graphs are provided in Fig. 8 and Fig. 11 for the case without and
with the time interface, respectively, to provide clarity in the analysis of
the simulated and measured results. To extract the time-boundary reflection
and transmission coefficients, it is first necessary to extract the spatial-
boundary reflection and transmission coefficients in the static case for both
State 1 and State 2.

Based on the signal flow diagram without the time boundary of Fig. 8, Fig. 9
compares the simulated vs. measured signals when the microstrip is in State 1
for the input $V_{\mathrm{in}}$ and output $V_{\mathrm{out}}$, shown in panels
(a) and (b) respectively. Using the datasets of these two pulses, we extract
the reflection and transmission coefficients at each spatial interface
following a procedure detailed in Supplementary Section 2. The simulated and
measured results compared to the theoretical values are summarized in Table 1,
with good agreement. The discrepancy for the measured
$\Gamma_{\mathrm{SB1}}^{(1)}$ is likely due to the loss in the SMA connector
since the spatial boundary is within the microstrip line. This loss is not
accounted for in the extraction process.

Figure 9:  Measured and simulated response to an input pulse with the microstrip in State 1. a, The response referenced at the input $V_{\mathrm{in}}$ to a 0.35 GHz bandwidth Gaussian pulse with the photodiodes in the OFF-state (State 1). b, The output response referenced at $V_{\mathrm{out}}$ . The dashed blue trace represents the simulated response with 24 loading elements in the model, confirming that the reflected pulses indicated as SB3 in the figure are indeed due to the output spatial boundary SB2 reflected to the input. Table 1: Spatial-Boundary Scattering Coefficients and Characteristic Impedance in State 1 | $\Gamma_{\mathrm{SB1}}^{(1)}$ | $T_{\mathrm{SB1}}^{(1)}$ | $\Gamma_{\mathrm{SB2}}^{(1)}$ | $T_{\mathrm{SB2}}^{(1)}$ |  $Z_{(1)}$ ($\Omega$)
---|---|---|---|---|---
Theory | 0.091 | 1.091 | -0.091 | 0.909 | 60
Simulation | 0.080 | 1.080 | -0.062 | 0.938 | 56.6
Measurement | 0.030 | 1.030 | -0.075 | 0.925 | 58.1
|  |  |  |  |

The reflected pulse from SB2 travels back down the transmission line, and is
recorded as a third spatial boundary SB3 in Fig. 9 (a). The scattering
coefficients of this third spatial boundary are assumed to be equivalent to
the coefficients for SB2. The dashed blue trace in Fig. 9 (a) represents the
simulated model with 24 loading elements, indicating the pulse at SB3 is
indeed due to the reflection off SB2 and transmitted through SB3 back to the
input.

Figs. 10 (a) and (b) compare the simulated vs. measured results for the static
case when the microstrip line is in State 2, with the ON-state photodiodes.
The optical pulse is turned on for approximately 52 ns for the measured
results to capture all dynamic events. The response at the input
$V_{\mathrm{in}}$ is plotted in (a), showing the reflected pulse from the
first spatial boundary SB1. The response at the output $V_{\mathrm{out}}$ is
plotted in (b), showing the transmitted pulse through the second spatial
boundary SB2. The method detailed in Supplementary Section 2 is again used to
extract the reflected and transmitted coefficients, summarized in Table 2 and
compared to the theoretical values with good agreement. The discrepancy in
measured $\Gamma_{\mathrm{SB1}}^{(2)}$ is again likely due to the SMA
connector loss, similar to the State 1 analysis above. The dashed blue trace
in (a) represents the simulated response with 24 elements, again demonstrating
the pulse at SB3 is due to the reflection off the output spatial boundary SB2
and captured at the input, where it is delayed with more loading elements. The
simulated and measured pulses at 13.6 ns and 10.7 ns, respectively, labeled
SB3, are due to this reflection.

Figure 10:  Measured and simulated response to an input pulse with the microstrip in State 2. a, The response referenced at the input $V_{\mathrm{in}}$ to a 0.35 GHz bandwidth Gaussian pulse with the photodiodes in the ON-state (State 2). b, The output response referenced at $V_{\mathrm{out}}$. The dashed blue trace again represents the simulated response with 24 loading elements in the ADS model. Table 2: Spatial-Boundary Scattering Coefficients and Characteristic Impedance in State 2 | $\Gamma_{\mathrm{SB1}}^{(2)}$ | $T_{\mathrm{SB1}}^{(2)}$ | $\Gamma_{\mathrm{SB2}}^{(2)}$ | $T_{\mathrm{SB2}}^{(2)}$ |  $Z_{2}$ ($\Omega$)
---|---|---|---|---|---
Theory | -0.111 | 0.888 | 0.111 | 1.111 | 40
Simulation | -0.114 | 0.886 | 0.128 | 1.128 | 38.6
Measurement | -0.058 | 0.942 | 0.061 | 1.061 | 44.2
|  |  |  |  |

The discrepancy of 2.9 ns between the simulated and measured pulses at SB3 is
likely due to differences in the simulated and measured phase velocity of
State 2. The phase velocities of the two states are extracted from the
dispersion diagram in Fig. 7 (b). For State 1, the simulated and measured
phase velocities $\nu_{\mathrm{p1}}$ are 9.6 m/s and 9.2 m/s, respectively.
For State 2, the simulated and measured velocities $\nu_{\mathrm{p2}}$ are
7.2$\times$107 m/s and 7.9$\times$107 m/s, respectively. The slower speed and
larger discrepancy in the simulated and measured phase velocities of State 2
corroborates this discrepancy.

Figure 11:  Signal flow graph for the time boundary experiment. The microstrip
line’s characteristic impedance and phase velocity is changed from State 1 to
State 2 at time $t$ = $t_{\mathrm{kick}}$ while the pulse is completely within
the DUT. Similar to the case without a time boundary in Fig. 8, the signal
enters from the source (AWG), is divided by the 6-dB power splitter (PS), and
is then scattered at the first spatial boundary SB1(1) in State 1. The
transmitted wave travels through the microstrip line in State 1, which is then
switched to State 2 while the pulse is completely within the DUT, generating a
time boundary TB1 with transmitted and reflected wave coefficients
$T_{\mathrm{TB1}}$ and $\Gamma_{\mathrm{TB1}}$, respectively. The reflected
signal travels back down the microstrip in State 2, and scatters on the first
spatial boundary SB1(2), now in State 2. This transmitted pulse then splits at
the PS and enters channel 1 of the scope ($V_{\mathrm{in}}$), as the measured
time-boundary reflection. The time-boundary transmitted signal travels down
the microstrip in State 2, and scatters at spatial boundary SB2(2), also in
State 2. This transmitted signal is then measured at $V_{\mathrm{out}}$
(channel 2 of the scope). The reflected signal travels back down the
microstrip line in State 2, scattering at SB1(2), split at the PS, and
measured at $V_{\mathrm{in}}$.

With the spatial coefficients known, the time-boundary coefficients are now
extracted. Referring to the signal flow graph with the time boundary in Fig.
11, the simulated and measured results are plotted in Fig. 12 (a) and (b)
referenced at the input voltage $V_{\mathrm{in}}$ and output voltage
$V_{\mathrm{out}}$, respectively. The time boundary is activated at
$t=t_{\mathrm{kick}}$, when the pulse has fully entered but has yet to exit
the microstrip trace. As mentioned previously in Section 3, the total length
of the microstrip line in State 1 was chosen to be just long enough to contain
the spatial spread of a single pulse, thus the pulse has travelled
approximately halfway through the line before the time-boundary is activated.

Referring to Fig. 12 (a), the reflected pulse due to the first spatial
boundary at SB1 is clearly visible with a positive amplitude due to the
microstrip line being in State 1. The reflected pulse due to the output
spatial boundary SB2 and indicated by SB3 at the input is also clearly visible
with positive amplitude due to the microstrip line now being in State 2.
However, we now see a pulse with negative amplitude between these two points,
indicated by TB1. This pulse is due to the time interface event triggered at
time $t_{\mathrm{kick}}$ = 6.85 ns. The time-boundary reflection and
transmission coefficients are extracted using the method described in
Supplementary Section 2, and are plotted along with the theoretical values in
Table 3, showing good agreement.

Figure 12:  Measured and simulated response of the time-boundary experiment. a, The response to a 0.35 GHz bandwidth Gaussian pulse referenced at the input $V_{\mathrm{in}}$ with a time boundary at time $t_{\mathrm{kick}}$ = 6.85 ns. b, Output response referenced at $V_{\mathrm{out}}$. The dashed blue trace again represents the simulated response with 24 loading elements in the ADS model confirming that the reflected pulse indicated as SB3 in the figure is due to the output spatial boundary SB2 reflected to the input. Table 3: Time-Boundary Scattering Coefficients | $\Gamma_{\mathrm{TB1}}$ | $T_{\mathrm{TB1}}$
---|---|---
Theory | -0.111 | 0.556
Simulation | -0.084 | 0.604
Measurement | -0.048 | 0.780
|  |

The slight discrepancy in the measured $\Gamma_{\mathrm{TB1}}$ is likely due
to losses and dispersion in the measured prototype unaccounted for in the
extraction process, as well as the finite switching speed of the photodiodes.
While their switching speed was confirmed to be 311 ps, as discussed above in
Section 3, the period of a 350 MHz signal is ${\sim}$2.9 ns. While the
switching time is faster than the period of the wave traveling through the
medium, a requirement for time-crystals [8], a faster switching time would
lead to a more abrupt time interface. This would lead to an increased
reflection amplitude similar to an equivalent abrupt spatial interface, as
explored in [26].

To further demonstrate the pulse reflection and transmission indicated by TB1
is due to a time boundary, Fig. 13 (a) and (b) plot the measured and simulated
response, respectively, of a single Gaussian pulse in which the triggering of
the time-interface $t_{\mathrm{kick}}$ is varied. In Fig. 13 (a), we see that
by delaying the triggering of the kick from $t_{1}$ to $t_{3}$, the time-
reflected pulse at TB1 is delayed accordingly. In Fig. 13 (b), we see a
similar response in the simulated model, thus confirming this pulse is
generated by the time-boundary event.

Figure 13:  Measured and simulated time reflection of a single Gaussian pulse
while adjusting the triggering of the time interface. a, Measured response
referenced at the input $V_{\mathrm{in}}$ while adjusting the triggering of
the time interface with three different steps at $t_{1}$ = 6.47 ns, $t_{2}$ =
6.75 ns, and $t_{3}$ = 6.97 ns. b, Simulated response corresponding well to
measurement.

###  4.3 Time Reflection of a Two-Peak Asymmetric Pulse

Figure 14:  Measured results of two sequential pulses fed into the microstrip
line in which the amplitude of the first pulse is varied. a, Demonstration of
a spatial reflection without a time interface. b Demonstration of a time
reflection with the time interface occurring at $t_{\mathrm{kick}}$ = 8.4 ns.

Fig. 14 illustrates the experimental demonstration of the time-reversal
property of reflected waves at a time-boundary discussed in Section 2 and
Section 3 above. The figure shows the measured input voltage versus time of a
two-peak asymmetric pulse fed into the microstrip and the corresponding
reflection signals without (a) and with the time interface (b). By varying the
amplitude of the first pulse, it is clear that (a) demonstrates a spatial
reflection, i.e., the leading pulse is the first to reflect due to the
impedance mismatch at the output and return to the input. In contrast, by
applying a time-boundary at time $t_{\mathrm{kick}}$ = 8.4 ns, the time at
which the pulse has fully entered the microstrip line but has yet to exit, (b)
demonstrates a time reflection. Here, the leading input peak becomes the
lagging peak due to the time reversal property of the time interface, a
homogeneous and near-instantaneous change in the effective characteristic
impedance of the microstrip. Likewise, after the time-boundary, the lagging
input pulse becomes the leading peak of the reflected wave. The FWHM bandwidth
of each peak is approximately 0.4 GHz.

###  4.4 Frequency Translation Demonstration

To demonstrate frequency translation at a time-boundary as discussed in
Section 2 with our prototype, the time reflection and transmission of an input
Gaussian pulse with a single carrier frequency was measured. Time-gated
Fourier transforms are applied independently over the input (red trace),
reflected (blue trace), and transmitted (green trace) to extract their
spectral and phase content, with the results plotted in Fig. 15. The spectral
phase versus frequency are plotted across the FWHM of each pulse.

Figure 15:  Measured results of a single carrier Gaussian input pulse
scattered at a time interface. The input pulse has a carrier frequency of 0.59
GHz incident at a time-boundary triggered at $t$ = $t_{\mathrm{kick}}$, with
the corresponding time-reflected and transmitted pulses demonstrating
frequency translation.

The input pulse in the frequency domain is centered at 0.59 GHz. The time-
reflected pulse is centered at 0.43 GHz, and the transmitted pulse is centered
at 0.51 GHz. This is conclusive evidence of frequency translation in our
prototype, and also demonstrates the reflection of an electromagnetic pulse
with the highest-to-date frequency from an optically-controlled ultra-fast
time interface. The frequency translation corresponds to an
$\omega_{2}/\omega_{1}$ ratio of 0.74 and 0.88 for the time-reflected and
transmitted waves, respectively, compared to the expected 0.67 from the theory
presented in Section 2 above. The discrepancy between measurement and theory
is thought to be due to the finite length of the transmission line. Part of
the pulse has begun to exit the microstrip line just before the photodiodes
have completely turned on, triggering the time interface.

Regarding the spectral phase plots in Fig. 15, the input pulse has a phase of
-81.7 degrees at 0.59 GHz, the time-reflected pulse has a phase of -196.6
degrees at 0.43 GHz, and the transmitted pulse has a phase of -273.4 degrees
at 0.51 GHz. Extrapolating the curves back to dc, the time-reflected signal
exhibits a close to 180-degree shift compared to the input and time-refracted
signals, corresponding to the theory prediction [26]. The slope of the time-
reflected signal is also shallower than the input signal, indicating negative
phase accumulation (counter-clockwise rotating phase vector) of the signal
after the time boundary. This interesting effect will be explored in the next
section.

###  4.5 Time Reversal and Phase Conjugation Demonstration

Figure 16:  Measured results of an input chirped pulse with a Gaussian
envelope and nonlinear phase and the corresponding time-reflected and
transmitted pulses demonstrating both frequency translation and time reversal.

This section provides experimental evidence of the phase-conjugation nature of
time-reflected waves [7, 6, 24, 8]. From the voltage relationship (S19)
derived in Supplementary Section 1, and considering the transmitted and
reflected waves have the same angular frequency magnitude but are 180 degrees
out of phase, i.e., $\omega_{2-T}=(-)\omega_{2-R}$, the transmitted and time-
reflected waves can be split up accordingly [1]:

$V_{\mathrm{2-T}}=V_{01}Te^{j(\omega_{\mathrm{2-T}}t-kx)}=V_{01}T[cos(\omega_{\mathrm{2-T}}t-kx)+jsin(\omega_{\mathrm{2-T}}t-kx)]$
(9)

$V_{\mathrm{2-R}}=V_{01}Re^{-j(\omega_{\mathrm{2-R}}t+kx)}=V_{01}R[cos(\omega_{\mathrm{2-R}}t+kx)-jsin(\omega_{\mathrm{2-R}}t+kx)].$
(10)

Here $V_{\mathrm{2-T}}$ and $V_{\mathrm{2-R}}$ represent the transmitted and
time-reflected waves, respectively, while $\omega_{\mathrm{2-T}}$ and
$\omega_{\mathrm{2-R}}$ represent the transmitted and reflected waves’ angular
frequency, respectively. These two waves have also been split into their
quadrature components to highlight the phase rotation for each.

We see that the transmitted wave in (9) rotates counter-clockwise, such that
the wave incurs positive phase as it propagates in the positive $x$ direction.
However, the phase rotation for the time-reflected wave in (10) is clockwise,
such that the wave incurs negative phase as it propagates in the negative $x$
direction. Thus, the time-reflected wave is essentially a time-reversed wave,
where any phase incurred in the wave before the time-reflection can be
reversed in the time-reflected wave after the time-reflection.

To measure this time-reversal property, a chirped pulse with a Gaussian
envelope and nonlinear phase was sent into our microstrip prototype and time
reflected. The details of the input chirped pulse can be found in
Supplementary Section 3. The experimental results are plotted in Fig. 16.
Again, frequency translation of the input pulse is observed, as expected, and
the phase versus frequency of both the input and transmitted pulses have a
negative slope. However, the phase of the time-reflected pulse has a positive
slope, indicating phase-conjugation of the reflected wave.

##  5 Conclusion

We experimentally demonstrate the reflection and transmission of
electromagnetic waves from a broadband microwave pulse with the highest-to-
date carrier frequency of 0.59 GHz due to an optically-controlled ultra-fast
time interface. The design and implementation of this time-boundary is enabled
by a reconfigurable periodically-loaded microstrip transmission line capable
of achieving near-instantaneous change in its effective impedance between two
states using commercially-available high-speed photodiodes with picosecond
switching rates. The prototype is optimized to incorporate advanced yet
realistic and accessible design solutions with practical advantages for real-
world applications in future microwave systems. Excellent agreement between
theory, simulation, and measurement is shown, validating the unique properties
of time-reflected waves at microwave frequencies, including the time-reversal,
frequency translation, and phase-conjugate nature of time-reflected waves.

Acknowledgments

This work is supported by the AFOSR under FY21-MURI award number
FA9550-21-1-0299 (Dr. Arje Nachman, Program Manager).

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## Supplementary Information

S1. Derivation of the Reflection and Transmission Coefficients of a Time
Boundary

Figure S1:  Distributed circuit diagram of a microwave transmission line for
the case when an additional capacitive loading is switched on at time $t$ = 0.

Fig. S1 shows the distributed circuit diagram of a microwave transmission line
(TL) used to derive the reflection $R$ and transmission $T$ coefficients due
to the scattering of a microwave signal at a time interface. This time
interface is generated by switching on a second shunt capacitive loading
within the TL, increasing the effective permittivity and decreasing the
characteristic impedance of the line. Before the switching event, only the
shunt capacitor $C_{\mathrm{a}}$ loads the TL, for a total capacitance of
$C_{1}=C_{\mathrm{a}}$. At time $t=0$, the switch is thrown, and the
additional shunt capacitance $C_{\mathrm{b}}$ is added to the circuit, for a
total capacitance of $C_{2}=C_{\mathrm{a}}+C_{\mathrm{b}}$. Referencing the
treatment in [26], the corresponding telegraphers equations at time $t>0$ are
derived beginning with [29]:

$v(x,t)=L\Delta x\frac{\partial i(x,t)}{\partial t}+v(x+\Delta x,t)$ (S1)

$i(x,t)=C_{\mathrm{a}}\Delta x\frac{\partial v(x+\Delta x,t)}{\partial
t}+C_{\mathrm{b}}\Delta x\frac{\partial v(x+\Delta x,t)}{\partial
t}+i(x+\Delta x,t)$ (S2)

Dividing both equations by $\Delta x$ and taking the limit $\Delta x\to 0$
yields the telegrapher’s equations:

$\frac{\partial v(x,t)}{\partial t}=-L\frac{\partial i(x,t)}{\partial t}$ (S3)

$\frac{\partial i(x,t)}{\partial t}=-C_{\mathrm{a}}\frac{\partial
v(x,t)}{\partial t}-C_{\mathrm{b}}\frac{\partial v(x,t)}{\partial t}$ (S4)

Multiplying (S3) by $\frac{\partial}{\partial x}$ and (S4) by
$\frac{\partial}{\partial t}$, and then plugging (S4) into (S3) yields the
equation:

$\frac{\partial^{2}v(x,t)}{\partial
t^{2}}=LC_{\mathrm{a}}\frac{\partial^{2}v(x,t)}{\partial
t^{2}}+LC_{\mathrm{b}}\frac{\partial^{2}v(x,t)}{\partial t^{2}}$ (S5)

A Laplace transform is now applied on the voltage $v(x,t)$ according to
$\mathcal{L}\\{v(x,t)\\}=\int_{0}^{\infty}v(x,t)\mathrm{e}^{-st}dt=V(x,s)$ and
$\mathcal{L}\\{\frac{\partial^{2}v(x,t)}{\partial
t^{2}}\\}=s^{2}V(x,s)-sv(x,0^{-})-\frac{\partial v(x,0^{-})}{\partial t}$
[31], and assuming a time-harmonic $\mathrm{e}^{j\omega t}$ variation of the
voltage and currents, the above equation becomes:

$\begin{split}\frac{\partial^{2}v(x,t)}{\partial
t^{2}}&=LC_{\mathrm{a}}\\{s^{2}V(x,s)-sv(x,0^{-})\\}\\\
&+LC_{\mathrm{b}}\\{s^{2}V(x,s)-sv(x,0^{-})-j\omega_{1}v(x,0^{-})\\}\end{split}$
(S6)

Since at $t<0$ the voltage drop across $C_{\mathrm{b}}$ is equal to zero, and
recognizing $C_{2}=C_{\mathrm{a}}+C_{\mathrm{b}}$ and $C_{1}=C_{\mathrm{a}}$,
the above equation can be simplified to the following wave equation for the
voltage:

$\left(\frac{\partial^{2}}{\partial
x^{2}}-LC_{2}s^{2}\right)V(x,s)=-LC_{1}(s+j\omega_{1})v(x,0^{-})$ (S7)

where $\omega_{1}$ is the angular frequency of the signal before the time
boundary. Assuming a harmonic variation in space $\mathrm{e}^{-jkx}$ gives the
voltage,

$V(x,s)=\frac{C_{1}(s+j\omega_{1})}{C_{2}(\omega_{2}^{2}+s^{2})}v(x,0^{-})$
(S8)

where $\omega_{2}$ is the angular frequency of the signal after the time
boundary, due to frequency translation. Since the TL is effectively a
homogeneous medium and there is no variation in $x$ across the time barrier,
the conservation of momentum between the two states yields,

$k=\omega_{1}\sqrt{LC_{1}}=\omega_{2}\sqrt{LC_{2}}$ (S9)

$\omega_{2}=\sqrt{\frac{C_{1}}{C_{2}}}\omega_{1}=\frac{Z_{2}}{Z_{1}}\omega_{1}$
(S10)

where $Z_{1}$ and $Z_{2}$ are the characteristic impedance of the TL before
and after the time boundary, respectively. Equation (S10) predicts the
frequency translation across the time boundary as a function of the ratio of
the two capacitive loading states, or the impedances of the two different
states. The above assumes a time-harmonic field, where the second order time
derivative on the right side of (S5) is equal to $-\omega^{2}$, and the second
order spatial derivative on the left side of (S5) is equal to $-k^{2}$. Since
momentum is conserved across the time interface, the spatial derivative on the
left side of (S5) is equivalent in both states, before and after the time
interface.

Using the initial value theorem $\lim_{t\to 0^{+}}f(t)=\lim_{s\to\infty}sF(s)$
[31] and (S8) for $V(x,s)$ above, and simplifying with $s_{1}=j\omega_{1}$ and
$s_{2}=j\omega_{2}$, the boundary condition across the time boundary is
obtained from,

$\displaystyle\lim_{t\to 0^{+}}v(x,t)$ $\displaystyle=v(x,0^{+})$ (S11)
$\displaystyle\lim_{s\to\infty}sV(x,s)$
$\displaystyle=\frac{s^{2}C_{1}(1+\nicefrac{{s_{1}}}{{s}})}{s^{2}C_{2}(1-(\nicefrac{{s_{2}}}{{s}})^{2})}v(x,0^{-})$
(S12)

which yields the first boundary condition:

$C_{1}v(x,0^{-})=C_{2}v(x,0^{+})$ (S13)

This boundary condition state that the total charge $Q$ remains constant
across the time boundary. Taking the time derivative $\frac{\partial}{\partial
t}$ of this first boundary condition yields:

$C_{1}\frac{\partial v(x,0^{-})}{\partial t}=C_{2}\frac{\partial
v(x,0^{+})}{\partial t}$ (S14)

which states that the current across the time boundary also remains constant.
Thus we get the second boundary condition across the time boundary as:

$I_{1}=I_{2}$ (S15)

Finally, the reflection and transmission coefficients can be derived from the
scattering equations before and after the switching event using the boundary
conditions derived above. Without losing generality, the switching event is
placed at time $t=0$. At $t<0$, before the switching event, the incident
voltage and current are,

$V_{1}=V_{01}\mathrm{e}^{s_{1}t-jkx}$ (S16)

$I_{1}=I_{01}\mathrm{e}^{s_{1}t-jkx}=\frac{V_{01}}{Z_{1}}\mathrm{e}^{s_{1}t-jkx}$
(S17)

where the voltage and current are related by the impedance in the initial
state:

$Z_{1}=\frac{V_{01}}{I_{01}}=\sqrt{\frac{L}{C_{1}}}$ (S18)

At $t>0$, after the switching event, the voltage and current can be
represented as:

$V_{2}=[T\mathrm{e}^{s_{2}t}+R\mathrm{e}^{-s_{2}t}]V_{01}\mathrm{e}^{-jkx}$
(S19)

$I_{2}=\frac{1}{Z_{2}}[T\mathrm{e}^{s_{2}t}-R\mathrm{e}^{-s_{2}t}]V_{01}\mathrm{e}^{-jkx}$
(S20)

where $R$ is the reflection coefficient, and $T$ the transmission coefficient,
and where the voltage and current are again related by the impedance, now in
the second state:

$Z_{2}=\frac{V_{02}}{I_{02}}=\sqrt{\frac{L}{C_{2}}}$ (S21)

It is also noted that the transmitted signal and reflected signal have similar
angular frequency magnitudes $|\omega_{2}|$, but are 180 degrees out of phase
in order to have distinct fields upon the time discontinuity [1]. This is
important when considering the phase relationship of the input, time
reflected, and time refracting signals, and more specifically, the time
reversal of the time-reflected wave.

Plugging the voltages and currents before and after the switching event into
the boundary conditions (S13) and (S15) at $t=0$ and solving yields the
following equations:

$\frac{C_{1}}{C_{2}}=T+R$ (S22)

$\frac{Z_{2}}{Z_{1}}=T-R$ (S23)

From these equations the relationship for the reflected and transmitted
coefficients in terms of either the capacitance’s of the two states $C_{1}$
and $C_{2}$, or the impedances of the two states $Z_{1}$ or $Z_{2}$, can be
derived as the following [26]:

$R=\frac{1}{2}\left[\frac{C_{1}}{C_{2}}-\sqrt{\frac{C_{1}}{C_{2}}}\right]=\frac{Z_{2}(Z_{2}-Z_{1})}{2Z_{1}^{2}}$
(S24)

$T=\frac{1}{2}\left[\frac{C_{1}}{C_{2}}+\sqrt{\frac{C_{1}}{C_{2}}}\right]=\frac{Z_{2}(Z_{2}+Z_{1})}{2Z_{1}^{2}}$
(S25)

S2. Extraction of Time-Boundary Reflection and Transmission Coefficients

Due to both spatial-boundaries and time-boundaries within the experiment, the
time-boundary reflection and transmission coefficients must be properly
extracted from the measured and simulated responses at $V_{\mathrm{in}}$ and
$V_{\mathrm{out}}$. Here we show the calculations and procedures used in order
to extract their magnitudes from the measured and simulated data. This
analysis assumes no insertion losses outside the microstrip line and power
splitter, i.e., the losses associated with the coaxial cables, which for the
frequency range of this work were deemed negligible.

First, the spatial-boundary reflection and transmission coefficients are
extracted from the measured or simulated responses in State 1 (photodiodes in
the OFF-state) and State 2 (photodiodes in the ON-state), both in steady-state
without a time-boundary. From the response at $V_{\mathrm{in}}$, the peak
voltage of the pulse at SB1, $V_{\mathrm{SB1}}$, as shown in Fig. 9 (a) for
State 1 and Fig. 10 (a) for State 2, along with the input voltage
$V_{\mathrm{input}}$, can be used to calculate the first spatial reflection
$\Gamma_{\mathrm{SB1}}$,

$V_{\mathrm{SB1}}=V_{\mathrm{input}}\,\Gamma_{\mathrm{SB1}}\,PS.$ (S26)

where $PS$ is the magnitude of the power splitter ratio including insertion
loss. Assuming conservation of energy at a spatial boundary [29], the
transmission coefficient can then be calculated from
$T_{\mathrm{SB1}}=1+\Gamma_{\mathrm{SB1}}$.

Next, from the output voltage response $V_{\mathrm{out}}$ shown in Fig. 9 (b)
for State 1, with the peak voltage of the pulse at SB2, $V_{\mathrm{SB2}}$,
the transmission coefficient at the second spatial-boundary $T_{\mathrm{SB2}}$
is extracted using the following equation:

$V_{\mathrm{SB2}}=V_{\mathrm{input}}\,T_{\mathrm{SB1}}\,\mathrm{Loss}\,T_{\mathrm{SB2}}.$
(S27)

where $\mathrm{Loss}$ is the voltage ratio of the transmission power loss
according to $\mathrm{Loss}=1-\lvert S_{11}\rvert^{2}-\lvert S_{21}\rvert^{2}$
[29], taken from the frequency response in Fig. 7. Similar to before, the
reflection coefficient $\Gamma_{\mathrm{SB2}}$ is calculated using the
conservation of energy as $\Gamma_{\mathrm{SB2}}=T_{\mathrm{SB2}}-1$ [29]. For
a Gaussian shaped pulse, a weighted average of the pulse in the frequency
domain is used to calculate the transmission loss across the FWHM bandwidth of
the pulse.

For State 2, due to increased frequency dependent loss and dispersion visible
in Fig. 7, the transmission loss is approximated to be
$V_{\mathrm{SB2}}/V_{\mathrm{input}}$, yielding $T_{\mathrm{SB2}}\approx
T_{\mathrm{SB1}}^{-1}$. Furthermore, for both states, the reflection and
transmission coefficients at SB3 are assumed to be equivalent to those at SB2
due to symmetry.

With the spatial reflection and transmission coefficients known, the
characteristic impedance of the microstrip line in both State 1 and State 2 is
calculated using the well known equation $\Gamma=(Z_{1}-Z_{0})/(Z_{1}+Z_{0})$
[29].

As detailed in the main text, when the characteristic impedance of the
microstrip line is changed from State 1 to State 2, a time-boundary is formed.
If a signal is propagating within the microstrip line during this time-
boundary, a time-scattering event occurs, generating a time-reflected and
time-transmitted wave. Fig. 11 demonstrates the signal flow with a time-
boundary generated by changing the impedance of the microstrip line from State
1 to State 2. From this, along with the simulated and measured responses at
$V_{\mathrm{in}}$ and $V_{\mathrm{out}}$ and the previously derived spatial-
boundary coefficients, we now outline the extraction process for the time-
boundary reflection and transmission coefficients.

Beginning with the input response $V_{\mathrm{in}}$ for the time-boundary
experiment shown in Fig. 12 (a), the peak voltage magnitude $V_{\mathrm{SB1}}$
of the pulse at SB1 is used to extract the time reflection coefficient
$\Gamma_{\mathrm{TB1}}$,

$V_{\mathrm{TB1}}=V_{\mathrm{input}}\,T_{\mathrm{SB1}}^{(1)}\,\mathrm{Loss}^{(1)}_{\text{1\textfractionsolidus
2}}\,\Gamma_{\mathrm{TB1}}\,\mathrm{Loss}^{(2)}_{\text{1/2}}\,T_{\mathrm{SB3}}^{(2)}\,PS$
(S28)

The superscript indicates which state the spatial-reflection coefficients and
the transmission loss are in; before the time-boundary, the microstrip is in
State 1, while after the boundary the microstrip is in State 2. The bandwidth
of the pulse also changes between State 1 and State 2 due to frequency
translation. Furthermore, we assume the signal has transmitted half the length
of the microstrip line before the time event at $t=t_{\mathrm{kick}}$, thus
only half the power loss is used for each state, as indicated by the
$\mathrm{Loss}_{1/2}$ term. This is a reasonable assumption considering the
microstrip line for this experiment was designed to be approximately the width
of the pulse in the time-domain considering the phase velocity of State 1. The
timing of the kick was carefully synchronized to occur right when the pulse
has fully entered the line, and just before it exits, according to the
discussion in Section 4. The actual length of travel of the pulse can be
calculated if more accuracy is required [26].

Finally, using the output response $V_{\mathrm{out}}$ for the time-boundary
experiment shown in Fig. 12 (b), the peak voltage magnitude $V_{\mathrm{SB2}}$
of the pulse at SB2 is used to extract the time transmission coefficient
$T_{\mathrm{TB1}}$ using the following equation:

$V_{\mathrm{SB2}}=V_{\mathrm{input}}\,T_{\mathrm{SB1}}^{(1)}\,\mathrm{Loss}^{(1)}_{1/2}\,T_{\mathrm{TB1}}\,\mathrm{Loss}^{(2)}_{1/2}\,T_{\mathrm{SB2}}^{(2)}$
(S29)

S3. Chirped Pulse Parameters for Time Boundary Experiment with Phase Variation

A chirped pulse with a linear instantaneous frequency $f(t)=ct+f_{0}$ has a
quadratic instantaneous phase:

$\phi(t)=2\pi(\frac{c}{2}t^{2}+f_{0}t)+\phi_{0}$ (S30)

where $c$ is the chirpiness, defined as:

$c=\frac{f_{1}-f_{0}}{T}$ (S31)

with $f_{0}$ the initial frequency, and $f_{1}$ the end frequency, of the
chirp, and $T$ the transition time between $f_{0}$ to $f_{1}$. The initial
phase at $t=0$ is represented by $\phi_{0}$. Considering a travelling wave
with spatial representation in a non-dispersive medium, with $t\rightarrow
t-x/\nu_{\mathrm{p}}$, the voltage of the chirped signal can then be
represented as:

$v(x,t)=cos(\pi c(t-\frac{x}{\nu_{\mathrm{p}}})^{2}+\omega t-kx)$ (S32)

where $\nu_{\mathrm{p}}$ is the phase velocity of the transmission line.


# Low Seesaw Scale Solution for $M_{W}$ and $(g-2)_{\mu}$

A. DE GIORGIa,∗ L. MERLOa S. POKORSKIb a) Departamento de Física Teórica and
Instituto de Física Teórica UAM/CSIC,
Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain b) Institute
of Theoretical Physics, Faculty of Physics,
University of Warsaw, Pasteura 5, PL 02-093, Warsaw, Poland

###### Abstract

In this short talk, we present a renormalizable model that can i) generate
neutrino masses via a low-scale seesaw mechanism and ii) solve the long-
standing $(g-2)_{\mu}$ and the more recent CDF II $M_{W}$-anomalies. This is
minimally achieved by introducing two sterile neutrinos and a single
electroweak-doublet vector-like lepton, with masses $<2$ TeV. We focus on the
one-generation scenario and the requirements to extend it to three
generations.

##  1 Introduction

| $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{L}$
---|---|---|---
$\ell_{L}$ | $\bf 2$ | $-1/2$ | 1
$\mu_{R}$ | 1 | 1 | 1
$H$ | $\bf 2$ | $+1/2$ | 0
$N_{R}$ | 1 | 1 | 1
$S_{R}$ | 1 | 1 | -1
$\psi_{L,R}$ | $\bf 2$ | $-1/2$ | 1
Table 1: Transformation properties under the gauge EW symmetry and their
lepton charges.

Here we summarize the work presented in Ref. [1].

Although we are aware of many problems and puzzles in the Standard Model,
their resolution and understanding seems far off due to the lack of
experimental data to guide us. However, it is fair to say that a single type
of extension of the SM cannot solve them all if taken seriously. This prompts
us to consider extensions of the SM that include at least two types of NP. If
minimal or single-field extensions already left open an enormous number of
possibilities, with two these are virtually infinite. This prompts us to
consider models where the new fields i) solve multiple problems or puzzles
simultaneously and ii) have characteristics that stand out mutually and make
the model more easily falsifiable. Although anomalies are generically
controversial and may, whether because they are the result of statistical
fluctuations, experimental or theoretical error, lead to closed roads, they
are very valuable to the scientific community. Both experimentally and
theoretically, they prompt us to look for new ways to do research. They push
us in constructing models to look for the most minimal extensions possible
that work well together by exploiting the peculiar characteristics of each.
This often results in flourishing areas, as in the case of Leptoquarks,
studied in great detail for the $B$-Anomalies.

Motivated by this, we propose here a model that can explain, with the addition
of only two vector-like leptons, neutrinos masses and two anomalies/tensions:

  1. 1.

Fermilab’s latest measurement of the anomalous magnetic moment of the muon,
the $(g-2)_{\mu}$, confirmed a deviation from the SM prediction of $\delta
a_{\mu}=(2.51\pm 0.59)\times 10^{-9}$ [2] ($4.2\sigma$);

  2. 2.

The measurement of the mass of the W boson by the CDF II collaboration [3]
$M_{W}=80.4335\pm 0.0094\ \text{GeV}$, which is in tension at $7\sigma$ with
the SM prediction and with the previous ones from LHCb and ATLAS.

##  2 The Model

The SM is extended by two vector-like leptons: an EW doublet and a singlet
under the SM gauge-group. The Yukawa sector of the Lagrangian reads

$\displaystyle-\mathscr{L}_{Y}=$
$\displaystyle\phantom{+}\overline{\ell_{L}}HY_{\mu}\mu_{R}+\overline{\ell_{L}}\widetilde{H}Y_{N}N_{R}+\epsilon\overline{\ell_{L}}\widetilde{H}Y_{S}S_{R}+\dfrac{1}{2}\mu\overline{S_{R}^{c}}S_{R}+\Lambda\overline{N_{R}^{c}}S_{R}++Y_{R}\overline{\psi_{L}}H\mu_{R}$
(1)
$\displaystyle+Y_{V}\overline{S_{R}^{c}}\widetilde{H}^{\dagger}\psi_{R}+Y^{\prime}_{V}\overline{\psi_{L}}\widetilde{H}N_{R}+M_{\psi}\overline{\psi_{L}}\psi_{R}+M_{L}\overline{\ell_{L}}\psi_{R}+\text{h.c.}\,,$

where $\ell_{L}$ and $\mu_{R}$ are the SM leptons, $H$ is the Higgs doublet,
$N_{R}$ and $S_{R}$ are the two sterile neutrinos, $\psi$ is the vector-like
EW doublet, and finally $\widetilde{H}\equiv i\sigma_{2}H^{\ast}$, with
$\sigma_{2}$ the second Pauli matrix. All the terms respect lepton number
conservation, except for those proportional to $\epsilon Y_{S}$ and $\mu$. We
work in perturbation theory expanding the results in powers of
$v/(\Lambda,M_{\psi},M_{L})$. For consistency we consider only the case
$\Lambda,\,M_{\psi,L}\gtrsim 500\ \text{GeV}$.

##  3 Phenomenology

### Neutrino Masses

The neutrino masses can be generated via a Low-Scale SS mechanism [4, 5, 6, 7,
8]. Once the EW symmetry is spontaneously broken, the mass terms can be
rewritten as

$-\mathscr{L}_{Y}\supset\dfrac{1}{2}\overline{\chi}\mathcal{M}_{\chi}\chi^{c}+\overline{\zeta_{L}}\mathcal{M}_{\zeta}\zeta_{R}+\text{h.c.}\,,$
(2)

with
$\chi\equiv(\nu_{L},\,N_{R}^{c},\,S_{R}^{c},\,\psi^{0}_{L},\,\psi_{R}^{0c})^{T}$,
$\zeta\equiv(\mu,\,\psi^{-})^{T}$ and

$\mathcal{M}_{\chi}=\left(\begin{matrix}0&m_{N}&\epsilon\,m_{S}&0&M_{L}\\\
m_{N}&0&\Lambda&m_{V^{\prime}}&0\\\ \epsilon\,m_{S}&\Lambda&\mu&0&m_{V}\\\
0&m_{V^{\prime}}&0&0&M_{\psi}\\\ M_{L}&0&m_{V}&M_{\psi}&0\\\
\end{matrix}\right)\,,\qquad\mathcal{M}_{\zeta}=\left(\begin{matrix}m_{\mu}&M_{L}\\\
m_{R}&M_{\psi}\end{matrix}\right)\,,$ (3)

where $m_{i}\equiv\dfrac{Y_{i}v}{\sqrt{2}}$. The parameter $M_{L}$ can be
eliminated by a rotation of
$\cos{\theta}\equiv\dfrac{M_{\psi}}{\widetilde{M}_{\psi}}$ with
$\widetilde{M}_{\psi}\equiv\sqrt{M_{\psi}^{2}+M_{L}^{2}}$. This amounts to a
redefinition of the original parameters,
$m_{\mu,R,N,V^{\prime}}\to\widetilde{m}_{\mu,R,N,V^{\prime}}$. We will denote
the fields in the mass basis with $\widehat{\cdot}$.

In this simplified model, only one active neutrino gets mass. We arbitrarily
choose that its mass corresponds to the square root of the atmospheric mass
squared difference, $\widehat{m}_{\nu}\sim\sqrt{\Delta m^{2}_{\text{atm}}}$
[9]. Given the range on $\widetilde{m}_{N}/\Lambda$ to explain the CDF II
measurement of $M_{W}$, one needs $\mu-15\,\epsilon\,Y_{s}\,v\cos\theta\approx
6\ \text{eV}$.

(a) Diagrams contributing to the $g-2$ of the muon at 1-loop in unitary gauge.

(b) The black curves represent the ratio $\delta m_{\mu}/m_{\mu}^{\exp}$.
$\delta a_{\mu}$ is set to the experimental central value.

Figure 1:

### CDF II $M_{W}$ Tension

The modification of the $W-\mu$ coupling induced by the mixing with the
sterile neutrinos has an impact on the computation of the muon $\beta$ decay

$\Gamma_{\mu}\simeq\dfrac{m^{\text{exp}\,5}_{\mu}G_{F}^{2}}{192\,\pi^{3}}\left(1-\dfrac{\widetilde{m}_{N}^{2}}{2\Lambda^{2}}\right)\equiv\dfrac{m^{\text{exp}\,5}_{\mu}G_{\mu}^{2}}{192\,\pi^{3}}\,,\qquad
G_{F}\simeq
G_{\mu}\left(1+\dfrac{\widetilde{m}_{N}^{2}}{2\Lambda^{2}}\right)\,,$ (4)

where $G_{F}$ is the Fermi constant parameter as defined in the Fermi
Lagrangian, and $G_{\mu}$ is the experimental determination extracted from the
muon lifetime. This implies a modification of the relation between the $W$
boson mass and the experimental determination of $G_{\mu}$ [10, 11],

$M_{W}\simeq
M_{Z}\,\sqrt{\dfrac{1}{2}+\sqrt{\dfrac{1}{4}-\dfrac{\pi\,\alpha_{\text{em}}}{\sqrt{2}\,G_{\mu}\,M_{Z}^{2}\,\left(1-\Delta
r\right)}\left(1-\dfrac{\widetilde{m}_{N}^{2}}{2\Lambda^{2}}\right)}}\,.$ (5)

The CDF II measurement is explained for
$\dfrac{\widetilde{m}_{N}^{2}}{\Lambda^{2}}\in[6.6,11.8]\crossproduct
10^{-3}\,\text{at}\leavevmode\nobreak\ 2\sigma$. Loop-level modifications are
subleading with respect to the tree-level one.

### Muon magnetic dipole moment, $(g-2)_{\mu}$

The NP contributions to $(g-2)_{\mu}$, are associated with the Feynman
diagrams in Fig. 1(a). The same diagrams without the external photon generate
the 1-loop correction to the muon mass, $\delta m_{\mu}$. At $1$-loop, the
chirally enhanced LO, i.e.
$\mathcal{O}(v^{2}/(\Lambda,\widetilde{M}_{\psi})^{2})$, contribution to
$\delta a_{\mu}$ vanishes [12, 13]. Defined
$F_{0}(x,y)\equiv\dfrac{3}{2}-\dfrac{x\log{y}-y\log{x}}{x-y}$, the NLO is
suppressed by four powers of the heavy neutral masses and reads

$\delta
a_{\mu}^{\text{CE-1L}}=\dfrac{3\,m^{\text{exp}}_{\mu}}{4\,\pi^{2}\,v^{2}}\dfrac{M_{W}^{2}}{\Lambda\widetilde{M}_{\psi}}\dfrac{\widetilde{m}_{N}\widetilde{m}_{R}}{\widetilde{M}_{\psi}}\left(\dfrac{m_{V}}{\widetilde{M}_{\psi}}+\dfrac{\widetilde{m}_{V^{\prime}}}{\Lambda}\right)\,F_{0}\left(\dfrac{\Lambda^{2}}{M_{W}^{2}},\dfrac{\widetilde{M}_{\psi}^{2}}{M_{W}^{2}}\right)\,.$
(6)

The $2$-loop diagrams turn out to be smaller than the $20\%$ of the LO
contribution in the considered parameter space.

##  4 Conclusions

The results are shown in Fig. 1(b). In the green(white)[red] region $\delta
m_{\mu}<0.3$($0.3<\delta m_{\mu}<1$)[$\delta m_{\mu}>1$] the tree-level mass.
In grey are shown the regions where $1<|Y_{V}|<5$ with
$\widetilde{Y}_{V^{\prime}}=0$, or $1<|\widetilde{Y}_{V^{\prime}}|<5$ with
$Y_{V}=0$. The correct value of $\delta a_{\mu}$ and of $M_{W}$ is achieved
with vectorlike leptons of masses $\lesssim 2$ TeV without the necessity of an
unnaturally large cancellation between tree- and loop-level contributions to
the muon mass. This allows for a very rich phenomenology at the TeV-scale at
collider. The one-generation scenario studied here is severely constrained but
can be eased in the more realistic three-generations model. In such a case,
further flavour structure must be included in the model to protect it from the
strong bounds from $\mu\to e\gamma$.

## Acknowledgments

A.d.G. and L.M. acknowledge partial financial support through the grant IFT
Centro de Excelencia Severo Ochoa No CEX2020-001007-S by the grant
PID2019-108892RB-I00 funded by MCIN/AEI/ 10.13039/501100011033, and the grant
agreement No 860881-HIDDeN. The research of S.P. has received partial
financial support through the grant DEC-2016/23/G/ST2/04301.

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