# Nonreciprocal Signal Growth in Space-Time Modulated Transmission Lines

Mohamed F. Hagag  Electronic Engineering Department, Military Technical
College, Cairo 11766, Egypt  Electronics and Communications Engineering
Department, American University in Cairo, Cairo 11835, Egypt  Thomas R. Jones
Elmore Family School of Electrical and Computer Engineering, Purdue
University, West Lafayette, Indiana 47907, USA  Karim Seddik  Electronics and
Communications Engineering Department, American University in Cairo, Cairo
11835, Egypt  Dimitrios Peroulis  Elmore Family School of Electrical and
Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA

###### Abstract

We investigate nonreciprocal signal amplification in a space-time-modulated
transmission line. By loading a transmission line with a sinusoidally time-
modulated capacitor, reciprocal momentum band gaps (MPG) appear in the unit
cell dispersion diagram (DD) at frequency ratio
$F_{signal}=0.5~{}F_{modulation}$. Due to the sole existence of complex
frequencies in the MPG, waves propagating with a momentum that lies within the
MPG are amplified. In addition to time modulation, the TL is space-modulated
by imposing a modulation phase shift $-\theta_{m}$ between successive unit
cells. As a result, forward and backward MPG locations in the DD shift from
$F_{signal}=0.5~{}F_{modulation}$ in opposite directions, causing
nonreciprocal signal growth frequencies. Nonreciprocal amplification is
confirmed by circuit modeling outcomes, which totally agree with DD results.

††preprint: APS/123-QED

##  I Introduction

Growing wave physics always gains high interest due to the continuous need for
high-power signals in all frequency regimes. Creating sufficient
nonlinearities and instabilities in a media allows growing waves due to energy
coupling between matters [1]. In the last centenary, the interaction between
plasma and charged-particle beams is utilized in the generation of high-power
microwave signals [2, 3]. Parametric amplification is another technique that
allows signal growth by pumping the system with a high-amplitude pump,
coupling the energy to a traveling signal. It is first used to generate high-
power signals in microwaves [4, 5] and recently in optical regime due to the
development of nonlinear crystal materials such as Lithium niobate (LiNbO3)
and lithium tantalate (LiTaO3) [6]. Lately, time-modulated media where the
media parameters vary periodically with time opens the road to new techniques
of nonlinearities and instabilities utilized in signal amplification and
generation [7, 8].

Periodically varying media parameters creates new types of artificial
electromagnetic media, generally called time crystals (TC). They exhibit
unusual physical effects such as momentum gaps, magnetless non-reciprocity [9,
10], parity-time symmetry [11], and topological aspects [7, 12]. Analog to
photonic crystals (PC) in which the media is space-modulated, creating
frequency band gaps [13, 14], momentum (k) band gaps (MPG) are created in the
dispersion diagram of TCs if the media parameters are time-modulated with a
sufficiently high speed. Only complex frequency exists within a MPG, allowing
the signals to grow [7, 8]. Consequently, all waves propagating in a TC with a
k which lays within the MPG will be amplified. TCs can be realized by time-
modulating permittivity, permeability, or both of the media and even by time-
modulating components loaded to a transmission line (TL) [15, 16]. When a
media is time and space modulated simultaneously, many exciting phenomena can
coexist, such as signal growth and non-reciprocity [17].

Here, we research the nonreciprocal signal growing in a TL with a space-time
modulated characteristic impedance $Z_{o}$. The loaded TL consists of T-shape
unit cells with a time-invariant TL as a series element and a time-modulated
capacitor (TMC) with a sinusoidal waveform as a shunt element. The TL is
space-modulated by forcing a modulation phase shift $-\theta_{m}$ between
sequential unit cells. The eigenvalue problem is solved, and the dispersion
diagram is plotted at different values of $\theta_{m}$, showing MPGs’
different locations for forward and backward propagation. To confirm the
nonreciprocal behavior, the space-time modulated TL is transient simulated
using Keysight Advanced Design System (ADS) software. The obtained results
confirm the nonreciprocal amplification and align with dispersion diagram
outcomes.

##  II Dispersion relation

The considered loaded transmission line (TL) unit cell is shown in Fig. 1(a).
It is a T-shape unit cell with two time-invariant (TI) TLs of length
$0.5m~{}\lambda_{m}$. $\lambda_{m}$ is the wavelength of the modulation signal
as series elements and the time-modulated capacitor (TMC) as a shunt element.
While the TMC is modulated in time following the function
$C(\omega,t)=C(\omega,t+lT)\ $ with $l\in\mathbb{Z}$, it is also space-
modulated by adding a successive modulation phase shift $\theta_{m}$ in
sequential unit cells. Following [18, 19, 20] and considering a constant phase
shift $-\theta_{m}$ in the modulation of TMC in each sequential unit cells,
the space-TMC can be expanded into a complex Fourier series as follows.

$C\left(\omega
t,\theta_{m}\right)=\sum_{s=-\infty}^{+\infty}{c_{s}(\omega)e^{js\omega_{m}t}e^{-js\theta_{m}}}$
(1)

where $c_{s}(\omega)$ are complex coefficients and $\omega_{m}$ is the angular
modulation frequency. A system with a TMC will possess an infinite number of
Floquet harmonics that follows

$\omega_{l}=\omega_{sig}+l\omega_{m}$ (2)

where $\omega_{sig}$ is the main signal angular frequency and $l$ is the
Floquet order. Across the TMC, the voltage or current follows the series below
[19]

$F\left(t\right)=\sum_{l=-N}^{+N}{f_{l}e^{j\omega_{l}t}}$ (3)

where $f_{l}$ are the complex coefficients of voltage and current. $N$ is an
integer that represents half the number of considered harmonics. As a result,
the total current can be found in terms of voltage harmonics as follows

$I\left(\omega
t,\theta_{m}\right)=\sum_{l=-\infty}^{\infty}\sum_{s=-\infty}^{\infty}{j~{}\left(\omega_{l}+s\omega_{m}\right)~{}c_{s}(}\omega_{l})\\\
\times e^{js\omega_{m}t}e^{-js\theta_{m}}~{}v_{l}~{}e^{j\omega_{l}t}$ (4)

The relation between current and voltage harmonics can be found using (3) and
(4) as follows.

$\sum_{r=-\infty}^{\infty}{i_{r}e^{jr\omega_{l}t}}=\sum_{l=-\infty}^{\infty}\sum_{s=-\infty}^{\infty}{j~{}\omega_{l+s}~{}c_{s}(}\omega_{l})\\\
\times e^{-js\theta_{m}}~{}v_{l}~{}e^{j\omega_{l+s}t}$ (5)

By equating the index $r$ by $l+s$ and matching $\omega_{r}$ frequency, we get

$i_{r}=\sum_{l=-\infty}^{\infty}{j~{}\omega_{r}~{}c_{r-l}(\omega_{l})~{}v_{l}~{}e^{-j(r-l)\theta_{m}}}$
(6)

Consequently, due to space modulation, the relation between the Y-matrix
elements in two successive unit cells $n$ and $n+1$ are related by

${Y_{(n+1)}}_{r}^{l}={Y_{(n)}}_{r}^{l}~{}e^{-i(r-l)\theta_{m}}$ (7)

Hence, the total impedance matrix is given by

$\bar{\bar{Y_{C}}}=j\bar{\bar{W}}P~{}\bar{\bar{\xi}}$ (8)

where
$\bar{\bar{W}}=diag(\omega_{-N},..,\omega_{-1},\omega_{s},\omega_{1},..,\omega_{N})$,
$\bar{\bar{\xi}}=diag(e^{-j(-N)\theta_{m}},..,e^{-j(-1)\theta_{m}},1,e^{-j(1)\theta_{m}},..,e^{-j(N)\theta_{m}})$.
$P$ is $(2N+1)\times(2N+1)$ in size which is mainly dependent on the
modulation waveform. Considering TMC with sinusoidal modulation that follows

$C\left(\omega,t\right)=C_{o}(\omega)\left(1+M_{D}\cos{\omega_{m}t}\right)$
(9)

where $M_{D}$ is the modulation depth and $C_{o}(\omega)$ is the capacitance
nominal value. $P$ is given by

$P=\\\
\left(\begin{matrix}\begin{matrix}c_{0}(\omega_{-N})&c_{-1}(\omega_{1-N})&0\\\
c_{1}(\omega_{-N})&c_{0}(\omega_{1-N})&c_{-1}(\omega_{2-N})\\\
0&c_{1}(\omega_{1-N})&c_{0}(\omega_{2-N})\\\
\end{matrix}&\begin{matrix}\cdots\\\ \cdots\\\ \cdots\\\
\end{matrix}&\begin{matrix}0\\\ 0\\\ 0\\\ \end{matrix}\\\
\begin{matrix}\vdots&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots&\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ddots\\\ \end{matrix}&\ddots&\vdots\\\ \begin{matrix}0\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ &0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ &0\\\
\end{matrix}&\cdots&c_{0}(\omega_{N})\\\ \end{matrix}\right)$ (10)

where $c_{\pm 1}(\omega){=0.5~{}c_{o}(\omega)\ M_{D}}$. According to [20],
space-TMC that has the admittance matrix shown in (8) will possess
transmission matrix its elements are multiplied by the space factor
$\bar{\bar{\xi}}$. Consequently, the transfer matrix of a space-TMC can be
obtained using

$T_{C}=\left[\begin{matrix}\bar{\bar{Ones}}&\bar{\bar{Zeros}}\\\
j\bar{\bar{W}}P&\bar{\bar{Ones}}\\\
\end{matrix}\right]\times\left[\begin{matrix}\bar{\bar{\xi}}&\bar{\bar{Zeros}}\\\
\bar{\bar{Zeros}}&\bar{\bar{\xi}}\\\ \end{matrix}\right]$ (11)

where $\bar{\bar{Ones}}$ and $\bar{\bar{Zeros}}$ are the unit and zero
matrices, receptively. The transfer matrix of the TI TL of length
$0.5~{}m~{}\lambda_{m}$ is given by

$T_{tl}=\left[\begin{matrix}\cos(\bar{\bar{\varphi}})&j\times
Z_{o}\times\sin(\bar{\bar{\varphi}})\\\ j\times
Y_{o}\times\sin(\bar{\bar{\varphi}})&\cos(\bar{\bar{\varphi}})\\\
\end{matrix}\right]$ (12)

where $Z_{o}$ and $Y_{o}$ are the TL characteristic impedance and admittance,
respectively and $\bar{\bar{\varphi}}$ is given by

$\bar{\bar{\varphi}}=2\times\pi\times m\times\frac{\bar{\bar{W}}}{\omega_{m}}$
(13)

The total transfer matrix is found using

$T_{t}=T_{tl}\times T_{C}\times T_{tl}$ (14)

Finally, the dispersion diagram (DD) can be plotted by solving

$T_{t}X=e^{\gamma d}X$ (15)

where $\gamma$, $X$, and $d$ are the propagation constant, the eigenvectors,
and the unit cell (UC) length, respectively. $\gamma$ is related to phase
constant $\beta$ and the attenuation constant $\alpha$ by

$\gamma=\alpha+i\beta$ (16)

##  III Signal growth in Non-reciprocal Dispersion diagram of Space-time
modulated TL

Figure 1:  (a) T-shape unit cell with a time-invariant TL of length
$0.5m~{}\lambda_{m}$ as series element and space-time modulated capacitor as
shunt element. Real and complex frequency dispersion diagram with $N=2$ for
the unit cell in (a) with $m=0.3$, $C_{o}=~{}4~{}$pF, $M_{D}=0.7$, $F_{m}=1$
GHz and TL $Z_{o}=$ $83~{}\Omega$ at (b) $\theta_{m}=~{}0$ rad, (c)
$\theta_{m}=~{}0.28$ rad, (d) $\theta_{m}=~{}0.57$ rad and (e)
$\theta_{m}=~{}0.86$ rad. (f) Unit cell electrical length variation with
frequency at $C_{o}$. (g) Signal growth frequency variation for forward and
backward propagation with $\theta_{m}$.

For the unit cell shown in Fig. 1(a), the Block impedance ($Z_{Bloch}$) of the
unit cell is chosen to be 50 $\Omega$ at $F_{sig}=0.5$ GHz. As result,
$C_{o}$, TL characteristic impedance $Z_{o}$ and $m$ are set to $4$ pF,
$83~{}\Omega$ and $0.3$, respectively. The dispersion digram (DD) is plotted
in Fig. 1(b) considering only time modulation ($F_{m}=1$ GHz and
$\theta_{m}=~{}0$) and $M_{D}=0.7$ with $N=2$. At $F_{sig}=0.5$ GHz
($F_{m}=2F_{sig}$), a phase matching occurs between the fundamental
$\omega_{sig}$ and the harmonic $\omega_{-1}$ in forward and backward
propagations. The matching phase equals the unit cell electrical length at
$C_{o}$ ($UCEL_{C_{o}}$). Consequently, momentum band gaps (MBG) for forward
and backward propagation are formed, in which no relation between real
frequency and phase exists [7, 8]. As shown in Fig. 1(b), within the MBG, only
complex frequency ($\omega_{sig}$ and $\omega_{-1}$) exists with a constant
real part equal $F_{sig}=0.5$ GHz ($F_{m}=2F_{sig}$) and varying imaginary
part. The imaginary part has positive (causes signal growing) and negative
(causes signal decaying) values with a maximum centered at the middle of the
gap. This behavior is similar to the exceptional point physics [21, 22]. The
complex frequency dispersion relation is plotted using the following equation
instead of (2)

$\Omega=\omega_{L}-i\sigma_{L}$ (17)

Next, in addition to time modulation, space modulation is considered by
imposing a constant phase shift ($-\theta_{m}$) to the modulation between
successive unit cells. According to (11) and focusing on the substantial
harmonics $\omega_{sig}$ and $\omega_{-1}$, while the dispersion lines of
fundamental $\omega_{sig}$ are not affected, the forward and backward
dispersion lines of harmonic $\omega_{-1}$ are shifted in a way that causes
the forward and backward MPGs to be settled at two different frequencies
following

${UCEL_{C_{o}}\ at\ \left(F_{f}\right)}-{UCEL_{C_{o}}\ at\
\left(F_{b}\right)}=\theta_{m}$ (18)

where $F_{f}$ and $F_{b}$ are the frequencies at which MBGs are formed at
forward and backward propagation, respectively. In addition, $F_{f}$ and
$F_{b}$ are related by

$\frac{F_{f}+F_{b}}{F_{m}}=1$ (19)

Figure 2: (a) Simulated nine unit cells with 1 W input power, TLs of length
$0.15\lambda_{m}$, $Z_{o}=83~{}\Omega$, $C_{o}=4$ pF, $F_{m}=1$ GHz,
$M_{D}=0.7$ loaded with a $50~{}\Omega$ load impedance. Forward (backward)
propagation is defined in the direction where the phase shift between
successive cells is $-\theta_{m}$ ($\theta_{m}$). Output forward and backward
power in case (b) input frequency F${}_{inp}=0.5$ GHz and $\theta_{m}=0$ rad,
(c) F${}_{inp}=0.55$ GHz and $\theta_{m}=0.28$ rad, (d) F${}_{inp}=0.45$ GHz
and $\theta_{m}=0.28$ rad, (e) F${}_{inp}=0.6$ GHz and $\theta_{m}=0.57$ rad,
(f) F${}_{inp}=0.4$ GHz and $\theta_{m}=0.57$ rad, (g) F${}_{inp}=0.65$ GHz
and $\theta_{m}=0.86$ rad, (h) F${}_{inp}=0.35$ GHz and $\theta_{m}=0.86$ rad.
At different Finp and with time dependence, (i) unit cell $Z_{Bloch}$, (j)
reflection coefficient considering $50~{}\Omega$ load and source.

Equation (18) works precisely at low values of $M_{D}$. However, as $M_{D}$
increases and the desired $\frac{F_{f}}{F_{m}}$ ($\frac{F_{b}}{F_{m}}$) shifts
from $0.5$ value, $\theta_{m}$ deviates slightly from (18) and can be easily
optimized. Because the elements forming the unit cell are not all time-variant
and the Unit cell $Z_{Bloch}$ does not ideally follow the capacitance
variation in a linear behavior, harmonics $\omega_{sig}$ and $\omega_{-1}$
matching phases shift from the MPG center, which causes this deviation. For
further investigation, DDs are plotted for $\frac{F_{f}}{F_{m}}$ values
$0.55$, $0.6$ and $0.65$ in Figs. 1(c-e), respectively. As shown,
$\frac{F_{b}}{F_{m}}$ exactly follows (19). On the other hand, the calculated
electrical lengths of the unit cell at different frequencies, shown in Fig.
1(f), are utilized in (18) to get the initial value of $\theta_{m}$ and then
optimized. At the exact value of $\theta_{m}$ associated with a desired
$\frac{F_{f}}{F_{m}}$ ($\frac{F_{b}}{F_{m}}$), complex frequencies exist with
a varying imaginary part and fixed real part define by the desired
$\frac{F_{f}}{F_{m}}$ ($\frac{F_{b}}{F_{m}}$). For $\frac{F_{f}}{F_{m}}$
($\frac{F_{b}}{F_{m}}$) values $0.55$ ($0.45$), $0.6$ ($0.4$) and $0.65$
($0.35$), $\theta_{m}$ optimized values are found to be $0.28$, $0.57$ and
$0.86$ rad, respectively. Moreover, as shown in Figs. 1(b-e), imaginary part
values of $\Omega$ almost are the same at different values of $\theta_{m}$.
Consequently, the same rate of signal growth could be achieved if the same
boundary conditions exist. In addition, as shown in Fig. 1(g),
$\frac{F_{f}}{F_{m}}$ and $\frac{F_{b}}{F_{m}}$ can take any value, (19) is
applied, lower or higher than $0.5$ by choosing the proper value and sign of
$\theta_{m}$.

##  IV Circuit Modeling

The scenarios presented in Figs.1(b-e) and discussed in section III will be
verified using transient simulations (TS) in ADS software. As illustrated in
Fig. 2(a), nine-unit cells are considered with 1 W input power source followed
by $50~{}\Omega$ isolator and $50~{}\Omega$ load impedance. TLs have a length
of $0.15\lambda_{m}$ and $Z_{o}=83~{}\Omega$. The capacitor has $C_{o}=4$ pF
and time-modulated with $F_{m}=1$ GHz and $M_{D}=0.7$. Output forward
(backward) power is the output power when connecting the source and the load
to the terminals, ensuring a $-\theta_{m}$ ($\theta_{m}$) phase shift between
successive cells. In all considered cases, TS output results are Fourier
transformed and plotted in Figs. 2(b-h).

For input frequency (Finp) equals $0.5$ GHz and $\theta_{m}=0$ rad, as shown
in Fig. 2(b), the output forward power (OFP) and output backward power (OBP)
are identical with a gain more than $13$ dB, congruent with Figs. 1(b).
Moreover, following Figs.1(c) and considering F${}_{inp}=0.55$ ($0.45$) GHz
with $\theta_{m}=0.28$ rad, OFP (OBP) is amplified to $11.7$ dB while no gain
is achieved in OBP (OFP) as illustrated in Figs. 2(c-d). Finally, by following
Figs. 1(d)/(e) and considering F${}_{inp}=0.6$/$0.65$ ($0.4$/$0.35$) GHz with
$\theta_{m}=0.57$/$0.86$ rad, OFP (OBP) is amplified to $11.4$/$10$ dB without
any gain in OBP (OFP) as shown in Figs. 2(e-f)/(g-h).

As discussed in section III, when space-time modulating the TL, it is expected
to have the same growth rate if the boundary conditions are the same. In the
simulations illustrated in Fig. 2, loaded TL is terminated in all cases, with
a fixed $50~{}\Omega$ impedance at the source and the load. The TL is matched
at $C_{o}$ and F${}_{inp}=0.5$ GHz to a fixed $50~{}\Omega$ impedance.
However, shifting from F${}_{inp}=0.5$ GHz changes the $Z_{Bloch}$ profile,
Fig. 2(i), with modulation period (Tm) which changes reflection coefficient
profile at the terminals, Figs. 2(j). Consequently, the boundary conditions
change, resulting in a growth rate variation shown in Fig. 2.

##  V Conclusion

Nonreciprocal signal growth in a space-time modulated TL is confirmed through
eigenvalue problem solution and circuit modeling. TL is time-modulated by
loading sinusoidally time-modulated capacitors and space-modulated by forcing
a modulation phase shift $-\theta_{m}$ between sequential unit cells. First,
the eigenvalue problem is formulated and solved. DDs are plotted at different
values of $\theta_{m}$. At $\theta_{m}=0$, only time-modulation is considered,
and reciprocal MPGs located at $F_{signal}=0.5~{}F_{modulation}$. Varying
$\theta_{m}$ causes MPGs to shift from $F_{signal}=0.5~{}F_{modulation}$ in a
nonreciprocal behavior. Second, for the same considered values of
$\theta_{m}$, space-time modulated TL is built in ADS software and transient
simulated. A very good agreement is found between DD and circuit modeling
results.

###### Acknowledgements.

This work has been completed under a research agreement between Purdue
University and The American University in Cairo.

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