A quantum graph approach to metamaterial design

Hello, everyone. I am pleased to introduce Professor Gregor Tanner from the University of Nottingham. Professor Tanner specializes in wave asymptotics, quantum chaos, and metamaterials design using quantum graph theories. He will present on the Quantum Graph Approach to Metamaterials Design.

Thank you, Bogdan, for organizing this event, and I appreciate Bryn's invitation, despite his absence today. The concept I will discuss originated a couple of years ago while we were exploring quantum graphs as a means to understand various properties of quantum systems and their connection to quantum chaos. This led us to consider whether we could apply this framework to study metamaterials, which are typically regular and periodic but possess unique characteristics at certain nodes.

Tristan Lawrie further developed this idea during his PhD, and much of my presentation today will focus on his contributions, which include numerous innovative concepts. This research began in collaboration with Dimitris Chronopoulos, who is currently at KU Leuven. Additionally, we have established a strong partnership with Gregory Chaplin and his team at the University of Exeter, and I will share some of their experimental findings later in the presentation.

Here, you can see several publications that resulted from this work, highlighting Tristan's significant contributions. He is currently a postdoctoral researcher at both Exeter and Nottingham.

Hello, everyone. I am pleased to introduce Professor Gregor Tanner from the University of Nottingham. Professor Tanner specializes in wave asymptotics, quantum chaos, and the design of metamaterials using quantum graph theories. He will present on the Quantum Graph Approach to Metamaterials Design.

Thank you, Bogdan, for organizing this event, and thank you, Bryn, for the invitation. Today's discussion stems from an idea we developed a few years ago while exploring quantum graphs as a means to understand various properties of quantum systems and their connection to quantum chaos. We wondered if this approach could be applied to the study of metamaterials, which are typically regular and periodic but possess unique properties at certain nodes or elements.

The majority of today's presentation will focus on the work of Tristan Lawrie, who has contributed significantly to this research during his PhD. I will highlight some of his innovative ideas later on. This research initially involved collaboration with Dimitris Chronopoulos, who is currently at KU Leuven, and we have also partnered closely with Gregory Chaplin and his team at the University of Exeter, whose experimental work I will discuss shortly.

Here, you can see several papers that have emerged from this collaboration, showcasing Tristan's substantial contributions. He is currently engaged in postdoctoral work at both Exeter and Nottingham.

In this presentation, I will outline our metamaterial design setup, present the experiments conducted by our collaborators, and discuss our recent focus on interfaces rather than just the materials themselves. This includes the introduction of concepts such as the Angular Filter, which endows these interfaces with interesting properties.

In this presentation, I will provide a brief overview of metamaterials, which I believe is familiar to many of you. I will discuss our approach to quantum graph construction of these metamaterials, focusing on the principles of beamforming, refraction, and reflection. These concepts are fundamental to the study of materials, but we will examine them within the discrete framework of graphs.

Additionally, I will touch on the topic of negative refraction and its relevance to our experiments. A significant focus will be on what I refer to as the K-space filter. While I will not delve deeply into other related topics, they remain important and interesting areas of study.

Metamaterials possess unique properties resulting from their fine local structure, which may include resonators or other components. This structure is typically small relative to the wavelength, allowing for a continuum treatment. However, due to this fine structure, metamaterials exhibit interesting dispersion relations and properties.

Modeling metamaterials often involves finite element methods, which can be time-consuming due to the need for meshing and periodic lattice considerations. While periodicity can simplify the modeling process, any modifications require remeshing. To address these challenges, we propose a rapid modeling approach that incorporates numerous parameters, offering significant degrees of freedom. This method enables the exploration of various metamaterial properties efficiently.

The quantum graph approach serves as an effective and quick method for testing ideas related to metamaterials, facilitating the transition from theoretical modeling to practical applications.

Metamaterials are materials with unique properties arising from their fine local structure, which may include resonators or other components. This structure is typically small relative to the wavelength, allowing the material to be treated as a continuum. However, the fine structure leads to intriguing dispersion relations and corresponding properties.

Modeling metamaterials is often achieved through finite element methods, which can be time-consuming due to the need for meshing in a periodic lattice. While periodicity allows for model simplification, any structural changes necessitate remeshing. To address these challenges, a rapid modeling approach was developed, incorporating numerous parameters to provide flexibility and retain essential metamaterial properties. This quantum graph method serves as an efficient way to explore ideas before transitioning to practical applications.

The accompanying illustration emphasizes the relationship between the atomic structure and wavelength. When the wavelength is comparable to the size of the structure, it can be resolved. As the wavelength increases, a continuum model emerges, which still captures aspects of the local structure, enhancing the material's intriguing characteristics.

In this section, we will explore the modeling of metamaterials. Instead of using traditional scatterers or resonant elements, we will represent them through one-dimensional connections between these elements. I will denote this special resonating element as γ, which may represent a single resonator or a more complex structure, such as a tree graph.

The foundational concept involves a regular lattice that serves as the underlying structure, with resonating elements layered on top. We will utilize the properties of these resonating elements to analyze the overall properties of the graph.

To proceed, we need to understand how waves propagate along the edges of this structure and how they scatter at the resonating elements. Typically, we analyze the interaction of incoming and outgoing waves through a scattering matrix. For illustration, consider a simple case with only two vertices, in contrast to the four vertices in a regular two-dimensional lattice.

At each vertex, we assume that the wave function on each edge maintains a consistent value, and we impose a condition on the derivative. By incorporating the parameter λ, we can derive the corresponding scattering matrix for this scenario, which represents the general case of our model.

To model the metamaterial, we establish one-dimensional connections between resonant elements rather than using discrete scatterers. This approach involves treating the system as a graph, where a regular lattice underlies a more complex structure, which may include resonators or intricate tree graphs. The properties of these resonating elements are utilized to analyze the characteristics of the graph.

We require information on wave propagation along the edges and the scattering behavior at the vertices. Typically, we consider incoming and outgoing waves, with a scattering matrix representing their interaction. In a simplified scenario with two vertices, we assume that the wave function is uniform across each edge, subject to specific derivative conditions. Introducing the parameter λ allows us to derive the corresponding scattering matrix, applicable in general cases.

In this context, V denotes the valency of the vertex, which is 2 in this instance and will increase to 4 later. As λ approaches 0, we observe flux conservation, corresponding to Neumann-Kirchhoff boundary conditions. Conversely, as λ approaches infinity, we encounter hard boundaries. These variations in λ influence the reflection and transmission of waves at the vertices. Our analysis will encompass one-dimensional, two-dimensional, and potentially three-dimensional lattices, typically employing Kirov-Neumann boundary conditions for the scattering matrices at the vertices.

V represents the valency of the vertex. In this instance, the valency is 2, but it will later increase to 4. As the parameter λ approaches 0, the system exhibits behavior akin to flux conservation, which aligns with the Neumann-Kirchhoff boundary condition. Conversely, when λ approaches infinity, the system reflects hard boundaries, indicating the presence of boundary conditions.

The relationship between λ and the behavior at the vertices is crucial; increasing λ results in more reflection and less transmission to the adjacent vertex. This establishes the foundational setup for our analysis. We will examine one-dimensional, two-dimensional, and potentially three-dimensional lattices, typically applying what are known as Kirov-Neumann boundary conditions. The scattering matrices at these vertices will take a specific form.

By employing block conditions and extending this framework to a lattice, we can derive relatively straightforward dispersion relationships. Here, N denotes the dimension, and we introduce the block wave number κ. The dispersion curve is described by the equation cos(kL) = cos(κJL), with summation over the dimensions. This framework effectively simulates free space propagation, although it incorporates some inherent band structure, which will be explored in further detail later.

In this section, we will explore the band structures in two dimensions, as illustrated in our dispersion curve. The lower portion of the curve typically exhibits a circular shape. As we approach the band edge, more complex features emerge.

The notation commonly used, particularly by our colleagues in Exeter, includes the γ XM relation, which identifies specific points on the dispersion surface. This notation provides a systematic way to describe the dispersion characteristics.

To analyze the dispersion surface, we solve the governing equations on a mesh using block conditions, which yield the wave numbers κ Y and κ X. The solutions obtained on this mesh reveal a pattern resembling a plane wave propagating through the structure.

The following slide illustrates the dispersion curves in two dimensions. In the lower energy region, the dispersion curve appears circular. As we approach the band edge, distinct features emerge. The notation commonly used, particularly by our colleagues in Exeter, refers to the γ-XM relation at specific points on the dispersion surface.

We can visualize the dispersion through contour plots. By solving the equation on a mesh with boundary conditions, we derive the values for κY and κX. This process allows us to identify solutions on the mesh, which resemble a plane wave propagating through the grid.

In low energy or low K space, the dispersion exhibits a circular wave pattern, derived from the expansion of the cosine function, leading to the relation K² = κX² + κY². This will serve as our free space solution. Future refinements will involve incorporating more complex elements into the graphs or modifying the edge length.

This slide presents a simple extension of our previous discussion. We observe two bands: the first band and the second band. By varying the dimensions in one direction, we set ( l_x = 2 ) along the x-direction and ( l_y = 1 ) along the y-direction. The first band remains similar to previous observations, while the second band exhibits a saddle structure. This saddle is critical as it facilitates negative refraction within that band.

During the presentation, please feel free to ask questions at any time. Your inquiries provide valuable feedback and help maintain engagement.

Next, we will explore wave propagation through this material, moving beyond infinite plane waves. We can analyze wave solutions based on a specific set of wave vectors, defined by ( \kappa_Y ) and ( \kappa_X ). For a fixed wave number ( K ), ( \kappa_X ) and ( \kappa_Y ) are interconnected through a dispersion relation. By fixing ( \kappa_Y ) and ( K ), ( \kappa_X ) becomes determined. This represents a specific solution within the mesh. By superimposing this solution with an appropriate coefficient and utilizing a Gaussian form for the parameters, we can generate a beam.

This slide illustrates the typical profile of a beam, represented by the α curve. As the beam is broadened, it takes on a cone-like shape, while a narrower beam resembles a plane wave. We can also shift the beam in various directions by selecting different solutions along the dispersion curve, which is essential for defining group velocity.

In reflection and transmission scenarios, the beam propagates according to the group velocity, which is determined by the gradient of the dispersion curve, rather than the phase velocity. This distinction is particularly significant in the context of negative refraction.

It is important to note that the underlying structure consists of one-dimensional waveguides interconnected at vertices. Locally, energy propagates along these one-dimensional waveguides, while globally, the waves behave as if they are plane waves traveling through the material.

The beam profile typically resembles an α curve. As the beam broadens, it takes on a cone-like shape, while a narrower beam approaches a plane wave configuration. By selecting different solutions along the dispersion curve, we can shift the beam in various directions, which is crucial for defining group velocity.

In the context of reflection and transmission, the beam propagates along the group velocity, which is determined by the gradient of the dispersion curve, rather than the phase velocity. This distinction is significant when considering phenomena such as negative refraction. Therefore, the beam's configuration is essential.

Underlying this concept is a network of one-dimensional (1D) waveguides interconnected at vertices. Locally, energy travels along these 1D waveguides, while globally, the waves behave as if they are plane waves within the material.

Now, let us examine the concepts of refraction and reflection in the context of two different metamaterials. We have metamaterial 1 and metamaterial 2, separated by boundary regions. This scenario necessitates a reevaluation of boundary conditions, differing from the continuous case where wave functions and normal derivatives are typically used. Instead, we must ensure that the wave functions and fluxes remain consistent across the boundary region. This problem can be addressed mathematically, leading to typical scenarios such as those involving infinite interfaces along the Y direction, with the X direction representing the transverse axis.

We are examining the phenomena of refraction and reflection at the interface of two distinct metamaterials. Each metamaterial has its own dispersion curve, which necessitates a reevaluation of the boundary conditions. Unlike the continuous case, where boundary conditions are typically expressed in terms of wave functions and their normal derivatives, we must ensure that the fluxes and wave functions remain consistent across the boundary region.

At the interface, the wave vector component ( \kappa_Y ) is preserved. As a plane wave approaches this interface, it experiences bending according to the corresponding ( \kappa_X ) value on the right side. This interaction illustrates the complexities involved in analyzing wave behavior in metamaterials with differing properties.

The parameter κ Y is preserved at the interface between two metamaterials with distinct dispersion curves. When a plane wave encounters this interface, it bends according to the corresponding κ X value on the right side of the graph.

In the context of beam equations, we can observe the effects of modifying one dimension of metamaterial 2, specifically the X direction. This adjustment alters the dispersion curve, resulting in reflection and transmission phenomena. Initially, all waves are transmitted due to the materials being equal. However, when we change the length in the wide direction, the dispersion curve shifts, leading to typical reflection and transmission behavior.

Notably, metamaterial 2 undergoes a transformation, developing a saddle point instead of a minimum, which creates a band gap resulting in complete reflection. This change is solely due to the alteration in the material's length. As we further modify the parameters, we reach a point where the band gap closes, leading to effects similar to negative refraction. In this scenario, an incoming ray is directed one way while the outgoing wave travels in the opposite direction, although the phase still moves in the original direction, resulting in negative reflection or transmission.

This straightforward approach demonstrates the influence of varying length scales in different metamaterials. We remain in the first band for metamaterial 1 but transition to the second band in metamaterial 2, which generates these intriguing effects. Researchers at Exeter have expressed interest in exploring this phenomenon further.

In this section, we will discuss the experimental setup used to investigate the properties of a metamaterial lattice. The researchers aimed to modify the lattice length to observe changes in wave behavior. The lattice structure, illustrated on the right, consists of four arms connected to the external environment. This connection is crucial for measuring surface waves, which allows for the analysis of wave propagation within the lattice.

To achieve a larger effective length, the researchers designed elements with varying leg lengths while maintaining a consistent lattice parameter. This experiment, conducted in the field of acoustics, was carried out by Timothy Starkey, Dale Moore, Peter Savage, and Gregory Chaplin, who are known for their expertise in this area.

The experimental setup features a plate with short legs on one side and long legs on the opposite side. Excitation is provided by a loudspeaker, while a scanning apparatus measures the surface waves on the plate. Additional components are included to minimize reflections, ensuring accurate measurements. This setup represents our first metamaterial configuration.

The experiment utilizes two distinct materials, each with unique structural features. The first material consists of a plate with short arms, which are assembled to create a surface with holes. The second material is thicker and also incorporates similar elements. Despite the differences in thickness, both materials present a uniform appearance on the surface.

The initial fired element model illustrates the dispersion curves for the short legs, represented by the Graph Model. This model displays the relationship between wave number and frequency. Our finite element model closely aligns with this representation.

For longer lengths, the dispersion characteristics appear compressed, resulting in a second band. Although the curves are similar, deviations become evident at higher frequencies or shorter wavelengths. The consistency observed in the lower frequencies further supports the validity of our pipe model as an effective representation of our graphs.

The initial analysis involved the fired element model, which produced dispersion curves for the short legs, represented in the Graph Model. These curves illustrate the relationship between wave number and frequency. The finite element model aligns closely with these findings. For longer lengths, the dispersion curves compress slightly, leading to a second band that appears similar to the first. However, at higher frequencies or shorter wavelengths, deviations become evident. Overall, this similarity suggests that our pipe model effectively represents the graph data.

Subsequently, measurements were conducted, focusing on surface waves generated by pulses. A Fourier transform was applied to analyze these waves. The resulting dispersion curves from the experiment, depicted in dark and light yellow, are compared to the predictions made by the finite element model, illustrated by the dashed lines. The experimental data reveals sound cones, indicating that any wave radiating from the surface is diminished, resulting in some loss of information. Notably, the experiment also demonstrates a negative group velocity, which aligns with predictions from the Graph Model.

In the remainder of this presentation, I will discuss a concept that has intrigued us: the Angular or K filter, which we have recently begun to understand and extend.

In this section, we will discuss the experimental setup and results related to surface wave measurements. The process involves generating pulses and measuring the resulting surface waves. A Fourier transform is then applied to analyze these waves. We compare the dispersion curves obtained from our experiments, represented by dark and light yellow lines, with predictions from our finite element model, shown as dashed curves.

The experimental results reveal the presence of sound cones, indicating that certain information is lost as waves radiate from the surface. Notably, we observed a negative group velocity, which aligns with our theoretical predictions using the Graph Model.

Next, we will explore a concept that has intrigued us: the Angular or K filter. This concept can be applied to various materials, including graphene. The setup involves two plates separated by a medium, such as air, connected through pipes. These pipes are designed to resonate and facilitate interactions between non-nearest neighbors in the structure.

As we delve deeper, we will examine how incident plane waves interact with this surface. Depending on the tuning of the surface, only specific angles allow the waves to transmit, while others result in reflection. This behavior resembles diffraction grating, where interference patterns emerge from multiple slits, leading to distinct transmission peaks. However, our setup differs from traditional diffraction, as it does not require a complete grating to produce these effects. Instead, we observe narrow peaks that can extend infinitely, with the slope indicating the focused nature of the incident beam.

When considering the interaction of a beam with a surface, it is essential to understand that the transmission pattern will shift with changes in the beam's angle, yet the fundamental characteristics remain constant. Transmission occurs only at specific angles; otherwise, the beam is reflected. This behavior distinguishes the phenomenon from traditional diffraction grating, as it is primarily a resonance effect.

We can visualize various configurations where we can tune the system to allow waves to pass through at designated angles. For instance, we might configure the system to permit transmission at zero degrees or at multiple specified angles. The design involves either a graph or a continuous material, interconnected through specific connectors. In this setup, we can connect to fourth neighbors and beyond, creating a network of resonant elements that influence the transmission properties of the surface.

This structure has been analyzed as a material that consists of one type connected in a different dimension. The dispersion curve exhibits maxima and minima, which depend on the extent of nearest neighbor coupling. However, our focus is on the interface rather than treating it as a bulk material.

In this analysis, I am removing the material on the left and right and concentrating on the interface. I will label the cells as M. The parameter μ represents the coupling between the M-th cell and the (M + μ)-th cell, where μ is an integer. The standard unit is defined by the length between these vertices or connectors.

This structure is characterized by a material of one type connected in a different dimension. The dispersion curve exhibits maximum and minimum values, influenced by the nearest neighbor coupling. However, our focus here is solely on the interface, rather than the material as a whole.

By removing the material on the left and right, we concentrate on the interface, which I will label as M. The parameter μ represents the coupling of the M cell to the M plus μ-th cell, where μ is an integer. The standard unit in this context is the length between these vertices or connectors.

Additionally, when we eliminate the bond, we can emphasize the interface further, introducing an extra parameter that denotes the length of the connector, which I refer to as μ.

We are examining two length scales: the difference between the underlying lattice constant and the connector length, denoted as ℓM.

Now, let's return to our Graph Model. At this vertex, we have four connections: one to the left, one to the right, one upward, and one downward. Each connection supports an incoming and outgoing wave, with the wave solutions on these edges described by the Helmholtz equation.

Due to the periodicity in the lattice and at the interface, we introduce the Bloch wave number, denoted as κ_y. This setup allows us to analyze how the interface scatters incoming waves, not just at a single vertex, but across the entire interface. We can derive a closed expression for this scattering behavior.

We have two length scales to consider: the difference between the underlying lattice constant and the connector length, denoted as ℓM.

Returning to our Graph Model, at a given vertex, we have four connections: left, right, upward, and downward. Each of these connections has an associated amplitude for incoming and outgoing waves. The wave solutions along these edges are governed by the Helmholtz equation. Additionally, due to the periodicity of the lattice at this interface, we introduce the Bloch wave number, κ_y.

The focus now is on how this interface scatters incoming waves, not just at a single vertex but across the entire interface. We can derive an expression for this scattering process.

Initially, we consider the amplitudes of incoming and outgoing waves at each vertex. The scattering matrix can be represented as having a valence of 4. For our analysis, we will set λ to zero, resulting in a term of 1/2 minus δ, PQ. The scattering matrix describes how the incoming wave is divided into transmitted and reflected components.

To simplify, we will disregard the degrees of freedom in the Y direction and derive the scattering matrix. This involves establishing connections between the incoming and outgoing amplitudes at each vertex using local scattering matrices, with λ set to zero for the left, right, up, and down connections.

Continuity conditions must be satisfied, meaning that the wave at position m plus μ, when set to zero, must equal the wave when moving along the line L μ downwards. This leads to a consistency condition, in addition to the Bloch condition.

Initially, I define the incoming and outgoing vertex amplitudes at each vertex. The scattering matrix can be represented as follows: for a valence of 4, I will set λ to zero, resulting in the expression ( \frac{1}{2} - \delta_{PQ} ). My focus is on describing the scattering matrix at the interface, which illustrates how the incoming wave is divided into transmitted and reflected components.

I will eliminate the degrees of freedom in the Y direction to derive this scattering matrix. The relationship between the incoming and outgoing amplitudes at each vertex is expressed in terms of local scattering matrices, with λ set to zero for the left, right, up, and down directions. There are specific continuity conditions to consider; for instance, the wave at position ( m + \mu ) at time ( t = 0 ) must match the wave propagating along the line ( L_\mu ). This establishes a consistency condition.

Furthermore, we observe a relationship between the outgoing and incoming amplitudes. Specifically, ( b_d^{out} ) and ( b_d^{in} ) are interconnected as we move from the outgoing to the incoming amplitudes over ( \mu ) steps. The accumulated phase along this length contributes to an equation that relates the left and right outgoing amplitudes to the left and right incoming amplitudes. This relationship is captured in a straightforward equation, which includes the transmission coefficient ( \tau_\mu T_\mu ) expressed as ( \frac{i \sin(K_l \mu)}{\kappa Y_\mu L - K_l \mu + i s K_l u u} ).

The two-dimensional scattering matrix at the interface exhibits intriguing properties. Singularities can occur in both the denominator and numerator of the matrix.

To determine the conditions for full transmission at the interface, we analyze the parameters ( K L_\mu ) and ( \kappa Y ). The condition for achieving full transmission is that ( \cos(K L_\mu) ) must equal ( \cos(\kappa Y L) ). This relationship is illustrated in the accompanying graph, where the x-axis represents ( K L_u ) and the y-axis represents ( \kappa Y ).

The black lines in the graph indicate regions of full transmission, while the red line represents a parameter related to the length connecting the next to nearest neighbors. By varying ( \mu ), we effectively modify the system. For a fixed ( K L_\mu ), the graph shows different values of ( \kappa ).

The parameter ( \kappa_Y ) represents the angle of the incoming beam or wave. At specific values of ( \kappa_Y ), full transmission occurs, while other angles result in reflection, as indicated by the red transmission line and green reflective waves.

By adjusting the length parameter of the nearest neighbor coupling, we modify the system's behavior. This adjustment leads to a critical point where transmission is maximized at ( \kappa_Y = 0 ), while all other angles result in complete reflection.

In certain conditions, specifically when ( \kappa_LY ) is a multiple of ( \pi ) times ( P ), full transmission is achieved. As the length is further altered, two distinct angles or values of ( \kappa_Y ) may yield transmission, while the remaining angles result in reflection.

Additionally, increasing the ( \mu ) parameter, which corresponds to extending the connection, enhances the occurrence of transmission peaks. This phenomenon resembles diffraction grating, but it primarily arises from the interface connection problem.

At the angles where transmission occurs, a resonance phenomenon is established within the system, characterized by periodic boundary conditions at the vertices. Each segment of the system exhibits resonance, contributing to the overall transmission behavior.

At specific values of κ LY, which are multiples of π times P times π, and under certain conditions for κ Ys, full transmission occurs. In contrast, if these conditions are not met, only reflection takes place. By adjusting the length of the connection, we can encounter scenarios with multiple angles or κ values that allow for transmission, while others result in reflection. Additionally, altering the μ parameter by extending the connection leads to an increase in transmission peaks. This phenomenon resembles diffraction grating, but it arises from a connection issue at the interface.

At the angle where transmission occurs, resonance phenomena develop within the pipes due to periodic boundary conditions at the vertices. Each pipe exhibits its own resonance. Depending on the connection to the external mesh, a maximum transmission is achieved. Conversely, at a minimum, no transmission occurs, resulting in negative values on the opposite side. When not transmitting, resonance still exists in each pipe; however, it leads to direct boundary conditions at the vertices. In this case, a nodal point must be present on the boundary, indicating that the mesh and interface do not couple. While solutions exist in the interior, the lack of coupling results in total reflection.

The connection to the external mesh determines the maximum and minimum transmission values. A minimum indicates that no transmission occurs, potentially resulting in a negative value on the opposite side. In the absence of transmission, resonance still exists within each pipe. However, this resonance leads to direct boundary conditions at the vertices, requiring a nodal point on the boundary. Consequently, the mesh and the interface do not couple. While solutions exist in the interior, there is no coupling between them, resulting in complete reflection.

In our investigation of a more continuous case, Gregory analyzed the continuous waveguide integrated with the pipes. We examined the continuous waveguide with a spiral overlay. When the system is properly tuned, we observe that transmission occurs primarily in specific directions. The interface effectively filters out certain directions based on parameter regimes. The dispersion curves illustrate that transmission is possible only at certain angles. Although the system functions with a continuum beneath and a spiral on top, its effectiveness diminishes due to excessive flow through the gaps between the pillars connecting to the interface.

We aimed to investigate the effectiveness of our approach in a continuum case. Gregory conducted an analysis by implementing a continuous waveguide, incorporating pipes and a spiral structure on top. By properly tuning the system, we observed that the incoming source primarily transmitted in specific directions, while other parameter regimes were filtered out by the interface.

The dispersion curves illustrate that transmission occurs only at certain angles. Although the system functions with a continuum beneath the spiral, its performance is somewhat diminished due to excessive flow occurring between the pillars connecting to the interface.

This phenomenon has practical applications. Tristan has advanced our understanding by exploring more complex 2D surfaces within 3D volumes, allowing for the imprinting of desired transmission and reflection characteristics in a two-dimensional format. One potential application is wave filtering, where specific waves are isolated. For instance, by identifying an interface that reflects vertically, we could enhance edge detection capabilities, effectively filtering out certain curvatures to delineate object boundaries.

The utility of this concept lies in its ability to generalize effects in a meaningful way. Tristan has enhanced this idea by exploring 2D surfaces that exist between 3D volumes. This approach allows for the selective imprinting of transmitted and reflected information in a two-dimensional format.

One potential application involves filtering specific waves. For instance, if an interface reflects vertically, it can be utilized for edge detection. By filtering out certain curvatures, we can identify where objects begin and end, thereby facilitating various applications in edge detection and object delineation.