# Space-Time Fresnel Prism

Zhiyu Li<EMAIL_ADDRESS>State Key Laboratory of Electrical
Insulation and Power Equipment, Department of Electrical Engineering, Xi’an
Jiaotong University, Xi’an, Shaanxi, China  Department of Electrical
Engineering, KU Leuven, Leuven, Belgium  Xikui Ma  State Key Laboratory of
Electrical Insulation and Power Equipment, Department of Electrical
Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, China  Amir Bahrami
Department of Electrical Engineering, KU Leuven, Leuven, Belgium  Zoé-Lise
Deck-Léger  Department of Electrical Engineering, Polytechnique Montréal,
Montréal, Quebec, Canada  Christophe Caloz  Department of Electrical
Engineering, KU Leuven, Leuven, Belgium

###### Abstract

Space-time modulation-based metamaterials have recently spurred considerable
interest, owing to the fundamental addition of the time dimension to the
medium parameters, and resulting novel properties and potential applications.
However, the implementation of most related structures – e.g., involving step,
slab or gradient discontinuities – has been hindered by the impossible
requirement of infinitely or prohibitively large device sizes. We provide here
a solution to this issue, consisting in a space-time transposition of the
conventional Fresnel prism, whereby a copy of the target modulation is
periodically re-injected at the input of a Fresnel-reduced finite structure,
so as to provide the same anharmonic and nonreciprocal frequency conversion as
the target space-time interface in the case of a modulation step. This
concept, which may readily extend to slab or gradient modulations, as well as
accelerated profiles for space-time chirping operations, may pave the way for
the practical development of a wide range of novel microwave and optical
space-time systems.

††preprint: APS/123-QED

##  I Introduction

Space-time systems are structures that are formed by the modulation of some
parameter (e.g., the refractive index) of a host medium in space and time or
only in time [1, 2, 3, 4, 5, 6, 7]. They represent fundamental extensions of
pure-space systems and have recently led to a wide diversity of novel
applications, including magnetless nonreciprocity [8, 9, 10, 11], time
reversal and refocusing [12, 13], inverse-prism scattering [14], temporal
coating [15], fundamental bound breaking [16, 17], dynamic crystals [18, 19]
and light deflection or bending [20, 21].

The modulation may be periodic [22, 23, 24, 25, 8, 26, 9, 27, 10, 11, 28, 17,
20, 29, 18, 19, 30, 31, 21] or aperiodic [32, 33, 34, 35, 36, 37, 12, 14, 16,
13, 15, 38, 39], depending on the application. While periodic-modulation
space-time systems may be implemented in finite-size structures, using Bloch-
Floquet impedance (reflection-less) terminations, aperiodic-modulation (e.g.,
step, slab, gradient) space-time systems are practically restricted by a
_fundamental issue of implementation size_ : those operating in the
continuous-wave regime [32, 35, 34, 39] would imply impractically large device
sizes, while those operating in the pulse-wave regime [33, 36, 37, 12, 14, 16,
13, 15, 38] would either require complex synchronization (see Sec. IX in [40])
or preclude extension to the continuous-wave regime.

We provide here a solution to the aforementioned issue of implementation size
in aperiodic space-time systems. This solution is based on a transposition of
the conventional Fresnel prism concept from two-dimensional space to space-
time. The corresponding prism is obtained by periodically re-injecting either
a copy of the target modulation or a judiciously adapted version of that
modulation into the input of the structure. As the conventional Fresnel prism,
the resulting space-time prism can provide drastic device size – and hence
also weight and cost – reduction.

##  II Space-Time Fresnel Prism Concept

The concept of the proposed Fresnel prism is depicted in Fig. 1, with the
conventional and space-time prisms being represented in Figs. 1(a) and (b),
respectively. Figure 1(a) shows the principle of the conventional Fresnel
prism [41, 42], whereby a bulky prism of refractive index $n$ (left panel) is
cut off into a sequence of similar low-profile sub-prisms (right panel) that
provides the same light deflection as the original prism with a much smaller
form factor ($d\ll D$).

Figure 1:  Illustration of (a) conventional (space-space) and (b) dynamic
(space-time) optical prisms (left panels) and their Fresnel counterparts
(right panels) with relevant optical structures and light rays. Reflection is
assumed to be negligible in both cases.

The space-time Fresnel prism is fundamentally different in nature and serves a
totally different purpose while applying the same size reduction principle.
The related concept and principle are described in Fig. 1(b). The target
space-time prism structure consists of a medium with refractive index $n_{2}$
that is dynamically transformed into a medium with a smaller refractive index
$n_{1}$ by a moving step modulation (left panel with inset). The purpose of
that device is to _anharmonically and directionally alter the frequency_ of
light [43, 44], which may have applications as diverse as versatile generation
and mixing, magnetless nonreciprocity and absorption, bound-breaking matching
and filtering, and classical or quantum processing [3]. Compared to the
conventional prism, the space-time prism replaces the spatial transverse
dimension ($x$) [Fig. 1(a)] by the time ($ct$) dimension, while maintaining
the longitudinal dimension ($z$). It can then be reduced to a Fresnel
structure similarly to the conventional prism, as illustrated in the figure
(right panel with inset), with the same benefit of drastic device size
reduction ($d\ll D$), while providing the same frequency transition operation
as the original space-time prism. We restrict here our attention to the case
of a single space-time interface, but the proposed concept straightforwardly
extends to the cases of a slab or a gradient modulation.

##  III Two Types of Space-Time
Fresnel Prism

The space-time Fresnel prism in Fig. 1(b) offers the benefit of simplicity
but, as we shall shortly see, it also has some drawbacks, which can be partly
mitigated. We shall therefore consider two versions of the space-time Fresnel
prism, whose operation principles are depicted in Fig. 2 (see Sec. X in [40])
and which will be detailed later: the initial prism in Fig. 1(b), Prism I,
shown in Fig. 2(a), and an improved version of it, Prism II, shown in Fig.
2(b).

Figure 2:  Operation principle of (a) the Space-Time (ST) Fresnel Prism I,
with Pure-Time (PT) sections [Fig. 1(b)], and (b) the space-time Fresnel Prism
II, with interluminal sections for greater conversion efficiency. The
sinusoidal waveforms and related arrows represent the continuous-wave regime;
the pulse (possibly multiple-pulse) regime would correspond to restricted
sections of these waveforms.

Prism I [Fig. 2(a)] involves two types of scattering modulation interfaces:
the desired space-time interfaces, which correspond to the oblique sections of
the structure and perform the desired space-time frequency conversion,
$\omega_{\mathrm{t}}^{\mathrm{ST}}=\omega_{\mathrm{i}}(1-\beta n_{1})/(1-\beta
n_{2})$ 111We may switch from
$\omega_{\mathrm{t}}^{\mathrm{ST}}>\omega_{\mathrm{i}}$ (blue shift) to
$\omega_{\mathrm{t}}^{\mathrm{ST}}<\omega_{\mathrm{i}}$ (red shift) by making
the ratio $n_{2}/n_{1}$ larger or smaller than 1, or by making the parameter
$\beta$ greater (co-directional modulation) or smaller (contra-directional
modulation) than zero. However, changing the ratio $n_{2}/n_{1}$ is preferable
to changing the velocity regime from $\beta>0$ to $\beta<0$ because $\beta>0$
implies a smaller reflection coefficient [3]., where $\beta=v_{\mathrm{m}}/c$
is the normalized modulation velocity [3], over the time intervals
$\tau^{\mathrm{ST}}$, and spurious pure-time interfaces, which correspond to
the horizontal sections and produce the undesired pure-time transitions
$\omega_{\mathrm{t}}^{\mathrm{PT}}=\omega_{\mathrm{i}}n_{1}/n_{2}$ [3] over
the intervals $\tau^{\mathrm{PT}}$ in the output waveform, as shown at the
right of the figure. Given its simplicity, this prism might be the solution of
choice in the pulse-wave regime, assuming synchronization of the incident
pulses with the space-time sections, but the pure-time intervals represent a
serious drawback in the continuous-wave regime, with conversion efficiency
$\eta_{\mathrm{I}}=\tau^{\mathrm{ST}}/(\tau^{\mathrm{PT}}+\tau^{\mathrm{ST}})=1-n_{2}\beta$
(see Sec. XI in [40]) decreasing to zero in the subluminal limit
$v_{\mathrm{m}}\rightarrow c/n_{2}$.

Prism II [Fig. 2(b)] is obtained by rotating and translating the vertex points
$X$ of Prism I [Fig. 2(a)] from the vertex points $O$ into the new vertex
points $X^{\prime}$ in the space-time diagram. The rotation, with the angle
set between the angles of the incident wave and transmitted wave trajectories,
eliminates the pure-time interface sections, while the translation extends the
new (interluminal, $c/n_{2}<v_{\mathrm{m}^{\prime}}<c/n_{1}$) interface
sections to the modulation final position. As the result of this operation,
the PT intervals have been replaced by silence intervals, of duration
$\tau^{\mathrm{silence}}$, as illustrated at the right of the figure. This may
a priori suggest that Prism II offers no benefit over Prism I, since it also
implies an undesired discontinuity in the output waveform. However, its
conversion efficiency,
$\eta_{\mathrm{II}}=\tau^{\mathrm{ST}}/(\tau^{\mathrm{silence}}+\tau^{\mathrm{ST}})=(1-n_{2}\beta)/(1-n_{1}\beta)$
(see Sec. XI in [40]) is greater, with an enhancement of
$\eta_{\mathrm{II}}/\eta_{\mathrm{I}}=1/(1-n_{1}\beta)\overset{v_{\mathrm{m}}\rightarrow
c/n_{2}}{=}1/(1-n_{1}/n_{2})$ that becomes important for small index contrast.

##  IV Spurious Effects Comparison

At this point, it may be interesting to note the following further differences
and similarities between the proposed space-time Fresnel prisms (Fig. 2) and
the conventional double-space prism [right panel of Fig. 1(a)]. In the latter
case, spurious diffraction occurs at the vertices of the structure, whereas no
such diffraction occurs in the former case since the related “rays”,
corresponding to space-time trajectories, have fixed incidence and scattering
directions [with slopes $n_{1}$ and $n_{2}$, see Fig. 1(b)], as opposed to
displaying a complex angular spectrum spread (see Sec. XII in [40]). On the
other hand, the horizontal sections of the conventional prism and Prism I,
although being of different natures, both induce spurious scattering, with
different deflection direction in the former case and with the aforementioned
pure-time frequency conversion in the latter case (see Sec. XII in [40]); the
situation of Prism II is essentially similar to that of Prism I insofar as
spurious scattering and consequent discontinuity are concerned, except for the
replacement of the horizontal pure-time sections by the interluminal space-
time sections.

##  V Phase Continuity Design

As seen above, both Prism I and Prism II suffer from spurious output waveform
discontinuity, although that issue may be largely mitigated in the latter
case. Can one devise a scheme that would completely eliminate the output
waveform – PT [Fig. 2(a)] or silence [Fig. 2(b)] interval – discontinuity for
applications requiring perfect continuity? We shall try next to address that
question.

Achieving such output waveform continuity would require both ensuring
continuity of the phases of the waveform at the starting and ending times of
the spurious intervals, i.e., $\phi_{\mathrm{e}}=\phi_{\mathrm{s}}$ in Fig. 2,
and interconnecting the corresponding points to close up the spurious gaps. In
the case of Prism II [Fig. 2(b)], the condition
$\phi_{\mathrm{e}}=\phi_{\mathrm{s}}$ is automatically satisfied from the fact
that the related trajectories (arrows of fixed phase) originate from the same
(unique) trajectories impinging on the $O$ vertices before splitting at the
input of the device. The re-injection period, $\tau_{\mathrm{II}}$ is related
to the device size $d_{\mathrm{II}}$ as (see Sec. XIII in [40])

$\tau_{\mathrm{II}}=d_{\mathrm{II}}(1/v_{\mathrm{m}}-1/v_{\mathrm{m}^{\prime}}).$
(1)

In contrast, Prism I [Fig. 2(a)] generally involves
$\phi_{\mathrm{e}}\neq\phi_{\mathrm{s}}$, but proper phase continuity,
$\phi_{\mathrm{e}}=\phi_{\mathrm{s}}+2\pi q$ ($q\in\mathbb{N}$), can still be
obtained under the prism size or temporal period constraint (see Sec. XIII in
[40])

$d_{\mathrm{I}}=q\frac{\lambda_{0}}{n_{1}}\quad\text{or}\quad\tau_{\mathrm{I}}=\frac{d_{\mathrm{I}}}{v_{\mathrm{m}}}=q\frac{cT_{0}}{n_{1}v_{\mathrm{m}}},$
(2)

where $\lambda_{0}=2\pi c/\omega_{\mathrm{i}}$ and
$T_{0}=2\pi/\omega_{\mathrm{i}}$ are the free-space incident wavelength and
period, respectively, with $\omega_{\mathrm{i}}$ being the incident wave
frequency.

##  VI Interconnection Schemes

Now that the phase continuity problem has been resolved, we may consider the
interconnection of the desired space-time intervals. Because the related
design for Prism I suffers from the disadvantage of dependency on the incident
frequency, according to Eq. (2)
[$d_{\mathrm{I}}=2\pi{}cq/(n_{1}\omega_{\mathrm{i}})$] and because it is
effectively the particular case of Prism II in the limit
$v_{\mathrm{m}}^{\prime}\rightarrow\infty$ (Fig. 2), we shall restrict here
our attention to the interconnection for Prism II and defer that for Prism I
to Sec. XIV in [40].

We propose here two interconnection schemes, which are depicted in Fig. 3.
Instead of suppressing the silence intervals $\tau^{\mathrm{silence}}$ in Fig.
2(b), which would be physically impossible, we space-time back-translate,
along the corresponding transmitted trajectories, the
$\phi_{\mathrm{e}}$-related outputs to the same times as the
$\phi_{\mathrm{s}}$-related points, as shown in Fig. 3(a). This operation
naturally requires some device size extension to the right compared to the
initial structure in Fig. 2(b), but the resulting length remains substantially
shorter than that of the target prism for a finite number of wave cycles (see
Sec. XV in [40]). Conceptually, this scheme could be implemented by
mechanically moving an output load backward in space by a distance
$d_{\mathrm{c}}$ at each starting phase time ($t_{\mathrm{s}m}$), as depicted
in Fig. 3(b). However, this approach would require impractically fast moving
parts in a real-world scenario. In contrast, the scheme presented in Fig.
3(c), using a sequence of switched delay lines with a common output load, is
perfectly realistic and accomplishes exactly the same operation. Figure 3(d)
plots the output waveforms without and with the two (identical)
interconnection schemes, illustrating how that the former continuous-phase
sections are interconnected to produce exactly the desired continuous output
waveform.

Figure 3: Interconnection schemes for suppressing the spurious silence
intervals in Prism II [Fig. 2(b)] with $N$ sub-prisms (here $N=3$). (a) Space-
time diagram required for describing the two schemes, with incidence, space-
time and silence zones. (b) Moving output load scheme. (c) Switched delay line
scheme, involving $N-1$ (here $N-1=2$) transmission line sections between $N$
switches (here $N=3$). (d) Output waveforms without and with interconnections.

The design of a switched-delay-line Prism II composed of $N$ sub-prisms [Figs.
3(a) and (c)] involves $N-1$ transmission line sections separated by $N$
switching points, with earlier switching times corresponding to the farther
switching positions. In designing the system, the switching points must be
placed at the positions (see Sec. XIV in [40])

$z_{\mathrm{s}m}=z_{\mathrm{f}}+(N-m)d_{\mathrm{c}},\quad m=1,2,3,...,N,$ (3a)
where $z_{\mathrm{f}}$ is the final position of the modulation and
$d_{\mathrm{c}}=\frac{(c/n_{1}-c/n_{2})(v_{\mathrm{m}^{\prime}}-v_{\mathrm{m}})}{v_{\mathrm{m}^{\prime}}(c/n_{1}-v_{\mathrm{m}})}d_{\mathrm{II}},$
(3b)

while the switching times must be set to the timings

$t_{\mathrm{s}m}=t_{\mathrm{s}1}+(m-1)\Delta t_{\mathrm{s}},\quad
m=1,2,3,...,N,$ (4a) where
$t_{\mathrm{s}1}=\tau_{\mathrm{II}}+\frac{d_{\mathrm{II}}+(N-2)d_{\mathrm{c}}}{c/n_{2}}$
(4b) is the switching time of the first switch and $\Delta
t_{\mathrm{s}}=\frac{n_{2}(v_{\mathrm{m}^{\prime}}-v_{\mathrm{m}})(c/n_{2}-v_{\mathrm{m}})}{n_{1}v_{\mathrm{m}}v_{\mathrm{m}^{\prime}}(c/n_{1}-v_{\mathrm{m}})}d_{\mathrm{II}}$
(4c) is the (uniform) switching interval between the switching times.

Note that none of Eqs. (1) and (3)-(4) involves the parameter
$\omega_{\mathrm{i}}$, which reveals that switched-delay-line Prism II design
offers the benefit, compared to its Prism I counterpart [Eq. (2)], of being
independent from the incident frequency and hence provides unlimited
operational bandwidth within a reasonably weakly dispersive region of
modulation media 222In case of need, one may resort to various dispersion
engineering or compensation techniques [60, 61] to enhance the bandwidth of
the prisms..

##  VII Simulation Results and Analysis

Figure 4 provides an illustration and validation of the proposed space-time
Fresnel prisms, based on full-wave Finite-Difference Time-Domain (FDTD)
simulations [47] utilizing the generalized Yee cell scheme recently reported
in [48, 49] (see Sec. XVI in [40] and animations in [50]). We selected here as
the excitation a traveling-wave harmonic plane wave with electric field
$E_{\mathrm{i}}=E_{0}\exp[-i(\omega_{\mathrm{i}}t-k_{1}z)]$ that is injected
into the structure at the space-time point $(z,ct)=(0,0)$ 333Although we
selected a harmonic plane wave excitation, the forthcoming discussion
straightforwardly extends to (modulated or unmodulated) pulse excitations of
different shapes.. Figures 4(a) and (b) plot the space-time evolution of
scattering for Prism I and Prism II, respectively, in the form of space-time
diagrams that were obtained by stacking in time successive spatial waveforms
obtained by the FDTD simulation. These results qualitatively demonstrate the
operation of the prism, with expected (here blue-shifted) frequency
transitions, spurious pure-time (here red-shifted frequency) intervals for
Prism I [Fig. 2(a)] and silence intervals [Fig. 2(b)] for Prism II. The slight
standing-wave pattern observed in the first medium is due to small reflection
near the space-time regions ($\overline{E}_{\mathrm{r}}=(n_{1}-n_{2})(1-\beta
n_{1})/[(n_{1}+n_{2})(1+\beta n_{1})]$ [3], with here
$|\overline{E}_{\mathrm{r}}|=0.097$ 444The reflection (although already
minimal) might be completely suppressed via either impedance matching
($\eta_{1}=\eta_{2}$) or oblique injection at the Brewster angle [35].).

Figure 4:  Full-wave simulated (FDTD) normalized electric field
($|\overline{E}_{x}|=|E_{x}/E_{0}|$) for (a) Prism I [PI, Fig. 2(a)] and (b)
Prism II [PII, Fig. 2(b)] with the parameters $\epsilon_{1}=1.1$,
$\epsilon_{2}=2$, $\mu=1$, $v_{\mathrm{m}}=0.2c$ and
$v_{\mathrm{m}^{\prime}}=0.8c$, and (c) corresponding output waveform Fourier
transforms, for the cases without and with interconnections (Fig. 3 for Prism
II and Sec. XIV in [40] for Prism I), and including comparison to the target
space-time interface [left panel of Fig. 1(b)]. The left bottom inset compares
the time-domain output waveforms for interconnected Prism II and the target
interface.

Figure 4(c) completes the qualitative results of Figs. 4(a) and (b) by
providing quantitative validation in terms of Fourier-transformed field
distributions. The results for the interconnected designs of both Prisms I and
II show excellent agreement with theory
($\overline{\omega}_{\mathrm{t}}=\omega_{\mathrm{t}}^{\mathrm{ST}}/\omega_{\mathrm{i}}=(1-\beta
n_{1})/(1-\beta n_{2})$ and $\overline{E}_{\mathrm{t}}=2n_{1}(1-\beta
n_{1})/[(n_{1}+n_{2})(1-\beta n_{2})]$ [3], with here
$\overline{\omega}_{\mathrm{t}}=1.102$ and $\overline{E}_{\mathrm{t}}=0.938$)
with frequency and magnitude errors of less than $0.5\%$ and $1.5\%$,
respectively. This minor discrepancy is due to a combination of the following
unaccounted-for effects: i) local multiple scattering in vertex regions (see
Sec. XII in [40]), ii) finite duration of the testing continuous wave and
resulting windowed fast-Fourier transform, and iii) FDTD discretization.

The simulation results in Fig. 4 showcase the capability of synchronized
Fresnel prisms to emulate a moving target interface. However, factors such as
noise, temperature fluctuations and interference with other devices, may
introduce jitter errors [53] in a real-life implementation. Figure 5 presents
simulation results that assess the effect of such jittering in the two
proposed prisms, with Figs. 5(a) and (b) showing the jittery waveforms of
Prism I and Prism II, respectively, and Fig. 5(c) showing related output
spectra and errors. It considers the worst-case scenario of systematic
modulation jitter mismatch (periodic advance-delay sequence) in the modulation
period, with jittering of up to $10\%$. The corresponding field amplitude
error does not exceed $5\%$, which is negligible in typical applications.

Figure 5:  Full-wave simulated (FDTD) normalized electric field
($|\overline{E}_{x}|=|E_{x}/E_{0}|$) illustrating the effect of modulation
jitter for (a) Prism I and (b) Prism II, with the same parameters as Fig. 4,
for a modulation jitter of $\Delta\tau/\tau_{\mathrm{I,II}}=10\%$, and (c)
corresponding output (after interconnection) spectra and jitter field
amplitude error
($|\overline{E}_{\mathrm{t}}^{\mathrm{jittery}}-\overline{E}_{\mathrm{t}}^{\mathrm{exact}}|/\overline{E}_{\mathrm{t}}^{\mathrm{exact}}$,
exact: target interface) for modulation jitters of
$\Delta\tau/\tau_{\mathrm{I,II}}=0\%$, $5\%$ and $10\%$.

##  VIII Conclusion and Discussion

We have presented the concept of space-time Fresnel prisms to address the
challenge of size reduction in anharmonic and nonreciprocal frequency
conversion provided by aperiodic-modulation space-time systems. Such
structures may be implemented in various technologies, including switched
transmission lines [54, 55, 56], nonlinear anisotropic perturbations [57, 58],
semiconductor free-carrier injection [59], and various other acoustic,
electronic, mechanical, plasmonic and optical modulations [5], with the usual
single modulation interface being replaced by two independent-velocity and
time-periodic modulation interfaces. The proposed Fresnel prism structures,
whose uniform (flat) space-time interfaces provide uniform space-time
frequency transitions, may be readily extended to Fresnel lens structures,
with accelerated (curved) space-time interfaces [49] providing unprecedented
type of space-time chirping functionalities.

###### Acknowledgements.

Z. Li acknowledges financial support from the China Scholarship Council (CSC,
No. 202106280230).

## References

  * Lurie [2007] K. A. Lurie,  _An Introduction to the Mathematical Theory of Dynamic Materials_ (Springer, 2007).
  * Caloz and Deck-Léger [2019] C. Caloz and Z.-L. Deck-Léger, Spacetime metamaterials—Part I: General concepts, IEEE Trans. Antennas Propag. 68, 1569 (2019).
  * Caloz and Deck-Léger [2019] C. Caloz and Z.-L. Deck-Léger, Spacetime metamaterials—Part II: Theory and applications, IEEE Trans. Antennas Propag. 68, 1583 (2019).
  * Engheta [2021] N. Engheta, Metamaterials with high degrees of freedom: space, time, and more, Nanophotonics 10, 639 (2021).
  * Caloz  _et al._ [2022] C. Caloz, Z.-L. Deck-Léger, A. Bahrami, O. C. Vicente, and Z. Li, Generalized space-time engineered modulation (GSTEM) metamaterials: A global and extended perspective., IEEE Antennas Propag. Mag. , 2 (2022).
  * Engheta [2023] N. Engheta, Four-dimensional optics using time-varying metamaterials, Science 379, 1190 (2023).
  * Yin  _et al._ [2023] S. Yin, E. Galiffi, G. Xu, and A. Alù, Scattering at temporal interfaces: An overview from an antennas and propagation engineering perspective., IEEE Antennas Propag. Mag. , 2 (2023).
  * Yu and Fan [2009] Z. Yu and S. Fan, Complete optical isolation created by indirect interband photonic transitions, Nat. Photonics 3, 91 (2009).
  * Estep  _et al._ [2014] N. A. Estep, D. L. Sounas, J. Soric, and A. Alù, Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops, Nat. Phys. 10, 923 (2014).
  * Taravati  _et al._ [2017] S. Taravati, N. Chamanara, and C. Caloz, Nonreciprocal electromagnetic scattering from a periodically space-time modulated slab and application to a quasisonic isolator, Phys. Rev. B 96, 165144 (2017).
  * Li  _et al._ [2018] H. Li, T. Kottos, and B. Shapiro, Floquet-network theory of nonreciprocal transport, Phys. Rev. Appl. 9, 044031 (2018).
  * Bacot  _et al._ [2016] V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, Time reversal and holography with spacetime transformations, Nat. Phys. 12, 972 (2016).
  * Deck-Léger  _et al._ [2018] Z.-L. Deck-Léger, A. Akbarzadeh, and C. Caloz, Wave deflection and shifted refocusing in a medium modulated by a superluminal rectangular pulse, Phys. Rev. B 97, 104305 (2018).
  * Akbarzadeh  _et al._ [2018] A. Akbarzadeh, N. Chamanara, and C. Caloz, Inverse prism based on temporal discontinuity and spatial dispersion, Opt. Lett. 43, 3297 (2018).
  * Pacheco-Peña and Engheta [2020] V. Pacheco-Peña and N. Engheta, Antireflection temporal coatings, Optica 7, 323 (2020).
  * Shlivinski and Hadad [2018] A. Shlivinski and Y. Hadad, Beyond the Bode-Fano bound: Wideband impedance matching for short pulses using temporal switching of transmission-line parameters, Phys. Rev. Lett. 121, 204301 (2018).
  * Li  _et al._ [2019] H. Li, A. Mekawy, and A. Alù, Beyond Chu’s limit with Floquet impedance matching, Phys. Rev. Lett. 123, 164102 (2019).
  * Deck-Léger  _et al._ [2019] Z.-L. Deck-Léger, N. Chamanara, M. Skorobogatiy, M. G. Silveirinha, and C. Caloz, Uniform-velocity spacetime crystals, Adv. Photonics 1, 056002 (2019).
  * Peng [2022] Y. Peng, Topological space-time crystal, Phys. Rev. Lett. 128, 186802 (2022).
  * Huidobro  _et al._ [2019] P. A. Huidobro, E. Galiffi, S. Guenneau, R. V. Craster, and J. B. Pendry, Fresnel drag in space–time-modulated metamaterials, Proc. Natl. Acad. Sci. U.S.A. 116, 24943 (2019).
  * Bahrami  _et al._ [2023a] A. Bahrami, Z.-L. Deck-Léger, and C. Caloz, Electrodynamics of metamaterials formed by accelerated modulation, Phys. Rev. Appl. 19, 054044 (2023a).
  * Tien [1958] P. K. Tien, Parametric amplification and frequency mixing in propagating circuits, J. Appl. Phys. 29, 1347 (1958).
  * Cassedy and Oliner [1963] E. S. Cassedy and A. A. Oliner, Dispersion relations in time-space periodic media: Part I—stable interactions, Proc. IEEE 51, 1342 (1963).
  * Cassedy [1967] E. S. Cassedy, Dispersion relations in time-space periodic media: Part II—unstable interactions, Proc. IEEE 55, 1154 (1967).
  * Reed  _et al._ [2003] E. J. Reed, M. Soljačić, and J. D. Joannopoulos, Color of shock waves in photonic crystals, Phys. Rev. Lett. 90, 203904 (2003).
  * Zurita-Sánchez  _et al._ [2009] J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function $\epsilon(t)$, Phys. Rev. A 79, 053821 (2009).
  * Martínez-Romero  _et al._ [2016] J. S. Martínez-Romero, O. M. Becerra-Fuentes, and P. Halevi, Temporal photonic crystals with modulations of both permittivity and permeability, Phys. Rev. A 93, 063813 (2016).
  * Mirmoosa  _et al._ [2019] M. S. Mirmoosa, G. A. Ptitcyn, V. S. Asadchy, and S. A. Tretyakov, Time-varying reactive elements for extreme accumulation of electromagnetic energy, Phys. Rev. Appl. 11, 014024 (2019).
  * Galiffi  _et al._ [2019] E. Galiffi, P. A. Huidobro, and J. B. Pendry, Broadband nonreciprocal amplification in luminal metamaterials, Phys. Rev. Lett. 123, 206101 (2019).
  * Pendry  _et al._ [2022] J. Pendry, E. Galiffi, and P. Huidobro, Photon conservation in trans-luminal metamaterials, Optica 9, 724 (2022).
  * Xu  _et al._ [2022] J. Xu, W. Mai, and D. H. Werner, Generalized temporal transfer matrix method: a systematic approach to solving electromagnetic wave scattering in temporally stratified structures, Nanophotonics 11, 1309 (2022).
  * Morgenthaler [1958] F. Morgenthaler, Velocity modulation of electromagnetic waves, IEEE Trans. Microw. Theory Techn. 6, 167 (1958).
  * Felsen and Whitman [1970] L. Felsen and G. Whitman, Wave propagation in time-varying media, IEEE Trans. Antennas Propag. 18, 242 (1970).
  * Fante [1971] R. Fante, Transmission of electromagnetic waves into time-varying media, IEEE Trans. Antennas Propag. 19, 417 (1971).
  * Kunz [1980] K. S. Kunz, Plane electromagnetic waves in moving media and reflections from moving interfaces, J. Appl. Phys. 51, 873 (1980).
  * Mendonça  _et al._ [2003] J. T. Mendonça, A. M. Martins, and A. Guerreiro, Temporal beam splitter and temporal interference, Phys. Rev. A 68, 043801 (2003).
  * Biancalana  _et al._ [2007] F. Biancalana, A. Amann, A. V. Uskov, and E. P. O’Reilly, Dynamics of light propagation in spatiotemporal dielectric structures, Phys. Rev. E 75, 046607 (2007).
  * Silbiger and Hadad [2023] O. Silbiger and Y. Hadad, Optimization-free filter and matched-filter design through spatial and temporal soft switching of the dielectric constant, Phys. Rev. Appl. 19, 014047 (2023).
  * Li  _et al._ [2023] Z. Li, X. Ma, A. Bahrami, Z.-L. Deck-Léger, and C. Caloz, Generalized total internal reflection at dynamic interfaces, Phys. Rev. B 107, 115129 (2023).
  * sup [a] See Supplemental Material at [URL will be inserted by publisher] for the derivations, additional figures and extra simulation results.
  * Saleh and Teich [2019] B. E. Saleh and M. C. Teich,  _Fundamentals of Photonics_ (John Wiley & Sons, 2019).
  * Golub [2006] I. Golub, Fresnel axicon, Opt. Lett. 31, 1890 (2006).
  * Landauer [1963] R. Landauer, Velocity modulation of propagating waves, J. Appl. Phys. 34, 1893 (1963).
  * Tsai and Auld [1967] C. Tsai and B. Auld, Wave interactions with moving boundaries, J. Appl. Phys. 38, 2106 (1967).
  * Note [1] We may switch from $\omega_{\mathrm{t}}^{\mathrm{ST}}>\omega_{\mathrm{i}}$ (blue shift) to $\omega_{\mathrm{t}}^{\mathrm{ST}}<\omega_{\mathrm{i}}$ (red shift) by making the ratio $n_{2}/n_{1}$ larger or smaller than 1, or by making the parameter $\beta$ greater (co-directional modulation) or smaller (contra-directional modulation) than zero. However, changing the ratio $n_{2}/n_{1}$ is preferable to changing the velocity regime from $\beta>0$ to $\beta<0$ because $\beta>0$ implies a smaller reflection coefficient [3].
  * Note [2] In case of need, one may resort to various dispersion engineering or compensation techniques [60, 61] to enhance the bandwidth of the prisms.
  * Taflove and Hagness [2005] A. Taflove and S. Hagness,  _Computational Electrodynamics: The Finite-Difference Time-Domain Method_ , Artech House antennas and propagation library (Artech House, 2005).
  * Deck-Léger  _et al._ [2023] Z.-L. Deck-Léger, A. Bahrami, Z. Li, and C. Caloz, Generalized FDTD scheme for the simulation of electromagnetic scattering in moving structures, Opt. Express 31, 23214 (2023).
  * Bahrami  _et al._ [2023b] A. Bahrami, Z.-L. Deck-Léger, Z. Li, and C. Caloz, Generalized FDTD scheme for moving electromagnetic structures with arbitrary space-time configurations (2023b), arXiv:2306.10035 .
  * sup [b] See Supplemental Material at [URL will be inserted by publisher] for the FDTD animations.
  * Note [3] Although we selected a harmonic plane wave excitation, the forthcoming discussion straightforwardly extends to (modulated or unmodulated) pulse excitations of different shapes.
  * Note [4] The reflection (although already minimal) might be completely suppressed via either impedance matching ($\eta_{1}=\eta_{2}$) or oblique injection at the Brewster angle [35].
  * Li [2007] M. P. Li,  _Jitter, Noise, and Signal Integrity at High-Speed_ (Pearson Education, 2007).
  * Ran  _et al._ [2015] J. Ran, Y. Zhang, X. Chen, K. Fang, J. Zhao, Y. Sun, and H. Chen, Realizing tunable inverse and normal Doppler shifts in reconfigurable RF metamaterials, Sci. Rep. 5, 11659 (2015).
  * Bahrami and Caloz [2021] A. Bahrami and C. Caloz, Arbitrary pulse shaping using nonuniform spacetime modulation, in  _2021 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting (APS/URSI)_ (2021) pp. 409–410.
  * Moussa  _et al._ [2023] H. Moussa, G. Xu, S. Yin, E. Galiffi, Y. Ra’di, and A. Alù, Observation of temporal reflection and broadband frequency translation at photonic time interfaces, Nat. Phys. , 1 (2023).
  * Philbin  _et al._ [2008] T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, Fiber-optical analog of the event horizon, Science 319, 1367 (2008).
  * Sloan  _et al._ [2022] J. Sloan, N. Rivera, J. D. Joannopoulos, and M. Soljačić, Controlling two-photon emission from superluminal and accelerating index perturbations, Nat. Phys. 18, 67 (2022).
  * Gaafar  _et al._ [2019] M. A. Gaafar, T. Baba, M. Eich, and A. Y. Petrov, Front-induced transitions, Nat. Photonics 13, 737 (2019).
  * Caloz [2011] C. Caloz, Metamaterial dispersion engineering concepts and applications, Proc. IEEE 99, 1711 (2011).
  * Fan  _et al._ [2020] R.-H. Fan, B. Xiong, R.-W. Peng, and M. Wang, Constructing metastructures with broadband electromagnetic functionality, Adv. Mater. 32, 1904646 (2020).
  * Kong [1990] J. A. Kong,  _Electromagnetic Wave Theory_ (Wiley-Interscience, 1990).
  * Lurie  _et al._ [2009] K. Lurie, D. Onofrei, and S. Weekes, Mathematical analysis of the waves propagation through a rectangular material structure in space–time, J. Math. Anal. Appl. 355, 180 (2009).
  * Sejdić  _et al._ [2009] E. Sejdić, I. Djurović, and J. Jiang, Time–frequency feature representation using energy concentration: An overview of recent advances, Digit. Signal Process. 19, 153 (2009).

###### Contents

  1. I Introduction
  2. II Space-Time Fresnel Prism Concept
  3. III Two Types of Space-Time  Fresnel Prism
  4. IV Spurious Effects Comparison
  5. V Phase Continuity Design
  6. VI Interconnection Schemes
  7. VII Simulation Results and Analysis
  8. VIII Conclusion and Discussion
  9. IX Pure-Time and Space-Time Discontinuities  in the Pulse and Continuous-Wave Regimes
  10. X Comparison of the Different Space-Time Structures
  11. XI Conversion Efficiency for the Two Prisms without Interconnections
  12. XII Comparison of Spurious Effects in the Conventional Fresnel Prism  and in the Space-Time Fresnel Prism
  13. XIII Phase-Matching Formulas for the Two Prisms
    1. XIII.1 Prism I [Eq. (2)]
    2. XIII.2 Prism II [Eq. (1)]
  14. XIV Design of the Interconnected Prisms
    1. XIV.1 Prism I
    2. XIV.2 Prism II [Eqs. (3)-(4)]
  15. XV Size Comparison of Interconnected Prism II  and the Target Moving Interface
  16. XVI Additional Simulation Results

##  IX Pure-Time and Space-Time Discontinuities
in the Pulse and Continuous-Wave Regimes

Figures 6(a) and (b) depict the wave scattering at a pure-time modulation step
and space-time modulation step in both the pulse-wave regime (left panels) and
continuous-wave regime (right panels). In the pulse-wave regime, a finite
implementation size is possible, but synchronization is required between the
incident wave and the modulation for scattering to occur. In the continuous-
wave regime, the size of the device would need to extend to infinity to
accommodate the infinite duration of the wave, as shown in the right panels of
the figure.

Figure 6:  Scattering at a (a) pure-time step interface and (b) subluminal
space-time step interface in the pulse-wave regime (left panels) and
Continuous Wave (CW) regime (right panels).

##  X Comparison of the Different Space-Time Structures

Figure 7 provides details on the operation principle of a space-time interface
and the two proposed space-time prisms.

Figure 7:  Detailed operation principle of a (a) space-time interface [Fig.
1(b), left panel], (b) Prism I [Fig. 2(a)] and (c) Prism II [Fig. 2(b)] in
space-time (top panels) and space-index (middle and bottom panels) dimensions.
IP: Initial Position; FP: Final Position; OP: Observation Position.

The top panels of Fig. 7 show 1+1 D space-time diagrams illustrating the
transmission behavior across an infinite space-time interface compared with
transmission across the two proposed types of space-time Fresnel prisms. Note
that in the case of the infinite interface, represented in Fig. 7(a), the load
must be positioned before the end of the modulation,
$z_{\mathrm{L}}<z_{\mathrm{f}}$ since $z_{\mathrm{f}}$ tends to infinity. In
contrast, in the case of Prisms I and II, which are respectively represented
in Figs. 7(b) and (c), the load may be positioned at the end of the (now
finite) structure, $z_{\mathrm{L}}>z_{\mathrm{f}}$.

The middle panels of Fig. 7 present a 1D optical waveguide (e.g., optical
fiber) analogy, with source, modulation and load, and with the refractive
indices varying along the axis of the waveguide.

The bottom panels of Fig. 7 represent the amplitude of the modulation and
waves at three specific times, $t_{a}$, $t_{b}$ and $t_{c}$, corresponding to
the space-time diagram in the top panels of the figure. The illustration
provides insight into the modulation process and demonstrates how the wave
amplitudes evolve over time and space. By examining the wave behavior at such
different instants, one can gain a deeper understanding of the space-time
characteristics and dynamics associated with the modulation.

##  XI Conversion Efficiency for the Two Prisms without Interconnections

Figure 8 shows space-time diagrams of Prism I and Prism II with relevant
information to derive their conversion efficiency formulas.

Figure 8: Space-time diagrams to derive the conversion efficiency formulas for
(a) Prism I and (b) Prism II.

To derive the conversion efficiency for the first type of space-time Fresnel
prism, depicted in Fig. 8(a), we compute the Pure-Time (PT) interval
$\tau^{\mathrm{PT}}$ and the Space-Time (ST) interval $\tau^{\mathrm{ST}}$,
using space-time geometrical relations, which yields

$\tau^{\mathrm{PT}}=t_{3}-t_{2}=\frac{d_{\mathrm{I}}}{c}n_{2}$ (5a) and
$\tau^{\mathrm{ST}}=(t_{2}-t_{0})-(t_{1}-t_{0})=\frac{d_{\mathrm{I}}}{c}(1/\beta-
n_{2}),$ (5b) respectively.

The conversion efficiency $\eta_{\mathrm{I}}$, is then found as

$\eta_{\mathrm{I}}=\frac{\tau^{\mathrm{ST}}}{\tau^{\mathrm{PT}}+\tau^{\mathrm{ST}}}=\frac{1/\beta-
n_{2}}{1/\beta}=1-n_{2}\beta,$ (6)

where the second identity is obtained by substituting Eq. (5) into Eq. (6).

To derive the conversion efficiency for the second type of space-time Fresnel
prism, shown in Fig. 8(b), we compute the silence interval
$\tau^{\mathrm{silence}}$ and the space-time interval $\tau^{\mathrm{ST}}$,
using again space-time geometrical relations, which yields

$\tau^{\mathrm{silence}}=(t_{5}-t_{0})-(t_{4}-t_{0})=\frac{d_{\mathrm{i}}}{c}(n_{2}-n_{1})$
(7a) and
$\tau^{\mathrm{ST}}=(t_{6}-t_{0})-(t_{5}-t_{0})=\frac{d_{\mathrm{i}}}{c}(1/\beta-
n_{2}),$ (7b) respectively, where $d_{\mathrm{i}}$ represents the catch-up
distance between the trajectory of the part of the incident wave interacting
with the left vertex of Prism II and the moving interface with velocity
$v_{\mathrm{m}}$.

The conversion efficiency $\eta_{\mathrm{II}}$, is then found as

$\eta_{\mathrm{II}}=\frac{\tau^{\mathrm{ST}}}{\tau^{\mathrm{silence}}+\tau^{\mathrm{ST}}}=\frac{1/\beta-
n_{2}}{1/\beta-n_{1}}=\frac{1-n_{2}\beta}{1-n_{1}\beta},$ (8)

where the second identity is obtained by substituting Eq. (7) into Eq. (8).

##  XII Comparison of Spurious Effects in the Conventional Fresnel Prism
and in the Space-Time Fresnel Prism

Figure 9 shows the space-space and space-time diagram for the desired and
spurious effects in the conventional Fresnel prism [right panel of Fig. 1(a)]
and the space-time Fresnel prism with pure-time sections [Fig. 2(a)],
respectively.

Figure 9:  Desired (bottom sub-prisms) and spurious (middle and top sub-
prisms) effects in (a) the conventional Fresnel prism [Fig. 1(a), right panel]
and (b) the space-time Fresnel prism with pure-time sections [Fig. 2(a)].

The arrows in the bottom sub-prisms show the desired deflection at a space-
space interface [Fig. 9(a)] and frequency transition at a space-time interface
[Fig. 9(b)], with reflections being neglected in both cases.

The arrows in the middle sub-prisms represent the diffraction occurring at the
apexes of the conventional Fresnel prism [Fig. 9(a)] (some diffraction, not
shown, also occurs at the vertices on the right of the apexes) and the fixed-
directional propagation occurring at the apexes of the space-time Fresnel
prism [Fig. 9(b)], with small reflections being also shown for the sake of
completeness.

The arrows in the top sub-prisms depict the spurious transmission and multiple
scattering occurring at the horizontal interfaces of both the conventional
Fresnel prism [Fig. 9(a)] and the space-time Fresnel prism [Fig. 9(b)]. Note
that, different from the conventional case, the multiple scattering in the
space-time Fresnel prism is restricted to the positive vertical ($+ct$)
direction due to causality.

##  XIII Phase-Matching Formulas for the Two Prisms

Figure 10 shows space-time diagrams of Prism I and Prism II with relevant
information to derive their phase matching design formulas. We selected here
as the excitation a traveling-wave harmonic plane wave of electric field
$E_{\mathrm{i}}=E_{0}\exp(-i\phi_{\mathrm{i}})$, with the phase
$\phi_{\mathrm{i}}(z,ct)=\omega_{\mathrm{i}}t-k_{\mathrm{i}}z$, but the
forthcoming derivations may be extended to (modulated or unmodulated) pulse
excitations of different shapes.

Figure 10:  Space-time diagrams to derive the phase continuity design formulas
for (a) Prism I and (b) Prism II.

###  XIII.1 Prism I [Eq. (2)]

To derive the size and temporal periods of Prism I [Fig. 10(a)] for phase
continuity, we may utilize the phase tracing method [62, 63], tracking back
the transmission phase from a given space plane to the incidence space plane
($z=0$), for simplifying the derivations. We find

$\phi_{\mathrm{s}}(z_{\mathrm{L}},ct_{3})=\phi(z_{\mathrm{f}},ct_{2})=\phi(0,ct_{0})\quad\textrm{and}\quad\phi_{\mathrm{e}}(z_{\mathrm{L}},ct_{4})=\phi(z_{\mathrm{i}},ct_{2})=\phi(0,ct_{1}).$
(9)

Successively substituting the last equalities of Eqs. (9) into the phase
continuity condition $\phi_{\mathrm{e}}-\phi_{\mathrm{s}}=2\pi q$
($q\in\mathbb{N}$) and setting the related phase difference to the
corresponding phase difference for the incident wave,
$\phi_{\mathrm{i}}(z,ct)=\omega_{\mathrm{i}}t-k_{\mathrm{i}}z$, in the plane
$z=0$, we obtain

$\phi_{\mathrm{e}}-\phi_{\mathrm{s}}=\phi(0,ct_{1})-\phi(0,ct_{0})=2\pi
q=\omega_{\mathrm{i}}(t_{1}-t_{0}),$ (10a) so that $t_{1}-t_{0}=2\pi
q/\omega_{\mathrm{i}}.$ (10b)

We obtain then the sought-after size and temporal period of Prism I using
space-time geometrical relations with Eq. (10b)

$d_{\mathrm{I}}=z_{\mathrm{f}}-z_{\mathrm{i}}=(t_{1}-t_{0})c/n_{1}=q\frac{\lambda_{0}}{n_{1}}\quad\text{and}\quad\tau_{\mathrm{I}}=\frac{d_{\mathrm{I}}}{v_{\mathrm{m}}}=q\frac{cT_{0}}{n_{1}v_{\mathrm{m}}},$
(11)

where $\lambda_{0}=2\pi c/\omega_{\mathrm{i}}$ and
$T_{0}=2\pi/\omega_{\mathrm{i}}$ are the free-space incident wavelength and
period, respectively.

###  XIII.2 Prism II [Eq. (1)]

For Prism II [Fig. 10(b)], we use again the phase tracing method but this time
applied to the silence intervals, which yields

$\phi_{\mathrm{s}}(z_{\mathrm{L}},ct_{7})=\phi(z_{\mathrm{f}},ct_{6})=\phi(0,ct^{-}_{1})$
(12a) and
$\phi_{\mathrm{e}}(z_{\mathrm{L}},ct_{4})=\phi(z_{\mathrm{i}},ct_{2})=\phi(0,ct^{+}_{1}),$
(12b)

where $t^{\mp}_{1}=t_{1}\mp\delta t$ with $\delta t\rightarrow 0$. Equation
(12) shows that the starting and ending phases of the silence intervals are
inherently continuous, so that the phase continuity condition
$\phi_{\mathrm{e}}=\phi_{\mathrm{s}}$ is automatically satisfied in Prism II.

For a given prism size $d_{\mathrm{II}}$, the temporal period is given by

$\tau_{\mathrm{II}}=t_{2}-0=(t_{6}-0)-(t_{5}-0)=\left(\frac{1}{v_{\mathrm{m}}}-\frac{1}{v_{\mathrm{m}^{\prime}}}\right)d_{\mathrm{II}}.$
(13)

##  XIV Design of the Interconnected Prisms

Figure 11 shows space-time diagrams of Prism I and Prism II with relevant
information to derive their interconnection design formulas.

Figure 11:  Space-time diagrams to derive the interconnection design formulas
for (a) Prism I and (b) Prism II with $N$ sub-prisms.

###  XIV.1 Prism I

The design of interconnected Prism I follows that of Prism II in the body of
the paper, based on space-time back-translating the points related to the
transmitted phases $\phi_{\mathrm{e}}$ along the transmission trajectories to
the same time levels as the $\phi_{\mathrm{s}}$-related points. We find, using
the space-time geometrical relations in Fig. 11(a), that the interconnection
distance $d_{\mathrm{c}}$ is equal to the prism size $d_{\mathrm{I}}$ [Eq.
(2)], which yields

$d_{\mathrm{c}}=d_{\mathrm{I}}=q\frac{\lambda_{0}}{n_{1}},\quad
q\in\mathbb{N}.$ (14)

Assuming that the first switch is positioned at $z_{\mathrm{s}1}$, the second
switch must to be positioned to its left at a distance of $d_{\mathrm{c}}$ for
phase continuity, i.e., at $z_{\mathrm{s}2}=z_{\mathrm{s}1}-d_{\mathrm{c}}$.
The position of the third switch follows as
$z_{\mathrm{s}3}=z_{\mathrm{s}2}-d_{\mathrm{c}}=z_{\mathrm{s}1}-2d_{\mathrm{c}}$,
and so on, until the last switch ($N^{\mathrm{th}}$) reaches the final
modulation position, $z_{\mathrm{s}N}=z_{\mathrm{f}}$. The $m^{\mathrm{th}}$
position of the switch is then given as

$z_{\mathrm{s}m}=z_{\mathrm{f}}+(N-m)d_{\mathrm{c}},\quad m=1,2,3,...,N,$ (15)

and the switching time of the $m^{\mathrm{th}}$ switch is hence

$t_{\mathrm{s}m}=t_{\mathrm{s}1}+(m-1)\Delta t_{\mathrm{s}},$ (16a) with the
switching time of the first switch being, according to Fig. 11(a),
$t_{\mathrm{s}1}=\tau_{\mathrm{I}}+\frac{z_{\mathrm{s}2}-z_{\mathrm{i}}}{c/n_{2}}=\left(\frac{1}{v_{\mathrm{m}}}+\frac{N-1}{c/n_{2}}\right)d_{\mathrm{I}},$
(16b) where the second identity is obtained by substituting Eq. (15) with
$m=2$ into $z_{\mathrm{s}2}$ and Eq. (11) into $\tau_{\mathrm{I}}$ and
rewriting the resulting equation in terms of $d_{\mathrm{I}}$, and where the
(uniform) switching interval between the switching times, according to Fig.
11(a), is $\Delta
t_{\mathrm{s}}=\tau_{\mathrm{I}}-\frac{d_{\mathrm{I}}}{c/n_{2}}=\left(\frac{1}{v_{\mathrm{m}}}-\frac{1}{c/n_{\mathrm{2}}}\right)d_{\mathrm{I}}.$
(16c)

###  XIV.2 Prism II [Eqs. (3)-(4)]

The derivation of the interconnection parameters $z_{\mathrm{s}m}$ [Eq. (3)]
and $t_{\mathrm{s}m}$ [Eq. (4)] for interconnected Prism II [Fig. 11(b)] is
fairly complicated and requires the space-time geometrical derivation of some
preliminary formulas, which are the duration of the interluminal interface,

$\Delta t^{\prime}=\frac{1}{v_{\mathrm{m}^{\prime}}}d_{\mathrm{II}},$ (17a)
the distance from the apex of the first two interfaces to the third interface,
$d^{\prime}=\Delta
t^{\prime}(v_{\mathrm{m}^{\prime}}-v_{\mathrm{m}})=\frac{v_{\mathrm{m}^{\prime}}-v_{\mathrm{m}}}{v_{\mathrm{m}^{\prime}}}d_{\mathrm{II}},$
(17b) where the second relation is obtained by using Eq. (17a), and the delay
of the incident wave trajectory between the apex of the first two interfaces
and the third interface, $\Delta
t_{\mathrm{i}}=\frac{d^{\prime}}{c/n_{1}-v_{\mathrm{m}}}=\frac{v_{\mathrm{m}^{\prime}}-v_{\mathrm{m}}}{v_{\mathrm{m}^{\prime}}(c/n_{1}-v_{\mathrm{m}})}d_{\mathrm{II}},$
(17c) where the second relation is obtained by using the second equality in
Eq. (17b).

The interconnection distance $d_{\mathrm{c}}$ is then obtained, by phase back-
tracing, as

$d_{\mathrm{c}}=\Delta
t_{\mathrm{i}}(c/n_{1}-c/n_{2})=\frac{(c/n_{1}-c/n_{2})(v_{\mathrm{m}^{\prime}}-v_{\mathrm{m}})}{v_{\mathrm{m}^{\prime}}(c/n_{1}-v_{\mathrm{m}})}d_{\mathrm{II}},$
(18)

where the second identity is obtained by substituting Eq. (17c) and rewriting
the equation in terms of $d_{\mathrm{II}}$.

The $m^{\mathrm{th}}$ position of the switch may next be expressed, similarly
to the case of Prism I, as

$z_{\mathrm{s}m}=z_{\mathrm{f}}+(N-m)d_{\mathrm{c}},\quad m=1,2,3,...,N,$ (19)

with the same switching time formula

$t_{\mathrm{s}m}=t_{\mathrm{s}1}+(m-1)\Delta t_{\mathrm{s}},$ (20a) but
different switching parameters
$t_{\mathrm{s}1}=\tau_{\mathrm{II}}+\frac{z_{\mathrm{s}2}-z_{\mathrm{i}}}{c/n_{2}}=\tau_{\mathrm{II}}+\frac{d_{\mathrm{II}}+(N-2)d_{\mathrm{c}}}{c/n_{2}},$
(20b) where the second identity is obtained by substituting Eq. (19) with
$m=2$ into $z_{\mathrm{s}2}$ with $\tau_{\mathrm{II}}$ and $d_{\mathrm{c}}$
given in Eqs. (13) and (18), respectively, and $\Delta
t_{\mathrm{s}}=\frac{c\Delta
t_{\mathrm{i}}}{n_{1}}\left(\frac{1}{v_{\mathrm{m}}}-\frac{n_{2}}{c}\right)=\frac{n_{2}(v_{\mathrm{m}^{\prime}}-v_{\mathrm{m}})(c/n_{2}-v_{\mathrm{m}})}{n_{1}v_{\mathrm{m}}v_{\mathrm{m}^{\prime}}(c/n_{1}-v_{\mathrm{m}})}d_{\mathrm{II}}.$
(20c)

##  XV Size Comparison of Interconnected Prism II
and the Target Moving Interface

We shall compare here the implementation size of interconnected Prism II [Fig.
3] with that of the target moving interface [left panel of Fig. 1(b)] for a
finite number of sub-prisms, $N$, for the switched-delay-line scheme.

Interconnected Prism II [Fig. 11(b)], composed of $N$ sub-prisms and involving
$N-1$ transmission line sections, has the extra delay-line size
$(N-1)d_{\mathrm{c}}$ compared to the initial Prism II size, $d_{\mathrm{II}}$
[Fig. 2(b)]. The total size of the interconnected Prism II is then obtained as

$D_{\mathrm{II}}=d_{\mathrm{II}}+(N-1)d_{\mathrm{c}},$ (21)

with $d_{\mathrm{c}}$ given in Eq. (18).

Within the time duration corresponding to $N$ sub-prisms in interconnected
Prism II, the target moving interface with velocity $v_{\mathrm{m}}$ reaches
the size [Fig. 11(b)]

$D=v_{\mathrm{m}}~{}(N\tau_{\mathrm{II}}+\Delta t^{\prime}),$ (22)

with $\tau_{\mathrm{II}}$ and $\Delta t^{\prime}$ given in Eqs. (13) and
(17a).

We may then compute the size reduction ratio as

$\gamma=\frac{D_{\mathrm{II}}}{D}=\frac{v_{\mathrm{m}^{\prime}}(c/n_{1}-v_{\mathrm{m}})+(N-1)(c/n_{1}-c/n_{2})(v_{\mathrm{m}^{\prime}}-v_{\mathrm{m}})}{(c/n_{1}-v_{\mathrm{m}})[Nv_{\mathrm{m}^{\prime}}-(N-1)v_{\mathrm{m}}]},$
(23)

where the second identity is obtained by substituting Eq. (18) into Eq. (21)
and Eqs. (13) and (17a) into Eq. (22), respectively.

For simplicity, we set the velocity of the interluminal interface to
$v_{\mathrm{m}^{\prime}}=c/n_{1}$. Then Eq. (23) reduces to

$\gamma=\frac{Nn_{2}/n_{1}-(N-1)}{Nn_{2}/n_{1}-(N-1)\beta n_{2}}.$ (24)

In the subluminal regime, where $\beta n_{2}<1$, the numerator of Eq. (24) is
always smaller than the denominator, so that the size reduction ratio is
always smaller than $1$. This implies that the implementation size of
interconnected Prism II is always smaller than the size of the target moving
interface.

To quantitatively visualise the reduction effect, Fig. 12 plots the reduction
ratio $\gamma$ versus the modulation strength $n_{2}/n_{1}$, the normalized
(to the wave velocity in the second medium) modulation velocity $\beta n_{2}$
and the logarithm of the number of sub-prisms. A structure with a small
reduction ratio can be realized by properly choosing the values of $n_{1}$,
$n_{2}$ and $\beta$.

Figure 12:  Isosurface plot for the size reduction ratio, $\gamma$ [Eq. (24)],
of interconnected Prism II versus the modulation strength $n_{2}/n_{1}$, the
normalized (to the wave velocity in the second medium) modulation velocity
$\beta n_{2}$ and the logarithm of the number of sub-prisms. The interluminal
interface velocity is chosen as $v_{\mathrm{m}^{\prime}}=c/n_{1}$ for
simplicity.

##  XVI Additional Simulation Results

To validate the phase continuity formula for Prism I, we present corresponding
field plots for two scenarios in Fig. 13: (i) $d=d_{\mathrm{I}}$ [phase
matching size, according to Eq. (2)], and (ii) $d\neq d_{\mathrm{I}}$.

Figure 13:  Full-wave simulated (FDTD) normalized electric field
($|\overline{E}_{x}|=|E_{x}/E_{0}|$) for Prism I (a) with the phase continuity
condition [Eq. (2)] satisfied and (b) with the phase continuity condition
unsatisfied, for the parameters $\epsilon_{1}=1.1$, $\epsilon_{2}=2$, $\mu=1$
and $v_{\mathrm{m}}=0.2c$. In both case, the insets show the field waveforms
at a specific space position $z_{\mathrm{L}}$ (right inset) and time point
$t_{0}$ (top inset).

Figure 14 shows the space-time diagrams for Prisms I and II over a larger
space-time area (more sub-prisms). For the sake of simplicity and clarity, the
space-time diagrams are cropped to a finite space window.

Figure 14:  Full-wave simulated (FDTD) normalized electric field
($|\overline{E}_{x}|=|E_{x}/E_{0}|$) for (a) Prism I [2(a)] and (b) Prism II
[2(b)] with the parameters $\epsilon_{1}=1.1$, $\epsilon_{2}=2$, $\mu=1$,
$v_{\mathrm{m}}=0.2c$ and $v_{\mathrm{m}^{\prime}}=0.8c$.

Figure 15 compares the output Fourier-transformed field distributions for the
space-time Prisms without and with interconnections as well as for the target
interface. Important differences – including significant frequency and
magnitude shifting as well as spurious sub-peaks – occur for the case of the
non-interconnected prisms, but this is because the related Fourier transforms
were taken across the entire output waveforms, including the spurious pure-
time and silence intervals, whereas the actual space-time waveform sections,
which may be separately used in some pulse regime applications, naturally
exhibit a much cleaner spectral response (obtainable by short-term Fourier
transformation [64]), as visualized in Figs. 14(a) and (b).

Figure 15:  Full-wave simulated (FDTD) normalized electric field
($|\overline{E}_{x}|=|E_{x}/E_{0}|$) for the output waveform Fourier
transforms for Prisms I and II in Fig. 14 without and with interconnections,
and for the target space-time interface [Fig. 1(b), left panel].