# Symmetry and Finite-size Effects in Quasi-optical Extraordinarily THz
Transmitting Arrays of Tilted Slots

Miguel Camacho, , Ajla Nekovic, Suzanna Freer, Pavel Penchev, Rafael R. Boix,
, Stefan Dimov and Miguel Navarro-Cía The work of M. Camacho was supported by
the Engineering and Physical Sciences Research Council (EPSRC) of the United
Kingdom, via the EPSRC Centre for Doctoral Training in Metamaterials [Grant
No. EP/L015331/1]. The work of S. Freer was supported by the EPSRC
[Studentship No. 2137478]. The work of R.R. Boix was supported by the
Ministerio de Ciencia, Innovación y Universidades [Grant TEC2017-84724-P]. The
work of M. Navarro-Cía was supported by the EPSRC [Grant No. EP/S018395/1],
the Royal Society [Grant No. RSG/R1/180040], and the University of Birmingham
[Birmingham Fellowship]. (Corresponding author: Miguel Navarro-Cía)M. Camacho
is with the Department of Electrical and Systems Engineering, University of
Pennsylvania, Philadelphia, PA 19104-6390, USA (e-mail:
mcamagu@seas.upenn.edu). Ajla Nekovic is with the Faculty of Electrical
Engineering, University of Sarajevo, Sarajevo 71000, Bosnia and HerzegovinaR.
R. Boix is with Microwaves Group, Department of Electronics and
Electromagnetism, College of Physics, University of Seville, Avda. Reina
Mercedes s/n, 41012, Seville, Spain (e-mail: boix@us.es)S. Freer and M.
Navarro-Cía are with the School of Physics and Astronomy, University of
Birmingham, Birmingham B15 2TT, United Kingdom (e-mail: M.Navarro-
Cia@bham.ac.uk)P. Penchev and S. Dimov are with the Department of Mechanical
Engineering, School of Engineering, University of Birmingham, Birmingham, B15
2TT, UK

###### Abstract

Extraordinarily transmitting arrays are promising candidates for quasi-optical
(QO) components due to their high frequency selectivity and beam scanning
capabilities owing to the leaky-wave mechanism involved. We show here how by
breaking certain unit cell and lattice symmetries, one can achieve a rich
family of transmission resonances associated with the leaky-wave dispersion
along the surface of the array. By combining two dimensional and one
dimensional periodic Method of Moments (MoM) calculations with QO Terahertz
(THz) time-domain measurements, we provide physical insight, numerical and
experimental demonstration of the different mechanisms involved in the
resonances associated with the extraordinary transmission peaks and how these
evolve with the number of slots. Thanks to the THz instrument used, we are
also able to explore the time-dependent emission of the different frequency
components involved.

###### Index Terms:

Extraordinary transmission, frequency selective surface, method of moments,
quasi-optics, terahertz, time-domain spectrometer.

##  I Introduction

In the early 1990s, the extraordinary optical tranmission (EOT) phenomenon
through subwavelength apertures was discovered [1, 2, 3] opening the door to
new and exciting physics. The transmission peak frequency and magnitude were
initially thought to invalidate Bethe’s prediction for subwavelength apertures
[4], although this comparison lacked a very meaningful feature of the
experimental sample: the periodicity. This characteristic held the key for the
explanation of the, at that point, unexplained mystery.

The relation between the high transmissivity and the periodicity became
apparent when the transmission spectrum was mapped for different planes of
incidence [5], finding that these peaks correspond to the excitation of
optical-frequency surface waves known as surface plasmons [6, 7]. These
surface waves, usually tightly bound to the surface of a plasmonic material,
can become leaky when supported by periodic structures, following the same
principle used nowadays by periodically-modulated leaky wave antennas [8] and
that can be traced back to [9]. This intuitive explanation was later confirmed
by means of plane wave expansion [10], showing that these surface waves are
capable of providing large transmission enhancements even for screens with
non-negligible thicknesses thanks to the evanescent coupling between surface
plasmons supported by opposite faces.

Figure 1: Schematic diagram of the freestanding subwavelength tilted slot
$N_{x}$ by $N_{y}$ array along with the collimated incident Gaussian beam.
Lattice periods a and b, and slot dimensions $l\times w$ rotated by an angle
$\alpha$.

Experiments similar to those published in [3] were conducted at microwave
regime, in the absence of plasmonic materials, shockingly finding similar
results [11, 12, 13, 14]. Also, EOT was reported at such regimes for
s-polarized illumination that cannot excite surface plasmons even if plasmonic
materials were used [15, 16]. Without plasmons, the explanation of microwave
extraordinary transmission was challenging. However, not one but two
complementary explanations were provided. The first one followed the same
logic as that one provided at optical frequencies: although metals do not
support plasmons at microwave frequencies, when properly geometrically
engineered, they can support analogous surface wave modes [17, 18], which
produce extraordinary transmission effects analogous to those of surface
plasmons [19]. It is worth noting that although these surface waves received a
large attention subsequently to their role in EOT, they were known by the
engineering community since the mid twentieth century [20, 21]. An alternative
explanation, based on impedance matching formulation well known for
transmission lines was provided over a decade ago [22, 23].

The development of extraordinary transmission was subsequent to similar
periodic structures, widely-used by the microwave community, known as
Frequency Selective Surfaces (FSS) [24, 25]. However, the latter make use of
the _natural_ resonance of the unit element in each until cell, defined this
as the one dictated by their electrical size, with little or no effects
arising from their arrangement [26]. This main difference is the root of
interesting phenomena, as imposing or breaking certain symmetries has drastic
effects on the existence of EOT peaks associated with the periodicities of the
system. For instance, when the unit cell presents a symmetry plane aligned
with the periodicity (electric wall, for instance), this inhibits the EOT peak
associated with such lattice vector [27]. As it was shown there, this means
that when finite arrays are considered, one can ignore the finite size effects
along one of the periodicities if the unit cell presents symmetries that
inhibit that resonance, as the surface wave along the short direction will not
couple to the radiation and therefore cannot sense the truncations of the
system [28, 29].

EOT arrays, with their narrow linewidth and strong suppression out of band,
have been shown to be excellent candidates for quasi-optical (QO) components
such as filters and wave plates [15, 30, 31]. However, the potential added by
the exploitation of the (lack of) symmetries is still to be explored. In an
effort to bridge this gap, in this paper we study the extraordinary
transmission phenomenon through arrays of non-symmetric slot arrays. This lack
of symmetry is two-fold: we make use of rectangular arrays (with periodicities
given by $a$ and $b$, with $a$ different from $b$), whose unit cell contains
slots that present a rotation with respect to the lattice vectors and also
insert a slot-shaped aperture which is not colinear with any of the lattice
vectors. In the first part of the paper we show how these two conditions are
required for us to explore the whole set of resonances associated with the two
different periodicities, and in the second part we experimentally demonstrate
the presence of these resonances, and show how finite effects also play an
important role when realistic collimated illuminations are used. Finally we
explore both the time-response of the system through spectrography and
illumination effects such as polarization and angle of incidence dependencies.

##  II Theory

Let us consider the array shown in Fig.1, which contains an array of slots of
dimensions $l\times w$, rotated with respect to the two periodicities, with
values $a$ and $b$. These slots are cut into an infinite perfectly-conducting
zero-thickness screen, which is a valid approximation for terahertz (THz)
frequencies, such as those studied here. This rectangular geometry for the
apertures is very convenient for the semi-analytical solution of the
scattering problem in terms of analytical basis functions, yet the phenomena
presented in the following can be found for any other geometry that lacks
mirror symmetry. Let us consider the case in which this array of slots is
illuminated by a plane wave propagating along the $\mathbf{\hat{z}}$
direction, whose electric field is given by $\mathbf{E}_{\text{i}}$. The
problem of the scattering by the array can be studied by means of an integral
equation for the tangential electric field in the slots,
$\mathbf{E}_{t}^{\mathrm{sc}}$, given by

Figure 2: Top figures show the transmission spectra at normal incidence for
the lowest Floquet mode for vertical, diagonal ($\alpha=45^{\circ}$) and
horizontal slots, with the electric field polarization perpendicular to the
long side of the slots, for (a) $a=$540\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ and $b=$480\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, (b)
$a=b=$540\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and (c)
$a=$540\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$b=$600\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Bottom figures show the real
part of the associated dispersion diagrams calculated for the diagonal slot
($\alpha=45^{\circ}$), over a color map representing a cut of the three
dimensional complex space value of the MoM matrix determinant. In the
dispersion diagrams, dashed lines represent the labelled lightline spatial
harmonics. All the data in this figure was obtained for
$l=$220\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$w=$175\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

$\displaystyle\mathbf{J}^{\mathrm{as}}+\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\\!\overline{\mathbf{G}}_{M}(x-x^{\prime},y-y^{\prime})$
$\displaystyle\cdot\mathbf{E}_{t}^{\mathrm{sc}}(x^{\prime},y^{\prime},z=0)\
dx^{\prime}dy^{\prime}=\mathbf{0}\qquad(x,y)\in\mathrm{slots}$ (1)

where $\overline{\mathbf{G}}_{M}(x,y)$ is a dyadic Green’s function, relating
the in-plane components of the electric field on the surface of the slots and
the surface current density on the metallic screen, and
$\mathbf{J}^{\mathrm{as}}$ is the surface current density on the perfectly
conducting plane in the absence of the slots (dictated by
$\mathbf{E}_{\text{i}}$). It is well-known that for the case of a doubly-
infinitely periodic array, the integral equation can be reduced to a single
unit cell defining a periodic dyadic Green’s function,
$\overline{\mathbf{G}}_{M}^{\mathrm{per}}$ which groups the contributions from
all the arrays through an infinite summation, in which the fields on the array
are considered Floquet-periodic as imposed by the impinging wave [32]. The
periodic expression of (II) is then given by

$\displaystyle\mathbf{J}^{\mathrm{as}}+\int_{0}^{a}\int_{0}^{b}\\!\overline{\mathbf{G}}_{M}^{\mathrm{per}}(x-x^{\prime},y-y^{\prime})$
$\displaystyle\cdot\mathbf{E}_{t}^{\mathrm{sc}}(x^{\prime},y^{\prime},z=0)\
dx^{\prime}dy^{\prime}=\mathbf{0}\qquad(x,y)\in\delta_{00}$ (2)

where the integral is limited to the single unit cell $\delta_{00}$ by
defining a two-dimensionally periodic Green’s function given by

$\overline{\mathbf{G}}_{M}^{\mathrm{per}}(x,y)=\sum_{m,n=-\infty}^{+\infty}\overline{\mathbf{G}}_{M}(x-ma,y-nb)\
\mathrm{e}^{\mathrm{j}(k_{x0}ma+k_{y0}nb)}$ (3)

where $k_{x0}$ and $k_{y0}$ are the phasing wavevectors defined as in [27].

Both equations (II) and (II) can be solved by means of the Method of Moments
(MoM) [33], which was used in [28] to tackle the problem of periodic non-
rotated slots, and therefore will be just summarized here. In this method, the
unknown tangential electric field $\mathbf{E}_{t}^{\mathrm{sc}}$ is expanded
as a linear combination of proposed basis functions. When this linear
expansion is introduced into either (II) or (II), one can build a system of
linear equations for the weights of those basis functions, that can be readily
solved by numerical means. The key of this method is the use of basis
functions that can render the features of the fields in the slot aperture,
reproducing the singular behaviour of the fields near the edges for both
polarization. Previous studies show that only a very reduced set of them is
needed (5 at most) when these are chosen to be Chebyshev polynomials weighted
by the singular square-root singular behaviour of the field for each
polarization [34, 35, 36].

Once the system of equations has been inverted, the total electric field on
the surface of the array can be directly calculated together with its Fourier
transform, which can be used to calculate the far-field and the scattering
parameters [32].

As discussed in the introduction, the presence of EOT phenomenon is linked to
the existence of surface waves supported by the array. Mathematically this
link is clear, as the existence of surface waves can be studied by looking for
the zeroes of the determinant of the matrix of the system of linear equations
resulting from the application of the MoM, $\left|\Delta^{\text{per}}\right|$,
which is a function of the geometrical parameters, the frequency _and_ the
phasing imposed between adjacent unit cells. This means that when the
excitation is close in frequency, phasing and polarization to that of the
surface wave, resonant behaviour is found. When the position of these zeroes
is tracked for a range of frequencies and phasings, one can build the
dispersion diagram of the surface modes [37].

Using the analysis techniques and the concepts presented here, we have studied
the transmission through a two dimensional periodic array of slots for three
different periodicity configurations, i.e. $a>b$, $a=b$ and $a<b$, as shown in
Fig. 2(a-c), which contain both the transmission spectra at normal incidence
(top figures) and the dispersion diagram of the leaky surface waves supported
by the array along the $x$ direction for the case of 45∘ slot rotation. Fig. 2
has been obtained for a doubly infinite periodic array of slots (i.e., through
the solution of Eqn. (II) with a periodic dyadic Green’s function).

In Fig. 2(b), the well-studied EOT through a square slot array is presented
for benchmarking purposes, for which we present both transmission spectra for
three different rotation angles at normal incidence and the dispersion diagram
for the 45∘ slot rotation. Given the equality between the lattice vectors
associated with both periodicities, their EOT resonances appear at the same
frequency, and the system has a single EOT peak as the screen zero-thickness
inhibit the so-called odd mode that appears very close to the frequency of the
first Wood-Rayleigh’s anomaly [22]. In terms of the surface wave dispersion
shown in the bottom figure, one can see only one mode crosses the broadside
direction.

When the periodicity along the $y$ direction is smaller than that along the
$x$, the degeneracy of the EOT peak is broken, and we would expect to find two
peaks in the transmission spectrum in Fig. 2(a). However, when the slot is
aligned with either of the periodicities and the impinging electric field is
perpendicular to that, one see that the peak associated with the lattice
vector parallel to the slot disappears. In contrast, both EOT peaks are
present for any intermediate value. The fact that the existence of one of the
periodicities can be completely hidden through the use of symmetries is
remarkable, and its explanation can be found both in the mathematics and in
the physics of the problem. Mathematically, the EOT is linked with the pole-
type divergence of the Green’s function at the onset of a diffraction lobe.
However, the symmetry of the problem can lead to a pole-zero cancellation when
the scattered electric field points perpendicularly to the in-plane wavevector
of the diffraction mode that is generating such pole divergence, via a zero in
the diagonal term of the dyadic Green’s function [27]. From the point of view
of the surface wave modes responsible for the EOT phenomenon, this absence of
EOT peaks under certain illuminations can be explained in terms of
polarization mismatch between the impinging radiation and the modes supported
by the array. Note that freestanding connected conducting structures, such as
arrays of apertures, support transverse magnetic (TM) modes, while
disconnected conducting surfaces such as arrays of patches, support transverse
electric (TE) modes. Therefore, if one wants to excite the surface mode
propagating along the $x$ direction, one would need to be able to excite a
magnetic component along the $y$ direction, and vice versa. In the presence of
symmetry (i.e., slot parallel to one of the axes), the impinging polarization
across or along the slot axis will be preserved, and only one of the two modes
will be excited. Therefore, the EOT peak associated with the not-excited mode
will be suppressed. However, if one considers a slot that is not aligned with
the axes, the slot field distribution will couple to both surface waves,
therefore exciting both EOT peaks.

In Fig. 2(c) we present the case in which the EOT associated with the
periodicity along $y$ appears at a frequency lower than that associated with
the periodicity along $x$. As seen earlier for Fig. 2(a), when the slot is
aligned with either of the axes, the periodicity along the direction parallel
to the long sides of the slot will not be present in the transmission
spectrum, although its presence can be noticed due to the loss in transmission
of the zero-th order Floquet (see the solid blue line of the top Fig. 2(c)),
due to the interesting fact that although the EOT peak does not appear, its
associated Floquet mode can couple to the energy of the scattered fields once
it is above its cut-off frequency, while avoiding the singularity of its
onset. In terms of the dispersion diagram, in the bottom Fig. 2(c), one can
see that the two modes associated with the surface waves propagating along $x$
and $y$ are pushed together by their corresponding lightlines, giving place to
a mode coupling through a Morse critical point [38].

The fact that the position of the two resonant peaks can be dictated by the
periodicities chosen along the $x$ and $y$ direction allows for a large
freedom in the design of quasi-optical filters. The electrical size of the
aperture can be then used to engineer the width of the resonances, as done in
standard single-resonance EOT filters [32]. In the particular case of tilted
slots, the angle $\alpha$ can be used for the design of the transmission
levels of the two peaks, as it tunes the coupling between the scattered field
and the two surface wave families, each associated to one periodicity.

As one can see, the EOT through tilted slot arrays opens a wide variety of
situations through the coupling of the impinging radiation into a wider
variety of Floquet modes, and therefore of EOT resonances. As it was shown in
[39], for the cases of highly symmetrically positioned slots, the finite size
along the non-excited EOT resonance can be disregarded, however, this is not
the case when one considers tilted slots. In the following let us focus on the
truncation effects that have to be considered for any realistic implementation
of quasi-optical filters based on this multi-EOT phenomenon.

##  III Measurements and Discussion

In order to verify the theoretical results and physical insight provided in
the previous section, we performed a series of experiments at THz frequencies
using QO experimentation techniques as discussed in the Appendix. The samples
under test were laser micromachined (further details about the fabrication can
also be found in the Appendix) subwavelength hole arrays on 10$\mu$m-thick
aluminium foils. The geometrical dimensions of the fabricated samples were
inspected with a microscope; the in-plane lattice periods were
$\mathrm{\textit{a}=540\pm$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$ and
$\mathrm{\textit{b}=600\pm$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$}$, and the
slot dimensions were $l=226\pm$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$w=176\pm$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

Figure 3: On-axis transmission through freestanding arrays of diagonal
($\alpha=45^{\circ}$) slots and varying number of rows $N_{y}$ with lattice
periods $\mathrm{\textit{a}\approx$540\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}$ and $\mathrm{\textit{b}\approx$600\text{\,}\mathrm{\SIUnitSymbolMicro
m}$}$, and slot dimensions $l\approx$220\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ and $w\approx$175\text{\,}\mathrm{\SIUnitSymbolMicro m}$$: measurements
(a) and MoM results (b). Figure 4: From left to right, spectrogram (i.e. time-
frequency-maps) of the detected waveform (co-polar measurement) for samples
with $N_{y}$ = 5, 11, 21 and 61 when the incident electric field is polarized
perpendicularly to the long side of the slots.

The transmission spectrum at normal incidence with the electric field
polarization perpendicular to the long side of the slots, $l$, increases with
the number of rows $N_{y}$ (Fig. 3) for diagonal slots. Notice that
$N_{x}\geqslant 65$ across the whole wafer in all fabricated samples, and
thus, the truncation along $x$ direction was assumed to have no impact in the
electromagnetic response. The MoM approach described in [35] can be then used
for the very efficient analysis of the structures involved in the experiments
by reducing the problem to a single strip of slots which is repeated
periodically along a single direction. Under this approach, integral equation
in (II) becomes

$\displaystyle\mathbf{J}^{\mathrm{as}}(x,y)+\sum_{j=0}^{N_{y}-1}\int\\!\\!\\!\int_{\eta_{0j}}\\!\overline{\mathbf{G}}_{\mathrm{1D}}^{\mathrm{per}}(x-x^{\prime},y-y^{\prime})$
$\displaystyle\cdot\mathbf{E}_{t}^{\text{sc}}(x^{\prime},y^{\prime},z=0)\
dx^{\prime}dy^{\prime}=\mathbf{0}\quad(x,y)\in\eta_{0j}$ (4)
$\displaystyle\qquad\qquad\qquad(j=0,\dots,N_{y}-1),$

where $\eta_{0j}$ represents the surface of the j-th slot within the unit
strip and $\overline{\mathbf{G}}_{\mathrm{1D}}^{\mathrm{per}}(x,y)$ is the 1-D
periodic dyadic Green’s function given in terms of the free space Green’s
function $\overline{\mathbf{G}}_{M}(x,y)$

$\overline{\mathbf{G}}_{\mathrm{1D}}^{\mathrm{per}}(x,y)=\sum_{i=-\infty}^{+\infty}\overline{\mathbf{G}}_{M}(x-ia,y)\mathrm{e}^{\mathrm{j}ik_{x0}a}.$
(5)

To introduce the illumination effects, a 1-D Gaussian beam profile is used as
the impinging wave for the calculation of $\mathbf{J}^{\mathrm{as}}(x,y)$, as
in [28]. In the experiment, almost total transmission is obtained when
$N_{y}=61$, whereas the MoM predicts total transmission. The slight
quantitative disagreement between the results for this $N_{y}$ is arguably, to
a large extent, due to the absence of loss (i.e. ohmic and scattering losses
due to surface imperfections) in the MoM, and, to a lower extent, due to the
two-dimensional MoM calculation in which the incident Gaussian beam has a
beam-waist of 4 mm along $y$ and infinite along $x$. For such array with
$N_{y}=61$, both the direct transmission and the contribution of the surface
mode propagating along the $y$ direction are saturated. Indeed, saturation of
the direct transmission actually happens already for $N_{y}=21$ given the
beam-waist of the Gaussian beam.

The existence of two transmission channels involved in the EOT (i.e. direct
transmission and leaky-wave-mediated transmission) becomes evident in the
time-frequency analysis of the scattering phenomenon, which allows for the
time-dependent analysis of the radiated waves [40]. To this end, Fig. 4 shows
the spectrograms for the different truncated arrays of diagonal slots with
varying number of slots along the $y$ direction, $N_{y}$. Regardless of
$N_{y}$, we find that a significant amount of the energy arriving at the
detector does it around 28 ps across the whole spectral window. This is
associated with the contribution of the direct transmission, as the whole-
array standing wave has not had enough time to form [32]. Meanwhile, there is
a prolonged presence of energy arriving beyond 28 ps only at the frequency of
the EOT peak. The larger $N_{y}$ is, the longer this ringing of energy lasts.
This is due to the fact that more elements are involved in the resonance,
increasing its quality factor (and therefore the frequency selectivity as
shown in Fig. 3). This resonance is formed as the superposition of counter-
propagating leaky waves, which are excited at a precise frequency, the EOT
frequency, dictated by the leaky-wave complex dispersion relation. Such a slow
build-up of the resonance is possible due fact that the crossing of the leaky-
wave dispersion with the broadside direction occurs at a zero slope (and
therefore zero group velocity). As it was shown in [41, 28], when Gaussian
beams are considered, the transmission peak is slightly red-shifted, meaning
that the resonance is mediated by slowly-propagating waves, but not completely
static ones, contrarily to that found when using plane waves in infinite
periodic arrays.

Figure 5: Transmission spectrum as a function of the angle of incidence for TE
(rotation of the sample along the E-axis) and TM polarization (rotation of the
sample along the H-axis), with black circles marking the largest transmission
for each angle of incidence. The electric field is perpendicular to the long
side of the diagonal ($\alpha=45^{\circ}$) slots.

We have also studied the complete angular dependence of the system by varying
both angle of incidence and polarization, as these allow to control the
matching of the impinging wave to the two leaky waves responsible for the two
different EOT resonances of the tilted slot array.

Fig. 5 shows the angle of incidence dependence of the measured transmission
spectrum under both TE and TM wave illumination, whose planes of incidence are
defined by the $z$ direction and the vectors shown in the insets of the
figure. In contrast to what we find when exploring the dispersion along the
$x$ axis in Fig. 2, at both planes of incidence associated with TE and TM
incidence, the two lightlines associated with the (1,0) and (0,1) harmonics
show very similar dispersion relation, enforcing a similar angle-dependence on
the EOT peaks. Additionally, one can see the excitation of higher order
spatial harmonics, such as the (-1,0) and (0,-1), whose associated resonant
frequencies increase with the angle of incidence. Although for the sake of
brevity we only show here the position of the peaks, we have found that the
relative transmissivity of the two EOT peaks can be controlled by choosing the
wave to be either TE or TM, therefore promoting the excitation of one of the
two leaky waves while suppressing the other. In particular, by using TE
oblique incidence the lower frequency resonance is favored, while the TM
oblique incidence enhances the higher EOT resonance. This effect can be
explained by the increasing wavevector matching between the impinging and the
leaky wave, which can ultimately lead to huge resonant amplitudes as
theoretically shown for bound surface waves supported by semi-infinite arrays
[29].

Figure 6: Measured on-axis transmission through freestanding arrays of
diagonal slots for different rotation of the $N_{y}$ = 61 sample along the
$z$-axis as indicated by the inset in the figure.

We have found experimentally, however, that the excitation of the two leaky
waves can also be controlled by playing with the field configuration of the
scattered fields on the slot. In Fig. 6, we show the normal incidence
transmission coefficient for varying angles of electric field polarization. We
find that the higher frequency EOT, associated with the periodicity along the
$x$ direction can be almost completely suppressed when the impinging
excitation is polarized along the $y$ direction. This is due to the
modification of the field distribution, as in contrast with all the cases
studied before, now there is a non-negligible electric field component excited
along the long side of the slot, which allow for the coupling with a larger
number of modes.

##  IV Conclusion

In this paper we have provided an extensive study of the EOT phenomenon
through periodic and finite arrays of tilted slots, which allow for a very
distinguishable simultaneous coupling to two different EOT resonances, in
contrast to the highly symmetric arrays traditionally studied in the
literature. We have shown that the two resonances can be linked to the
existence of two different leaky waves, propagating along perpendicularly to
each other, as demonstrated by the calculation of their dispersion diagrams
and also by showing their matching (or absence of matching) to the scattered
fields under certain symmetries. By performing experiments at terahertz
frequencies we have both demonstrated the validity of our theory, but have
also shown the role that finite effects play in the quality factor of the
theoretically highly selective EOT resonance. Thanks to the time-domain
spectroscopy, we are able to show the consequences that the dispersion
characteristics of the leaky-waves involved have in the time response
associated with each frequency component of the scattered fields, namely its
slow resonance build-up. By exploring the angular dependence under two
differently polarized illuminations, we have shown the off-axis dispersion of
the leaky waves which is dominated by the linear lightline harmonic dispersion
and also the ability to suppress or enhance the EOT resonances via off-axial
excitation of the slots. The insight provided here sets the design guidelines
for advanced quasi-optical components based on extraordinary transmission
phenomenon.

## Appendix

Fabrication. The laser fabrication trials were performed on a state-of-art
laser micromachining system that integrates a femtosecond (fs) laser source
(Amplitude Systems Satsuma model) with a maximum average power of 5W, central
emission wavelength of 1030 nm, pulse duration of 310 fs and maximum pulse
repetition rate of 500 KHz. A stack of five mechanical axes (three linear and
two rotary stages) were used to move the samples with very high accuracy and
precision, i.e. the positioning resolutions of the linear and rotary stages as
stated by the manufacturer were 0.25 $\mu$m and 45 $\mu$rad, respectively
[42]. The laser system was also equipped with a high dynamics 3D optical scan
head for structuring with scanning speeds up to 2 m/s and thus to obtain
relatively high material ablation rates. A 100 mm focal length telecentric
lens is mounted at the exit of the beam path to achieve a focused beam spot
diameter of 30 $\mu$m. The achievable processing accuracy and repeatability
with the 3D scan head is better than $\pm 10\mu m$ for all scanning speeds
used in the experiments, while its processing volume is 30 mm (X) $\times$ 30
mm (Y) $\times$ 10 mm (Z) [43]. The specific laser parameters used to produce
the filters are detailed in Table 1 below.

TABLE I: Laser parameters for the fabrication of the finite size tilted slot arrays. Laser Parameter | Unit | Value
---|---|---
Power | W | 3.9
Frequency | kHz | 500
Pulse energy |  $\mu$J | 7.8
Scanning speed | m/s | 2
Pulse duration | fs | 310
Beam spot diameter |  $\mu$m | 30
Laser beam polarization | - | Circular
Hatch style | - | Random
Hatch pitch |  $\mu$m | 4
Ablation rate per layer |  $\mu$m | 1

Measurements. The samples were characterized with the all fibre-coupled THz
time-domain spectrometer TERA K15 from Menlo Systems in a QO configuration
without purging. A couple of TPX planoconvex lenses (effective focal length
$\approx$54 mm) were used to work with a collimated beam whose frequency-
dependent beam-waist at the sample position was estimated to be 4.1 mm and 4.0
mm along the E\- and H-plane at 0.5 THz, respectively. For the angle-resolved
measurements of Fig. 5 (with an angular step of 2.5∘ until 10∘ and 5∘ until
30∘) and Fig. 6, the $N_{y}$ = 61 sample was installed on a manual rotation
stage and on a high-precision rotation mount. The lock-in constant was set to
300 ms for the angle-resolved measurements of Fig. 5 and 100 ms for the rest,
whereas the total temporal length of the recorded waveforms was at least 260
ps to have a spectral resolution of 3.8 GHz in the worst case. Calibration was
done by comparing the measurements with the line-of-sight configuration (i.e.
emitter and detector on-axis) without the sample on the sample holder. To
generate the spectrograms (i.e. short-time Fourier transform) in Fig. 4, each
waveform is divided into segments of length 8.9 ps and an overlap of 98%,
which are windowed with a Hann window.

## Acknowledgements

The authors would like to thank Dr. J. Churm (University of Birmingham) for
measuring the unit cell dimensions of the fabricated truncated arrays.

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