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\chapter{Zigzag Metamaterials} \label{Chapter:ZZvar}

As summarised in the previous chapter, the concept of artificially engineered media (known as metamaterials) has revolutionised the field of optics, photonics and materials, extended the boundaries of microfabrication and stimulated the development of novel characterisation techniques. The demonstration of anomalous reflection and refraction of light by metasurfaces opened another exciting chapter in optical engineering \cite{yu2013flat}. 

One class of such metasurfaces, namely non-diffracting wavevector selective metasurfaces \cite{fedotov2015wavevector}, exhibit optical transmission spectra with a strong amplitude correlation with the angle of incidence. The zigzag design, belonging to this class, was explored in this study. Different variations of this design were simulated and fabricated to further explore its properties. These designs were optically characterised and showed differences when compared with the simulated spectral responses. To understand these differences the zigzag metasurfaces were characterised using a coherent light source in the near-IR.

\section{Design and fabrication of zigzag metasurfaces} \label{ZZdesign}

The first 2 metasurfaces that were investigated consisted of arrays of continuous zigzag nano-wires (ZZnW) and their inverse, continuous zigzag nano-slits (ZZnS). The unit cell of the metasurface was $650nm \times 500nm$ which rendered the zigzag metasurfaces immune to scattering into high-order diffraction modes in the near-IR. Each unit cell contained a single chevron of the zigzag structure. The chevrons arms were $125nm$ wide and at $45^{\circ}$ to the base of the unit cell and at $90^{\circ}$ to each other. A $32 \times 40$ array of this unit cell formed the full metasurface making an active area of $20.8{\mu}m \times 20{\mu}m$.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=12cm]{Figures/C3/ZZ slits sem full.png}
  \caption{SEM image of entire continuous zigzag slits metasurface. {\color{blue}{Scale bar is 5 $\mu$m.}}}
  \label{Figure:ZZSEMFull}
\end{figure}

A Helios Nanolab 600 DualBeam FIB/SEM was used to mill the metasurfaces into an $80nm$ thick film of amorphous gold sputtered onto a $0.5mm$ thick fused-quartz substrate. This machine incorporates a field-emission SEM with imaging resolution down to $1nm$ and a gallium ion gun with a milling resolution of $20nm$.

To further investigate the properties of the zigzag metasurfaces additional samples were fabricated with variations in the zigzag geometry. The first of these was to break the continuous structures to see how they are affected by blocking the propagation of plasmons along the zigzag wires and slits. To do this, gaps were placed at each apex in the zigzag geometry to form metasurfaces of isolated wires or slits. The gaps were $50nm$ wide for the wire samples and $100nm$ wide for the slit samples. This difference was required to ensure the gaps in the slit samples remained intact during the milling process. In addition to the fully isolated designs another variation was made with gaps placed at every other apex to form metasurfaces of separated chevrons. These samples are shown in Figure \ref{Figure:SEMZZVars}. The next change was to make the angle between the zigzag arms sharper. Samples were fabricated with the zigzag arms at $51^{\circ}$ and $59^{\circ}$ to the base of the unit cell with the width of the zigzag arms remaining constant. These had unit cells with dimensions of $600nm \times 600nm$ and $500nm \times 650nm$ respectively. The angle variations for the slits are shown later in Figure \ref{Figure:ZZAV_slits_SEM}. Continuous, isolated and chevron wires and slits were fabricated with each of these sharper angles.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/SEM ZZ paper pic.jpg}
  \caption{SEM images of fabricated planar metamaterials featuring (a) continuous zigzag nanowires, (b) continuous zigzag nanoslits, (c) broken zizag nanowires, and (d) broken zigzag nanoslits. Yellow boxes indicate elementary unit cells of the metamaterials. {\color{blue}{Scale bars are $1 \mu m$.}}}
  \label{Figure:SEMZZVars}
\end{figure}

\section{Optical characterisation}

{\color{blue}{*** Add details about CRAIC microphotospectrometer stitching artifact at 900nm  - paragraph added***}}

The transmission and reflection spectra of the metasurfaces were recorded using a commercial near-IR microspectrophotometer. It was developed by CRAIC Technologies based on a ZEISS Axio microscope and features a cooled near-IR CCD array with spectral resolution of $0.8nm$ and a Köhler illumination system with a broadband linear polariser. The latter incorporated a tungsten-halogen lamp and was tuned to a ${\times}15$ reflective objective with NA 0.28, producing broadband incoherent plane-wave illumination at near normal incidence. The spectra were collected through a set of square apertures of various sizes installed in the image plane of the microscope. The largest aperture that fit within the active sample area was used for each measurement, most commonly the 11µm square. The measured transmission spectra were normalised to the transmission of the bare substrate or with the sample removed where this was not possible. The reflection from a bare gold area of the samples was used to normalise the reflection spectra. A dark scan was also used to remove any stray incidental light from the measurements. For reflection this was done by replacing the sample with a piece of Vantablack which has near $0\%$ reflectivity and for transmission by blocking the illumination path.

{\color{blue}{Some of the spectra presented in the following figures include a minor technical artefact present at $900nm$ where the detectors change in the spectrometer from one sensitive to visible light, to a cooled detector for infrared radiation. This is a common issue in spectrometers with a large spectral range and can be reduced through careful calibration of the machine.}}

\section{Simulation method} \label{Sim}

The commercial simulation software COMSOL Multiphysics was used to model the zigzag metasurfaces. The model consisted of a single unit of the gold metasurface placed on top of fused quartz and with air above. The quartz and air extend $2{\mu}m$ away from the gold with refractive indices of $1.45$ \cite{malitson1965interspecimen} and $1$ respectively. The real and imaginary refractive indices from tabulated data \cite{palik1998handbook} were used to define the properties of the gold. The 4 boundaries surrounding the metasurface were defined as continuous boundary conditions, effectively making the model an infinite plane. The top and bottom of the model were defined as ports and were terminated using perfectly matched layers to eliminate any reflections. Different volumes in the metasurface plane were selected to be gold to model the inverted and isolated structures. An identical model with no gold present was used as a reference to normalise the results. Further details can be found in Appendix \ref{Chapter:COMSOL}.

\section{Transmission spectra of zigzag metasurfaces}

Figure \ref{Figure:SEMZZSlitVars} shows the details of the three designs of slits explored in this study. The inverse of these formed the wire samples discussed later.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=6cm]{Figures/C3/slits vars.png}
  \caption{SEM images of the 3 zigzag slit variations investigated. {\color{blue}{Each image contains 9 unit cells in a $3 \times 3$ array with a total size of $1.95 \mu m \times 1.5 \mu m$. }}}
  \label{Figure:SEMZZSlitVars}
\end{figure}

%In all the previous works on nanostructured metamaterials, where incoherent illumination with the above (or similar) methodology and instrumentation had been employed, a good agreement between the theory and experiment was reported. However, the metamaterials in those studies had one feature in common- their designs were based on piecewise rather than continuous patterns. Thus, the effect we observed must have been overlooked in the interpretation of past experiments. To verify this conclusion, we fabricated and subjected to the same testing procedure as above, a set of reference samples, which resembled ZZnW and ZZnS metasurfaces with broken zigzags. Their pattern was derived from the original by introducing a $50nm$, for wires, or $100nm$, for slits, wide split in every corner of the zigzags as seen in Figure \ref{Figure:SEMZZVars}.

\subsection{Spectra of zigzag slits}

The following figures show either the reflection or the transmission spectra of the three types of zigzag structures considered here: Continuous, Isolated and Chevron illuminated by incoherent light. The first set of figures presented the slits format as shown in Figure \ref{Figure:SEMZZSlitVars}. 

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_variants_slits_Pol=para.png}
  \caption{Transmission spectra from three zigzag variants. Polarisation parallel to slits. {\color{blue}{***Colours fixed.***}}}
  \label{Figure:ZZV_slits_para_TR}
\end{figure}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag slit variants - Reflection - Polarisation parallel to slits.png}
  \caption{Reflection spectra from the 3 zigzag variants. Polarisation parallel to slits.}
  \label{Figure:ZZV_slits_para_RE}
\end{figure}

As shown in Figure \ref{Figure:ZZV_slits_para_RE}, the continuous slits have a distinct double dip in reflection around $1100nm$. The chevron slits have a less pronounced double dip at a slightly lower wavelength. The isolated slits have a single dip in reflection at $950nm$. Figure \ref{Figure:ZZV_slits_para_TR} shows features in transmission that line up with the features observed in the reflection spectra. The continuous slits have a distinct double peak, while the chevron slits have a much less pronounced double peak. The isolated slits show a single peak in transmission.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_variants_slits_Pol=ortho.png}
  \caption{Transmission spectra through the three zigzag variants. Polarisation perpendicular to the slits. {\color{blue}{***Colours fixed***.}}}
  \label{Figure:ZZV_slits_ortho_TR}
\end{figure}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag slit variants - Reflection - Polarisation orthogonal to slits.png}
  \caption{Reflection spectra of three zigzag slit variants with incident polarisation perpendicular to slits.}
  \label{Figure:ZZV_slits_ortho_RE}
\end{figure}

Figure \ref{Figure:ZZV_slits_ortho_RE} spectra indicate that both the isolated and chevron slits have a strong dip in reflection at $1300nm$ and $1900nm$, respectively.

Figure \ref{Figure:ZZV_slits_ortho_TR} presented the spectra with the isolated and chevron slits that have peaks in the transmission spectra that line up with the dips in reflection on the previous slide. The peak for the isolated slits is much lower and shows absorption of around $40\%$ when compared with the dip in reflection. Chevron slits only show around $5\%$ absorption.

\subsection{Spectra of zigzag wires}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=6cm]{Figures/C3/wires vars.png}
  \caption{SEM images of the 3 zigzag wire variations investigated. {\color{blue}{Each image contains 9 unit cells in a $3 \times 3$ array with a total size of $1.95 \mu m \times 1.5 \mu m$. }}}
  \label{Figure:SEMZZwireVars}
\end{figure}

The next figures show the spectra of the inverse metasurface consisting of nano-wires. These can be considered as the positive complement to the negative designs shown previously which consisted of nano-slits. 3 variations on the design were again investigated. These are shown in Figure \ref{Figure:SEMZZwireVars} and were continuous, isolated and chevron zigzag nano-wires.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_variants_wires_Pol=ortho.png}
  \caption{Transmission spectra from the zigzag wire variants. Polarisation orthogonal to the wires. {\color{blue}{***Colours fixed***.}}}
  \label{Figure:ZZV_wires_ortho_TR}
\end{figure}

Figure \ref{Figure:ZZV_wires_ortho_RE} shows the reflection spectra with the polarisation perpendicular to the nano-wires. The continuous wires show a double peak in transmission around $1150nm$. The chevron wires only have a single peak here. The isolated wires have a single peak at a slightly lower wavelength.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag wire variants - Reflection - Polarisation orthogonal to wires.png}
  \caption{Reflection spectra from the three wire variants. Polarisation orthogonal to the wires.}
  \label{Figure:ZZV_wires_ortho_RE}
\end{figure}

Figure \ref{Figure:ZZV_wires_ortho_TR} shows that the continuous wires have a double dip in the transmission while the chevron and isolated wires only have a single dip. These features correspond to the reflection features seen on the previous slide.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_variants_wires_Pol=para.png}
  \caption{Transmission spectra from the zigzag wire variants. Polarisation parallel to the wires. {\color{blue}{***Colours fixed.***}}}
  \label{Figure:ZZV_wires_para_TR}
\end{figure}

Figure \ref{Figure:ZZV_wires_para_RE} shows that the isolated wires have a peak in reflection at $1350nm$. The chevron wires have a peak around $1900nm$. The continuous wires have a peak at $700nm$ and show a rise in reflection towards the end of the wavelength.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag wire variants - Reflection - Polarisation parallel to wires.png}
  \caption{Reflection spectra from the zigzag wire variants. Polarisation parallel to the wires.}
  \label{Figure:ZZV_wires_para_RE}
\end{figure}

Figure \ref{Figure:ZZV_wires_para_TR} shows that the isolated wires have a dip in transmission at $1350nm$ which shows around $30\%$ absorption when compared with the reflection spectra. The chevron wires have a dip at $1900nm$ with an absorption of about $20\%$. The continuous wires show the same split resonance feature that is seen with the continuous slits sample at $1 \mu m$. While this is the same resonance feature it is much broader and the split is not as deep for the wires as their geometry has not been optimised for this polarisation.

\section{Discussion}

The figures presented in the previous section demonstrate the emergence of the split resonance for the continuous wires and slits, and for the chevron slits. The relationship between the reflection and transmission spectra for each polarisation can be seen and the effect of absorption is also observable. The transmission and reflection spectra of the continuous, isolated and chevron metasurfaces for both polarisation orientations form a comprehensive dataset for the exploration of these effects.

The discussion of results starts with a comparison between the experimental results and simulated spectra.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13.5cm]{Figures/C3/zigzag_modelling_variations.png}
  \caption{ Transmission spectra for the continuous and isolated slits and wires, as indicated in the insets. Measured spectra are shown as solid lines and the simulated spectra as dashed lined. Note the colours: black, red, green and blue indicate the four designs indicated in the insets.}
  \label{Figure:ZZSim}
\end{figure}

\subsection{Comparison between modelling and experiments} \label{VFsim}

The results of the COMSOL simulations for each metasurface design are shown in the Figure \ref{Figure:ZZSim} alongside their corresponding experimental results. The method used was described previously in Section \ref{Sim} and further details can be found in Appendix \ref{Chapter:COMSOL}. There is a large disparity between the experimentally obtained spectra and those from the modelling. The experimental result for transmission through the zigzag slits, plotted as a solid black line, shows a double peak when the incident polarisation is parallel to the slits of the surface. This is not seen in the modelled spectra, plotted with a dashed line, which shows a single resonance peak. An analogous effect is observed for transmission through the zigzag wires when the polarisation is orthogonal to the wires, shown again in Figure \ref{Figure:ZZSim}. Here a double dip is observed experimentally where a single dip is found when the structure is modelled.

The double dip feature seen for the zigzag wires corresponds to the double peak feature for the zigzag slits. The feature for the zigzag wires is red shifted by $100nm$ from that of the slits. This is due to the two metasurfaces not being the exact inverse of each other. This can be seen in the SEM images in Figure \ref{Figure:ZZSim} where the slits and wires milled in the gold are roughly $150nm$ and $100nm$ wide, respectively. This is due to imperfections from the fabrication techniques.

The simulations for the chevron structures show a single peak or dip for the slits and wires respectively. The experimental results for the chevron wires agree with this while the chevron slits show a less pronounced double peak. For the isolated slits and wires the experimentally obtained spectra agree with the results from the simulations, both showing a single peak or dip. These result shows that the effect is inherent to the larger superstructure of the surface and not the individual unit cells.

{\color{blue}{***Clarify role in simulations***}}

{\color{blue}{
The results of my simulations, presented in Figure \ref{Figure:ZZSim}, agree with the experimental spectra observed from the isolated wires and slits but not for the continuous wires and slits where the distinctive split resonance is missing. One apparent difference between my simulations and our experimental set up is the coherence of the incident light. In the microphotospectrometer, the light originates from an incandescent white light source, meaning the light is incoherent. In contrast, the light in my simulations is in the form of a plane wave and is therefore spatially coherent.
}}

To investigate this difference further a larger and more complex COMSOL model was constructed by Dr Vassili Fedotov to simulate the effect of incoherent illumination. Instead of using periodic boundaries, a complete, albeit smaller, version of the continuous zigzag slits was created. It contained $10 \times 12$ unit cells, measuring $6.6 \times 6.2 {\mu}m$. All the simulation’s boundaries were terminated using scattering boundary conditions. Light was input through one of the faces parallel to the surface and the power was captured at the opposite face. Unlike before the input face was split into 9 different regions which each produced an individual light beamlet. Each of these beamlets had a cross-section measuring $2.4 \times 2.4 {\mu}m$. This size corresponds to the spatial coherence length of incandescent white light at a wavelength of $1.1{\mu}m$. The response to incoherent light was simulated by running the model 9 times, each time engaging only one of the beamlets, and then calculating the sum of the 9 transmission spectra. By running the model with all the beamlets engaged, coherent illumination could be produced.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=10cm]{Figures/C3/cohVSin_sim.PNG}
  \caption{Modelled spectral response of the continuous zigzag slits when illuminated using incoherent and coherent light. Insert shows the polarisation orientation relative to the slits.}
  \label{Figure:CohSIM}
\end{figure}

{\color{blue}{The results of Dr Fedotov's}} simulations are shown in Figure \ref{Figure:CohSIM}. As can be seen, for incoherent light, the model reproduced the signature double peak feature that we observed experimentally. When the model was set to produce coherent light the same response as was seen in our previous simulations was observed.

\subsection{Spectral response of the zigzag slits}

The simulated spectral response of the metasurfaces for all three types of designs, continuous, isolated as well as chevron, contained a single resonance spanning from $0.9{\mu}m$ to $1.5{\mu}m$ and centred at $1.1{\mu}m$. This resonance corresponds to the excitation of the fundamental $\lambda/2$- current mode – a standing wave of charge (plasma) oscillations, which builds up in every straight segment of the zigzag pattern once the half of its period can fit the length of the segment.

In the wavelength range $0.9 - 1.5{\mu}m$  the spectral feature, shown in blue curve in Figures \ref{Figure:ZZV_slits_para_TR} and \ref{Figure:ZZV_wires_ortho_RE}, for example for the continuous slits design, includes a pair of resonances separated by a gap of about $0.2{\mu}m$. This feature is also visible, but less pronounced for the case of the chevron slits, shown in red curve, but is absent for the case of isolated slits (green curve) Intriguingly, the overall profile of the spectral response in each case resembles closely that of Fano resonances normally exhibited by metasurfaces with substantially more complex patterning. More intriguingly, the results of our measurements disagree – even at the qualitative level – with the predictions of rigorous numerical modelling that informed the actual design of the metasurfaces.

Given the complementarity of the two metamaterials, the profiles of the respective split resonances were also complementary (as dictated by Babinet’s principle): peaks in one spectrum corresponded to dips in the other, and vice versa. Intriguingly, the results of our measurements disagree, even at the qualitative level, with the predictions of rigorous numerical modelling that informed the design of our metamaterials (dotted curves in Figure \ref{Figure:ZZSim}). Indeed, the modelled response in each case displayed a whole resonance centred at $1.1{\mu}m$ \cite{munk2005frequency}. It resulted from the excitation of the fundamental $\lambda/2$-current mode, the most common localised (dipolar) resonant mode that had been particularly favoured by microwave and RF antennas and planar metamaterials of various designs \cite{munk2005frequency,vardaxoglou1997frequency}. In the zigzag metasurfaces the mode corresponded to a standing wave of charge oscillations, which built up locally in every straight segment of the zigzag pattern, once the half of the wave’s period fitted the length of the segment (see insets to Figure \ref{Figure:ZZSim}).  That rendered each segment as an independent half-wavelength resonator, which determined the response of the planar metamaterials. The noted strong (and rather unexpected) discrepancy called for careful examination and verification of the methodology used.

{\color{blue}{*** Correct reference now replaces the Benz paper in the text above, namely Munk 2005 ***}}

Importantly, since the resonant mode supported by the continuous pattern had its nodes located in the corners of the zigzags, the described modification did not affect the nature of the resonant response in the resulting nanostructures. The localised distributions of charge density sustained by the reference metamaterials at their resonance were very similar to those calculated for the ZZnW and ZZnS metasurfaces. Also, as in the case of continuous zigzags, the transmission spectra predicted for piecewise zigzags featured a whole resonance spanning from $0.9{\mu}m$ to $1.5{\mu}m$, though its centre appeared blueshifted by about $50nm$ due to physical shortening of the broken segments. In spite of the apparent similarities between the two cases, it was possible to reproduce experimentally the main features of the response predicted numerically for the reference samples.  

Some quantitative mismatch resulted from an uncertainty in specifying the dielectric function of gold and the difficulty of reproducing fine features of piecewise zigzags during the fabrication. The apparent discrepancy between the calculated and measured spectra in Figure \ref{Figure:ZZSim} below $0.9{\mu}m$ signifies the appearance of the first-order diffraction modes in the substrate due to the $660nm$ period of the metasurfaces (which acted as conventional diffraction gratings in that wavelength range for light propagating in the substrate). The discrepancy results from the fact that those diffraction modes existed only in the substrate, which for practical reasons was modelled as a semi-infinite slab while in the experiment it had a finite thickness.

\subsection{Origin of transmission suppression by ZZnS metasurface under incoherent illumination} \label{planewave}

The analysis shown above indicates that the discrepancy between the theory and experiment we observed earlier with the ZZnW and ZZnS metasurfaces corresponded to a genuine effect, previously unseen in metamaterials and somewhat exclusive to the continuous pattern. This discrepancy could only have resulted from the difference between the simulated and actual illumination conditions, which thus far had been routinely disregarded in the study of metamaterials: the common modelling approach assumed the complete coherence of incident light, while the spectroscopic measurements involved spatially incoherent light.

The nature of the discovered effect and, more specifically, the origin of the split resonance can be deduced from the distribution of electric field near the ZZnS Metasurface at the split’s centre wavelength. Light wave polarised in the plane of the cross-section (||-polarization), which propagates upward as a beamlet until scattered by the nanostructure. While the incident wave is confined laterally to the area of the patch, the transmitted and reflected waves can be pictured as spreading along the plane of the metamaterial far beyond these confines. The resulting field configuration indicates that the mechanism of light scattering by the ZZnS metasurface is extremely nonlocal. Since planar metamaterials do not diffract as a whole (i.e. do not support high-order diffraction modes) this mechanism must involve plasmon waves, which due to the continuity of the pattern can leak from a locally excited unit cell and propagate up and down the zigzags. In the ZZnS metasurface these waves are guided in the form of a mode confined to the nanoslits. It transports the excitation via the zigzag channels to other parts of the metamaterial, where it is radiated and interferes with the fields scattered there locally. 

A strong local response plays a crucial role here not only by selecting the actual mode that mediates nonlocal scattering but also by rendering it nondispersive. As explained earlier, at the resonance of the zigzag metasurfaces the period of the plasmon waves has to fit inside one unit cell exactly.

Since in the near-IR the dispersion of surface plasmons in gold is close to linear, their wavelength can be estimated as $\lambda_g \approx \lambda\sqrt{(1+\epsilon_s)/2}$, where $\epsilon_s$ is the dielectric constant of the substrate. Correspondingly, at the resonance $\lambda_g/2 = 0.44{\mu}m$ matches almost exactly the length of zigzag segments, $0.45{\mu}m$.

This ensures that only $\lambda/2$-mode is sustained by the zigzag channels and all the unit cells engaged in nonlocal scattering will be driven to the resonance radiating most strongly. More importantly, this guarantees that the radiated fields are all in phase and, hence, add up to form planar wavefronts that stretch wide over the metamaterial, significantly increasing the spatial coherence of scattered light. Nothing of this kind happens outside the resonance.

The total transmitted field then arises as a superposition of the planar wavefronts propagating in the forward direction, which are sourced by different unit cells exposed to light. In the case of incoherent illumination, the exposed unit cells are excited with random phases and, hence, the planar wavefronts they produce do not interfere, preventing the transmission of a ZZnS metasurface from reaching maximum. Under coherent illumination all unit cells oscillate in sync, which ensures constructive interference of the scattered planar wavefronts and, as a result, maximal transmission.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/zz wavevectors.png}
  \caption{Panels (a) and (b) show schematically how the field transmitted by a ZZnS metasurface is formed under incoherent (a) and coherent (b) illumination. $\varphi_i$ – is the phase of the excitation of $i^{th}$ unit cell. $E_i^{sca}$  is the amplitude of the planar wavefront that locally excited $i^{th}$ unit cell produces via non-local scattering.}
  \label{Figure:ZZvectors}
\end{figure}

The effect could also be summarised as follows. The field transmitted by a ZZnS metasurface arises as a superposition of the planar wavefronts propagating in the forward direction, which are sourced by different unit cells exposed to light. In the case of incoherent illumination, the exposed unit cells are excited with random phases and, hence, the planar wavefronts they produce do not interfere, preventing the transmission of the metamaterial from reaching its maximum (as illustrated in Figure \ref{Figure:ZZvectors}a). This is evident from the expression for transmitted light intensity written down below in terms of the planar waves scattered non-locally as shown in Equation \ref{Itr}:

\begin{equation} \label{Itr}
    I^{tr} = \frac{c\epsilon_0}{2}\left(\sum_{i=1}^N |E_i^{sca}|^2 + 2Re\sum_{i=1}^N\sum_{ \genfrac{}{}{0pt}{}{j=1}{j \neq i}}^N E_i^{sca}E_j^{sca^*}\right)
\end{equation}

where N is the number of unit cells, which are engaged in non-local scattering by each locally excited unit cell, and $E_i^{sca}$ is the complex amplitude of the planar wave sourced by $i^{th}$ unit cell through non-local scattering. Given that the phases of $E_i^{sca}$ are random, all cross-terms in Equation \ref{Itr} cancel out. Correspondingly, under coherent illumination all unit cells oscillate in sync with the exact same phase, which ensures that the cross-terms in Equation \ref{Itr} all add up (constructive interference of the scattered planar wavefronts) and, as a result, maximum transmission is achieved (see Figure \ref{Figure:ZZvectors}b).

{\color{blue}{*** Add figure from simulation explanation plane wave emission from the zigzag metasurface - {\textit{Added a new figure and a short explanation ***}}}} 

{\color{blue}{
Figure \ref{Figure:Plane_zigzag} shows a cross-section of the simulated ZZnS metasurface discussed earlier, in Section \ref{VFsim}, while it is excited from the center only with light polarised parallel to the zigzag slits at a wavelength of $1.12 \mu m$. The plane of the cross-section is parallel to the slits with the blue to red colours representing the real part of the electric field. The direction of incidence is indicated by the white arrow with the white dashed lines indicating the lateral extent of the incident wave. 

}}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=14cm]{Figures/C3/Plane_zig-zag.jpg}
  \caption{{\color{blue}{Simulation of the zigzag metasurfaces response to plane wave, incoherent illumination at $1.12 \mu m$ (from \cite{ZZcoh}).}}}
  \label{Figure:Plane_zigzag}
\end{figure}



\subsection{Zigzag nanowires under incoherent illumination} \label{ZZnW}

The results obtained for the wires version of metasurfaces agree and are consistent with the spectra measured for the slits version. However, the evidence of the split resonance is less pronounced, indicating perhaps stronger absorption. Furthermore, while in Figures \ref{Figure:ZZV_slits_para_TR} and \ref{Figure:ZZV_slits_para_RE}, there was some evidence of the split for the chevron slits, in the corresponding case of wires, Figures \ref{Figure:ZZV_wires_ortho_TR} and \ref{Figure:ZZV_wires_ortho_RE}, the split shows a smaller contrast and, furthermore, it is absent for the case of chevron wires.

In this case of a ZZnW metasurface the scattering process at the resonance is very similar to that revealed by the case of slits. The only differences are that nonlocal scattering is mediated by the plasmonic mode of the nanowires, while the field scattered forward propagates alongside the field of the incident wave.

Correspondingly, nonlocal scattering controls reflection of the ZZnW metasurface in the exact same way as it controls transmission of the complementary ZZnS design (in agreement with Babinet’s principle): it reduces the intensity of light reflected by the ZZnW metasurface under incoherent illumination. This translates into an enhancement of transmission, as it was observed in the experiment.

\section{The effect of coherent illumination on the transmission spectra of the zigzag metasurfaces}

{\color{blue}{
The zigzag metasurfaces have been modelled under both coherent and incoherent illumination but the spectra have so far only been recorded experimentally under incoherent illumination. The final piece missing was to measure the spectra under coherent illumination. To achieve this, 2 new metasurfaces were fabricated on a gold coated fused-quartz substrate. These consisted of a continuous ZZnS metasurface and a continuous ZZnW metasurface using the same design as discussed earlier, in Section \ref{ZZdesign}. Both metasurfaces were $20 \times 20 \mu m$ and were spaced further apart than usual to ensure that they could be illuminated individually. Additionally, a window matching the dimensions of the metasurfaces was milled into the gold to be used as a reference.
}}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=14cm]{Figures/C3/ZZCoh results.png}
  \caption{Transmission spectra of (a) ZZnW metasurface and (b) ZZnS metasurfaces measured under incoherent (solid lines) and coherent (crosses) illumination at normal incidence.}
  \label{Figure:CohResults}
\end{figure}

The experiment was carried out in Professor Pavlos Lagoudakis lab and was led by Dr Tamsin Cookson. A quasi-CW optical parametric oscillator (Chameleon Compact OPO) by Coherent with a tuning range of $1$ - $1.35{\mu}m$ was used. The transmission response of the metasurfaces to coherent light was measured using a combination of the wavelength-tuneable laser and a broadband power meter. The Linearly polarised OPO output was focused on the samples to a spot with a diameter of $50{\mu}m$ by an achromatic lens with an $100mm$ focal length. Another $100mm$ lens was used to collect the transmitted light and direct it towards the power meter. The spectra were acquired with a step of $25nm$ and normalised to the transmission of the fused-quartz window. {\color{blue}{The recorded spectra, shown in Figure \ref{Figure:CohResults}, are missing the split resonance observed with incoherent illumination and feature a single resonance centred around $1.1{\mu}m$.}}

%\begin{figure}[!htb]
%  \centering
%  \includegraphics[width=12cm]{Figures/C3/Coh vc incoh art.png}
%  \caption{Sketch of how the metasurface interacts with coherent and incoherent light.}
%  \label{Figure:CohArt}
%\end{figure}

\section{The effect of a sharper angle between the zigzag arms}

The angle between the zigzag arms was varied to explore the effect on the split resonance discussed earlier. The original design, with arms at $45^{\circ}$ to the horizontal, was fabricated alongside 2 new designs with their arms at $51^{\circ}$ and $59^{\circ}$ to the horizontal. Figure \ref{Figure:ZZAV_slits_SEM} shows SEM images of the different zigzag slit structures used.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=10cm]{Figures/C3/ZZangleSEM.png}
  \caption{SEM images of the zigzag slit metasurfaces used. {\color{blue}{Images are not to the same scale.}}}
  \label{Figure:ZZAV_slits_SEM}
\end{figure}

The following figures capture the main experimental results collected, with each angle variation shown in a different colour for the continuous, isolated and chevron designs. The design of these structures was discussed earlier in Section \ref{ZZdesign}. As in the previous experiment, the incident light polarisation was set to be either perpendicular or parallel to the zigzag rows.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_angle_variation_norm_slits_Pol=para.png}
  \caption{Transmission spectra of the three continuous zigzag slit metasurfaces with polarisation parallel to the zigzags.}
  \label{Figure:ZZAV_norm_slits_para}
\end{figure}

The results for the continuous zigzag slit metasurfaces are shown in Figure \ref{Figure:ZZAV_norm_slits_para}. As the angle of the zigzags was made tighter the split in the resonance is seen to reduce. The wavelength of the split is shifted to a higher wavelength for the $51 ^\circ$ design.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_angle_variation_chev_slits_Pol=para.png}
  \caption{Transmission spectra of the three chevron zigzag slit metasurfaces with polarisation parallel to the zigzags.}
  \label{Figure:ZZAV_chev_slits_para}
\end{figure}

Figure \ref{Figure:ZZAV_chev_slits_para} shows the spectra from the chevron slit zigzag designs. The maximum transmission at the resonance increases and the split appears to move away from the central resonance to lower wavelengths for the larger angles.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_angle_variation_iso_slits_Pol=para.png}
  \caption{Transmission spectra of the three isolated zigzag slit metasurfaces with polarisation parallel to the zigzags.}
  \label{Figure:ZZAV_iso_slits_para}
\end{figure}

All three of the isolated slit metasurfaces showed a well defined single resonance peak in their spectra. The wavelength of the resonance moves to higher wavelengths for the 2 larger angles investigated.

%\begin{figure}[!htb]
%  \centering
%  \includegraphics[width=10cm]{Figures/C3/Zigzag_angle_variation_norm_slits_Pol=ortho.png}
%  \caption{Transmission spectra of the three continuous zigzag slit metasurfaces with polarisation perpendicular to the zigzags.}
%  \label{Figure:ZZAV_norm_slits_ortho}
%\end{figure}

%\begin{figure}[!htb]
%  \centering
%  \includegraphics[width=10cm]{Figures/C3/Zigzag_angle_variation_chev_slits_Pol=ortho.png}
%  \caption{Transmission spectra of the three chevron zigzag slit metasurfaces with polarisation perpendicular to the zigzags.}
%  \label{Figure:ZZAV_chev_slits_ortho}
%\end{figure}

%\begin{figure}[!htb]
%  \centering
%  \includegraphics[width=10cm]{Figures/C3/Zigzag_angle_variation_iso_slits_Pol=ortho.png}
%  \caption{Transmission spectra of the three isolated zigzag slit metasurfaces with polarisation perpendicular to the zigzags.}
%  \label{Figure:ZZAV_iso_slits_ortho}
%\end{figure}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_angle_variation_norm_wires_Pol=ortho.png}
  \caption{Transmission spectra of the three continuous zigzag wire metasurfaces with polarisation perpendicular to the zigzags.}
  \label{Figure:ZZAV_norm_wires_ortho}
\end{figure}

The next figures explore the zigzag wire metasurface designs. Figure \ref{Figure:ZZAV_norm_wires_ortho} shows the results for the continuous zigzag wire metasurfaces. The double dip feature for $45 ^\circ$ which can be seen to be reduced as the angle is increased with an additional dip appearing at a slightly lower wavelength.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_angle_variation_norm_wires_Pol=para.png}
  \caption{Transmission spectra of the three continuous zigzag wire metasurfaces with polarisation parallel to the zigzags.}
  \label{Figure:ZZAV_norm_wires_para}
\end{figure}

Looking at the opposite polarisation orientation for the continuous zigzag wire designs the same split resonance feature observed for the continuous zigzag slit metasurfaces is seen. It is broader and less defined for the wires designs as they were not optimised for this polarisation. As the angle of the zigzags is increased the split in the resonance becomes narrower for $51 ^\circ$ and then almost disappears for the $59 ^\circ$ sample.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_angle_variation_chev_wires_Pol=ortho.png}
  \caption{Transmission spectra of the three chevron zigzag wire metasurfaces with polarisation perpendicular to the zigzags.}
  \label{Figure:ZZAV_chev_wires_ortho}
\end{figure}

For the chevron wire structures a single dip is observed for the $45 ^\circ$ design with an additional dip appearing at lower wavelengths for the larger angles. This is shown in Figure \ref{Figure:ZZAV_chev_wires_ortho}.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag_angle_variation_iso_wires_Pol=ortho.png}
  \caption{Transmission spectra of the three isolated zigzag wire metasurfaces with polarisation perpendicular to the zigzags.}
  \label{Figure:ZZAV_iso_wires_ortho}
\end{figure}

Figure \ref{Figure:ZZAV_iso_wires_ortho} shows the spectra for the isolated wire designs. For $45 ^\circ$ and $51 ^\circ$ a single dip can be seen. For $59 ^\circ$ a second dip appears for lower wavelengths.

%\begin{figure}[!htb]
%  \centering
%  \includegraphics[width=10cm]{Figures/C3/Zigzag_angle_variation_chev_wires_Pol=para.png}
%  \caption{Transmission spectra of the three chevron zigzag wire metasurfaces with polarisation parallel to the zigzags.}
%  \label{Figure:ZZAV_chev_wires_para}
%\end{figure}

%\begin{figure}[!htb]
%  \centering
%  \includegraphics[width=10cm]{Figures/C3/Zigzag_angle_variation_iso_wires_Pol=para.png}
%  \caption{Transmission spectra of the three isolated zigzag wire metasurfaces with polarisation parallel to the zigzags.}
%  \label{Figure:ZZAV_iso_wires_para}
%\end{figure}

{\color{blue}{*** Explanation of reduction in effect - Paragraph added ***

The reduction of the split resonance feature, as the angle is increased, can be explained by increased losses stemming from the sharper corners of the zigzag geometry and by the larger path length the plasmons must take to travel along the zigzags. These increased losses mean that the plasmons originating from different points along the zigzags have less chances to interfere with each other leading to a reduction of the split resonance observed. 
}}

{\color{blue}{*** Explain frequency difference for 51deg continuous ZZnS - Paragraph added ***

The centre wavelength of the resonance is determined by the size and the geometry of the unit cell, as discussed earlier. In general, each unit acts as a resonator thus the resonance can be controlled by altering its design, in particular the length of the arms. In this part of the experimental work, different angles between the arms were considered, but in doing so the length of the arms did not remain constant. Hence, multiple aspects of the designs were altered, making more precise conclusions about the effect of the varying angle more challenging. An extension of this project would need to consider these aspects separately.
}}

Concluding this part of the project, the split resonance feature was observed in the new designs with different angles. As the angle was increased it was seen that the split in the resonance decreased in contrast. However, exploring these different designs and angles, with the aim of determining the optimum design did not lead to a stronger effect, thus the original design proved to be the most successful.

\section{Varying the angle of incident illumination}

To explore further this effect, a separate study was carried to explore the dependence of the spectral response on the experimental conditions, namely the tilt of the samples at different degrees and planes with respect to the incident light. The aim was to investigate how the split resonance in the optical transmission spectra varied with the angle of incidence.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=8cm]{Figures/C3/tilt_diagram.jpg}
  \caption{Rotation axis with zigzags down the slope, (a), and along slope ,(b).}
  \label{Figure:tilt_def}
\end{figure}

The following figures present the impact of tilting the sample on the spectra. Figures \ref{Figure:tilt_slits_down_slope} and \ref{Figure:tilt_wires_down_slope} considers the case of slits/wires being tilted so that the zigzags are running “down the slope”. Figures \ref{Figure:tilt_slits_along_slope} and \ref{Figure:tilt_wires_along_slope} show the case where the zigzags are running "along the slope". The axis of these rotations are defined in Figure \ref{Figure:tilt_def}. 

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag slits parallel polarisation - Small angle variation.png}
  \caption{Spectra of continuous zigzag slit metasurface as inclination angle is increased with the zigzags going down the slope. Polarisation parallel to the slits.}
  \label{Figure:tilt_slits_down_slope}
\end{figure}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag wires perpendicular polarisation - Small angle variation.png}
  \caption{Spectra of continuous zigzag wire metasurface as inclination angle is increased with the zigzags running along the slope. Polarisation perpendicular to the wires.}
  \label{Figure:tilt_wires_down_slope}
\end{figure}

For both the nanowire and nanoslit samples, shown in Figures \ref{Figure:tilt_slits_down_slope} and \ref{Figure:tilt_wires_down_slope}, the split in the resonance becomes broader and deeper as the incidence angle is increased. The central wavelength of the split becomes slightly red-shifted at the larger angles.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag slits parallel polarisation - Slits along the slope.png}
  \caption{Spectra of continuous zigzag slit metasurface as inclination angle is increased with the zigzags running along the slope. Polarisation parallel to the slits.}
  \label{Figure:tilt_slits_along_slope}
\end{figure}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=13cm]{Figures/C3/Zigzag wires perpendicular polarisation - Wires along the slope.png}
  \caption{Spectra of continuous zigzag wire metasurface as inclination angle is increased with the zigzags running along the slope. Polarisation perpendicular to the wires.}
  \label{Figure:tilt_wires_along_slope}
\end{figure}

Figures \ref{Figure:tilt_slits_along_slope} and \ref{Figure:tilt_wires_along_slope} show the spectra as the sample was inclined on the opposite axis. In this case, the dependence on this tilt angle is much less pronounced. The slits, shown in Figure \ref{Figure:tilt_slits_along_slope}, show an overall decrease in transmission around the split resonance with the lower wavelength half having the largest change. In Figure \ref{Figure:tilt_wires_along_slope} we can see that the nanowires metasurface shows very little change when the tilt angle was increased with the largest decrease in transmission being below $900nm$ away from the split resonance.

{\color{blue}{***Explain increased splitting as sample tilted***

The enlarging of the split in the resonance observed for the zigzag slit samples can be explained by the effect discussed previously, in Section \ref{planewave} and shown in Figure \ref{Figure:Plane_zigzag}, where the metasurface scatters light of the resonant frequency in the form of a plane wave. This scattered plane wave will propagate perpendicular to the plane of the metasurface and therefore, as the angle of incidence is increased, more light of the resonant frequency will be redirected from the beam path through the spectrometer leading to a reduction in the observed transmission. As discussed in Section \ref{ZZnW}, the ZZnW metasurface acts as the inverse to the ZZnS which explains the enhancement of the transmission observed as the angle of incidence is increased.

Changing the angle of incidence of the light illuminating the sample was another experimental parameter that was explored to further investigate the observed split in the resonance. This part of the experimental work was intended as a proof of principle investigation to provide an additional control on the observed split, in particular the mechanism of the directional selectivity and its effect on the spectra. A more in-depth understanding of the trend observed requires follow-on modelling that would constitute future work and the extension of this project.
}}

\newpage

\section{Conclusions}

In this chapter, planar metamaterials were shown as a powerful paradigm of optical engineering, which enables one to control the flow of light across structured material interfaces in the absence of high-order diffraction modes. Here, planar metamaterials, formed by nanopatterned metal films, respond differently to incoherent light, depending on the design of the zigzag pattern. 

Experimentally verified, and confirmed via rigorous numerical modelling that the optical response of metallic planar metamaterials (homogeneous metasurfaces that do not diffract as a whole) may vary dramatically with the spatial coherence of incident light. 
%This visualisation of this effect is shown in Figure \ref{Figure:CohArt} for the case of nondispersive delocalised plasmons supported by continuous metallic fabric of the metamaterials. 
Correspondingly, any continuous periodic metamaterial pattern should exhibit the effect provided that its unit cells locally support the resonant excitation of the symmetric $\lambda/2$-dipole mode.

In future work, different shapes of the nanostructure could be explored, for example, an array of meanders, the so-called fish-scale pattern \cite{fedotov2005planar} could be among the potential candidates. Further investigation into variations in the zigzag design could also include shallower angles of the zigzags or the case half way between the slits and wires designs, where the surface is its own inverse. Further simulations should be undertaken to model the response of the zigzag metasurfaces as the angle of incident light is varied. A more complex simulation method using COMSOL reproduced the signature double peak feature which could be explored further. 

{\color{blue}{
Following this investigation of the different zigzag designs, and their influence on the transmitted and reflected light, the next part of the project expanded on the exploration of nano-structured metamaterials, focusing on those consisting of achiral units and their potential to induce optical activity.
}}