Two-Dimensional Photonic Crystals Applied in High-Performance Meta-Systems

Abstract

Two-dimensional photonic crystals (2D PhCs) are nanostructure arrays arranged periodically or quasi-periodically, holding great promise as components for integrated and on-chip photonic platforms. The 2D PhCs can be considered as a special type of metasurfaces with periodicity, exhibiting versatile capabilities in the manipulation of electromagnetic waves. In this chapter, we present a summary of recent research trends and challenges related to the application of 2D PhCs as metasurfaces. Initially, we introduce the concepts and principles of Mie resonance, bound states in the continuum, and Fano resonance. Subsequently, we delve into some of the significant applications of 2D-PhC meta-systems, namely structural color generation, polarization manipulation and holography, reviewing their respective advancements. Finally, we offer an outlook on the challenges and potential future developments of 2D-PhC meta-systems to provide guidance for future investigations.

1. Introduction

The study of photonic crystals (PhCs) has emerged as a vibrant field within the past two decades. Alongside PhCs, there exists another concept known as metamaterials, which also play a significant role in manipulating electromagnetic waves. While metamaterials share similarities with PhCs, they are not constrained by periodicity and typically operate in the microwave range, spanning millimeters to meters in wavelength. Consequently, metamaterials often tackle macroscopic problems without the need for quantum mechanics considerations. Notably, one characteristic of metamaterials is their structural scales being much smaller than the operating wavelength. For instance, invisible cloaks rely on the formation of LC circuits, comprising circular components acting as inductors and parallel components serving as capacitors. The overall LC circuit resonates at a specific frequency, functioning as a dipole harmonic oscillator known as Lorentz resonance. Due to their operation near resonance frequencies, metamaterials exhibit significant absorption and narrow working bands. The study of invisible cloaks gained prominence in 2008, with some researchers employing frequency domain-based methods to achieve precise calculations. However, real-world applications face challenges, such as the presence of dispersion when epsilon is negative, limited bandwidth, and pronounced absorption near the center. These limitations are influenced by the Kramers-Kronig relation, rendering the overall system impractical due to high losses. To address these issues, two-dimensional (2D) metamaterials known as metasurfaces have been developed. These thin surface or slab structures, with low loss characteristics, find applications in various fields. For example, metasurfaces have been utilized in frequency-selective radomes to enhance radar functionality, particularly in stealth fighter jets. Many metasurfaces exhibit periodicity, enabling diverse functionalities such as nonlinearity, structural color generation, polarization manipulation, and topological states. Metasurfaces with periodicity can also be referred to as 2D-PhC meta-systems, requiring the application of theories such as Mie resonance, bound states in the continuum (BIC), Fano resonance, and photonic topology to investigate their properties.

This chapter offers a comprehensive overview of the current research trends and challenges surrounding 2D-PhC meta-systems. To elucidate the mechanisms by which these meta-systems manipulate electromagnetic waves, we delve into the fundamental concepts and principles of Mie resonance, BIC, and Fano resonance. Subsequently, we explore the versatile capabilities of 2D-PhC meta-systems through three specific applications: structural colors, polarization manipulation, holography. Lastly, we provide an outlook on the challenges and potential future developments that will guide further investigations in the realm of 2D-PhC meta-systems.

2. Principles of 2D PhC meta-system

2.1 Mie resonance

According to the theory of Mie scattering, when scatters like cylinders or microspheres interact with plane waves, their scattering field can be conceptualized as an amalgamation of the radiation fields generated by multiple poles [1]. In cases where the incident light’s wavelength is similar to the size of the nanostructure, the high-order radiation field tends to have lower intensity compared to the low-order radiation field. Hence, it is common practice to disregard the high-order radiation field and focus solely on the low-order modes. In the cylindrical coordinate system, these low-order modes comprise resonant modes associated with magnetic and electric dipoles, respectively. This implies that nano resonators can be viewed as dipole resonators, with the contributions from sub-terms of dipoles playing a crucial role in the scattering field [2, 3]. Consequently, the scattering fields generated by magnetic and electric dipoles can be harnessed to interfere with each other, enabling the modulation of incident field scattering in a unidirectional manner, akin to Huygens scattering. The polarizability of electric and magnetic dipoles can be described by the Mie coefficient in the formula’s expansion. Subsequently, we delve into the field distributions of electric and magnetic dipole modes, as well as the resulting field distribution upon their interference. In meta-systems, the design and construction of metasurfaces necessitate an analysis rooted in the coupling mode theory of Huygens scattering.

In 2015, Manuel Decker conducted an extensive investigation on the principle of light modulation in Mie scattering metasurfaces, which is briefly outlined below and illustrated in Figure 1 [2]. Consider a scenario where a pair of electric and magnetic dipoles are aligned along the x and y axes, respectively, characterized by their respective polarizabilities a_e and a_m. The Lorentz resonance frequencies are denoted as ω_e and ω_m, while the attenuation coefficients are represented by r_e and r_m. Utilizing the coupling mode theory, it can be derived that when the incident light is polarized along the x-axis, the angular frequency of the incident light, denoted as ω, can be expressed as follows:

Figure 1.

(a) Metasurface formed by arrays of nano dielectric pillars. (b) Magnetic dipole and electric dipole resonance modes [2].

$$\mathbf{E}\left(z,t\right)={\overrightarrow{E}}_x{e}^{-{\mathit{ik}}_{d}z-\mathit{i\omega t}}$$E1

In the equation, $k_d=n_d\omega /c_0$ represents the wave vector of the incident light in a medium with a refractive index of n_d. The interaction between the dipoles gives rise to the magnetic dipole moment and electric dipole moment, which can be expressed as follows:

$$\overrightarrow{m}=a^m\left({\overrightarrow{H}}_y+k_d^2{\epsilon }_d{G}^0\overrightarrow{m}\right)$$E2

$$\overrightarrow{p}=a^e\left({\overrightarrow{E}}_x+\frac{k_d^2}{{\epsilon }_0}{G}^0\overrightarrow{p}\right)$$E3

The interaction among the silicon pillars can be considered as the coupling effect of the entire periodic structure with a single resonance mode. Consequently, the overall structure can be characterized by effective magnetic polarizability and effective electric polarizability. By utilizing the expressions for the dipole moments, it becomes possible to calculate the far-field response of the total field outside the metasurface, which includes the scattering fields originating from the electrical and magnetic responses of the micropillars, as well as the interaction with the incident field. When the electric field is polarized along the x-axis or y-axis, the final transmission coefficient of the surface can be obtained as follows:

$$t=1+\frac{{\mathit{ik}}_d}{2A}\left(a_{\mathit{eff}}^e+a_{\mathit{eff}}^m\right),a_{\mathit{eff}}^e=1/\left(\frac{{\epsilon }_0{\epsilon }_d}{a^e}-k_d^2{G}^0\right),a_{\mathit{eff}}^m=1/\left(\frac{1}{a^e}-k_d^2{G}^0\right)$$E4

The effective polarizability can be represented by a Lorentz-type expression, given by:

$$a_{\mathit{eff}}^e=\frac{a_0^e}{{\omega }_e^2-{\omega }^2-2i{\gamma }_e\omega }$$E5

$$a_{\mathit{eff}}^m=\frac{a_0^m}{{\omega }_m^2-{\omega }^2-2i{\gamma }_m\omega }$$E6

When the angular frequencies ω_e and ω_m are not equal (${\omega }_e\ne {\omega }_m$), the transmission will be zero at these frequencies.

$$T={\left|t\left({\omega }_{e,m}\right)\right|}^2={\left|1+\frac{{\mathit{ik}}_d}{2A}{a}_{\mathit{eff}}^{e,m}\left({\omega }_{e,m}\right)\right|}^2=0$$E7

Therefore, the coefficient can be computed as follows:

$$a_0^{e,m}=\frac{4{\mathit{Ac}}_0}{n_d}{\gamma }_{e,m}$$E8

By combining the Eqs. (4)–(8), the transmission coefficient of the metasurface can be expressed as follows:

$$t=1+\frac{2i{\gamma }_e\omega }{{\omega }_e^2-{\omega }^2-2i{\gamma }_e\omega }+\frac{2i{\gamma }_m\omega }{{\omega }_m^2-{\omega }^2-2i{\gamma }_m\omega }$$E9

The reflection coefficient of the metasurface can be obtained by using the same method, as

$$r=\frac{2i{\gamma }_e\omega }{{\omega }_e^2-{\omega }^2-2i{\gamma }_e\omega }-\frac{2i{\gamma }_m\omega }{{\omega }_m^2-{\omega }^2-2i{\gamma }_m\omega }$$E10

Utilizing the above expressions, curves can be plotted of the transmission and reflection coefficients.

Here we analyze the resonant peaks of the transmission and reflection curves. When the resonant peaks of the electric dipole and magnetic dipole do not coincide, resulting in their spatial separation, the radiation fields from these dipoles interfere destructively with the incident field. As a consequence, two transmission valleys are formed, leading to a substantial decrease in overall amplitude intensity. Regarding the phase characteristics, both the electric and magnetic dipoles introduce a phase change of π. When the resonances of the magnetic dipole and electric dipole are separated, the resultant phase change, obtained through vector addition. Notably, the π phase change occurs individually at the resonances of the magnetic dipole and the electric dipole. Conversely, when the electric dipole and magnetic dipole resonate at the same wavelength, one of the dipoles is nullified by the interference with the incident light, while the remaining dipole radiates light with an intensity equivalent to that of the incident field. Consequently, the amplitude intensity remains constant at one. Regarding the phase behavior, when the electric and magnetic dipoles share the same wavelength, the phase difference between them amounts to 2π.

Extensive research has demonstrated that only a limited number of metasurfaces are capable of achieving phase modulation within the range of 0 to 2π. The majority of metasurfaces heavily rely on the manipulation of geometric parameters, particularly by altering the diameter size and periodicity, to induce changes in both amplitude and phase.

The research conducted by Manuel Decker et al. has also revealed that the resonance wavelengths of electric and magnetic dipoles can be altered by adjusting the diameters of the silicon pillars. Specifically, at a particular wavelength, if the change in the silicon pillar diameters can induce the same phase, it indicates that the structural parameters of the particles are conducive to achieving the same resonance wavelength for the electromagnetic dipoles. The corresponding phase change when the diameter of the silicon pillars does not favor the attainment of the same resonance wavelength. However, once the diameter of the silicon pillars reaches a certain value, the electric dipole and magnetic dipole will exhibit the same resonance wavelength. As the diameter varies, the electromagnetic dipoles will possess different resonance wavelengths.

The majority of metasurfaces utilizing Mie resonance rely on manipulating the structural parameters of the nanostructures. By establishing the relationships between the field amplitude and the diameter of the nanostructure, as well as the phase of the field, the diameter of the nanostructure, and the wavelength, one can generate a phase change diagram characterized by high transmission by selecting appropriate points. This phase change diagram serves as the foundation for realizing various functionalities within metasurfaces based on Mie resonance.

2.2 BICs

2.2.1 Concept of BICs

In the realm of quantum mechanics, a bound state typically refers to a discrete state wherein the energy of an electron is lower than the potential well barrier (Figure 2(a), E < 0), causing the electron to be confined within the well. The wave function associated with this state has zero probability at infinity. On the other hand, when an electron possesses sufficient escape energy (Figure 2(a), E > 0), its energy becomes coupled outward, and the wave function extends to infinity, forming an extended wave. However, in 1929, von Neumann and Wigner proposed a special case depicted in Figure 2(b). In a three-dimensional (3D) potential well, the electron’s wave function can expand beyond the boundaries of the potential well, yet the energy of the electron remains unaffected. Instead, the electron exists as a bound state in the extended region, known as BICs. These electronic BICs are artificial eigenstates, and their confinement is highly susceptible to spatial disturbances, making them challenging to achieve experimentally [4, 5, 6]. Nonetheless, their exceptional properties, distinct from those predicted by traditional wave theory, have sparked significant research interest.

Figure 2.

Extended and bound states in quantum wells. (a) In quantum wells: When E > 0, there are extended states, and when E < 0, there are discrete states. (b) Position of the BIC.

According to traditional wave theory, the presence of a bound state in an open system is determined by the frequency of the wave. If the wave’s frequency falls below the lowest frequency of the continuous spectrum, it exists as a discrete bound state since there is no available radiation path. Conversely, if the wave’s frequency lies within the continuous spectrum, it manifests as oscillations in the radiation region and eventually dissipates into infinity. However, the concept of BICs challenges the conventional understanding of state binding mechanisms. BICs exhibit frequencies within the continuous spectrum, yet possess an infinite-high quality factor and do not experience radiation leakage. This phenomenon is vividly depicted in Figure 3 [7].

Figure 3.

Schematic diagram of BIC in the spectrum [7].

Figure 3 illustrates a sinusoidal wave oscillating at an angular frequency ω, along with its corresponding field distribution in different states. The spectrum depicted consists of a discrete state represented by the green curve and an extended state denoted by the blue curve. The green discrete state, akin to the discrete mode at E < 0 in Figure 2(a), is commonly known as the bound state below the lowest frequency of the continuous spectrum. For instance, in an optical fiber, when the waveguide mode satisfies the condition of total internal reflection, the energy becomes confined within the waveguide, forming a discrete state with a frequency below the lowest frequency of the radiating waves. It is important to note that within the blue continuous spectrum, an orange curve represents leakage resonances capable of coupling with radiation waves and escaping outside the system. For instance, when an incident wave undergoes modulation by a grating and satisfies the phase matching condition with the propagation constant inside the waveguide, a leakage mode is formed, which can couple with the radiation wave. Such resonances can be characterized by complex frequencies, expressed as $\omega ={\omega }_0-\mathit{i\gamma }$, where ω₀ represents the resonant frequency and r represents the radiated loss. When the radiation loss γ approaches zero, this resonance decouples from the radiation wave and becomes a bound state within the continuous spectrum, known as a BIC. This corresponds to the red curve illustrated in Figure 3.

In conclusion, BICs are unique wave functions that propagate to infinity while their energy resides within the eigenstate of the continuous spectrum as a bound state. These states correspond to special resonances [8] that exhibit no energy dissipation in the radiative continuum region. They can be considered as dark states, characterized by extremely narrow resonance linewidths and infinite quality factors ($Q={\omega }_0/2\gamma$) in real space resonances. BICs represent anomalous solutions of the wave equations that describe light or matter, as they possess discrete and spatially bounded characteristics, while coexisting within the radiation region [9].

2.2.2 Classification of BIC

The identification of BICs in a system can be facilitated by examining their frequency position and decoupling from the radiation region. When the frequency lies above the lowest frequency of the radiating waves and remains decoupled from the radiation, it signifies the presence of BICs. Conversely, if the frequency is coupled to the radiation, it represents a bound state similar to a waveguide mode. BICs can be classified into two categories based on the decoupling mechanism between the eigenstate and radiation waves. The first category is symmetric protected BICs, which arise from the orthogonal eigenmodes and radiation modes, forming a bound state that remains decoupled from radiation waves [10, 11]. The second category is interference-canceling BICs, which enter the radiation channel through the interferometric cancelation between different modes, preventing coupling to the bound state of the radiation wave [12].

Symmetrically protected BICs are commonly observed in systems exhibiting reflection or rotational symmetry. They arise as bound states in the extended region, resulting from the complete decoupling of eigenmodes and radiation modes at high symmetry positions or Γ point within the Brillouin zone, driven by symmetry mismatch. For instance, consider the PhC plate depicted in Figure 4 [7]. The entire structure exhibits periodicity in the x and y directions, while the wave expands and radiates in the z direction. When a rotational symmetry of 180 degrees along the z-axis (C₂) is maintained, the electromagnetic field of the extended wave assumes an odd mode. Simultaneously, the eigenmodes within the structure correspond to even modes that are orthogonal to the odd mode, leading to the formation of symmetrically protected BICs.

Figure 4.

Symmetrically protected BICs in a PhC slab [7].

Ideal BICs possess an infinitely large quality factor, representing dark states in real space that cannot be directly excited by incident light. Consequently, they correspond to spectral points where resonances disappear. For instance, consider the periodic silicon nanorod array depicted in Figure 5 [13]. This structure exhibits C₂ rotational symmetry, and when the two elliptical nanorods within each unit cell maintain a high degree of symmetry, the intrinsic mode of the structure around 1560 nm corresponds to a BIC. In the spectrum, a transmission valley with an extremely narrow linewidth, indicated by a straight line, can be observed. However, when an in-plane perturbation is introduced, the BIC with an infinitely large quality factor transforms into a quasi-BIC with a finite linewidth. Moreover, the linewidth of the quasi-BIC increases with the magnitude of the perturbation, as confirmed by the color map of the transmission spectrum. This phenomenon reveals that by breaking the in-plane symmetry, BICs can be converted into quasi-BICs, and the radiation loss of quasi-BICs can be controlled by adjusting the perturbation. Consequently, this provides a new degree of freedom for manipulating the interaction between light and matter.

Figure 5.

Periodic silicon nanorod array structure to generate quasi-BICs [13].

BICs based on interference cancelation, unlike symmetrically protected BICs, do not require altering the system’s symmetry. Instead, this type of BIC is achieved through specific parameter adjustments, which enable the interference and subsequent cancelation of all modes entering the radiation channel. Consequently, they decouple from the radiation region. Temporal Coupled-Mode Theory (TCMT) [14] can be employed to describe this scenario. Assuming there is no external radiated light, let the amplitudes of the two resonant states be denoted as $A={\left(A_1,A_2\right)}^{\mathrm{T}}$, satisfying the time-containing Hamiltonian equation $i\partial A/\partial t=\mathit{HA}$, with the Hamiltonian H given by [13, 14, 15]:

$$H=\left(\begin{array}{cc}{\omega }_0& \kappa \\ \kappa & {\omega }_0\end{array}\right)-\mathit{i\gamma }\left(\begin{array}{cc}1& e^{\mathit{i\psi }}\\ e^{\mathit{i\psi }}& 1\end{array}\right)$$E11

In the equation, κ represents the near-field coupling coefficient between the two resonant modes, γ denotes the emissivity of a single resonant mode, and ψ represents the phase difference between the two resonant modes. Subsequently, the two eigenfrequencies of the Hamiltonian can be expressed as:

$${\omega }_{\pm }={\omega }_0\pm \kappa -\mathit{i\gamma }\left(1\pm e^{\mathit{i\psi }}\right)$$E12

When the phase difference ψ is an integer multiple of π, one of the eigenfrequencies is denoted as ${\omega }_0\pm \kappa -2\mathit{i\gamma }$. It is observed that its radiation loss doubles compared to the usual case. On the other hand, the other eigenfrequency, ${\omega }_0\pm \kappa$, is a purely real number with zero imaginary part, indicating the absence of radiation loss. This eigenfrequency corresponds to a BIC that is decoupled from the radiation region. This type of BIC is often found in structures featuring two resonators, resembling a Fabry-Pérot cavity, hence referred to as Fabry-Pérot BIC. Figure 6 illustrates this concept, where adjusting the distance between the two resonant cavities is akin to modifying the phase difference between the two resonant modes, $\psi =\mathit{kd}$. When this adjustment satisfies the condition that the wave oscillates back and forth between the two resonators once, the phase difference undergoes a complete change of 2π, resulting in the formation of a bound state within the continuous spectrum.

Figure 6.

Mechanism of a Fabry-Pérot BIC [7].

In the described double-resonator structure of the Fabry-Pérot cavities, there exists a special case where $\psi =\mathit{kd}=0$, yielding a real number solution. This situation arises when the distance between the two resonators is zero, meaning that two leakage resonance modes within a single resonator can interfere with each other. Through phase cancelation, a BIC is formed. This behavior can also be understood within the framework of TCMT. Assuming that the two resonance modes reside within a resonant cavity and enter the same radiation channel, the amplitude Hamiltonian can be expressed as [7, 16, 17]:

$$H=\left(\begin{array}{cc}{\omega }_1& \kappa \\ \kappa & {\omega }_2\end{array}\right)-i\left(\begin{array}{cc}{\gamma }_1& \sqrt{{\gamma }_1{\gamma }_2}\\ \sqrt{{\gamma }_1{\gamma }_2}& {\gamma }_2\end{array}\right)$$E13

In the equation, ω₁ and ω₂ represent the resonant frequencies of the two modes, while γ₁ and γ₂ denote the radiation losses associated with each resonant mode, respectively. As these two resonant modes can radiate into a common radiation channel, a coupling term $\sqrt{{\gamma }_1{\gamma }_2}$ arises between the modes. Solving the eigenequations reveals that the solution for a BIC necessitates satisfying the following condition:

$$\kappa \left({\gamma }_1-{\gamma }_2\right)=\sqrt{{\gamma }_1{\gamma }_2}\left({\omega }_1-{\omega }_2\right)$$E14

One of the eigenvalues in the equation becomes a purely real number, corresponding to a BIC. This specific type of BIC, referred to as Friedrich-Wintgen BIC, is named after Friedrich and Wintgen who proposed formula (14) [7]. Generally, Friedrich-Wintgen BICs are located at non-Γ points. To illustrate, consider the 1D grating waveguide structure. This structure comprises two modes: a plasmonic mode supported by a silver grating and a photonic mode in the upper SiO₂ waveguide. As the incidence angle is varied, these two modes enter a strongly coupled region, exhibiting an inverse crossover phenomenon. At an incidence angle of 11.4°, the two modes interfere and mutually cancel each other, resulting in the formation of a BIC [18].

Both Fabry-Pérot BICs and Friedrich-Wintgen BICs are bound states resulting from interference cancelation among multiple resonance modes. Additionally, there is a case where bound states emerge through interference cancelation between multiple groups of radiated waves within a single resonance mode, known as single-resonance BICs. One of the examples is a PhC plate that illuminated by an oblique angle planar wave at 37°. At the non-Γ point above the light cone, there exists a BIC located at k_x = 0.27 with an infinitely large quality factor, indicating a single-resonance BIC. This is attributed to the presence of a single resonance mode with TM polarization within the energy band [12].

In practice, BICs may undergo transformation into quasi-BICs due to structural limitations, material absorption losses, and external disturbances [19]. These quasi-BICs, also referred to as ultra-microcavity modes, exhibit finite radiation losses. Today, such quasi-BICs have been employed in various periodic photonic systems to achieve high-Q resonances [20, 21, 22, 23].

2.3 Fano resonance

2.3.1 Fano resonance in atomic physics

In 1935, Beutler made the remarkable discovery of unusually sharp absorption peaks in the noble gas absorption spectra. Fascinated by these anomalous spikes, Fano was the first to describe this resonance phenomenon with an asymmetric linear pattern. Photoionization of an atom can occur through various mechanisms. One method involves directly exciting the inner-shell electrons above the ionization threshold, where the atom absorbs a photon and subsequently releases an electron, expressed as:

$$A+\mathrm{\hslash }v\to A^{+}+e$$E15

Another approach is known as self-ionization [24], where atoms are excited to quasi-discrete energy levels and undergo ionization by emitting an electron into a continuous state, given by:

$$A+\mathrm{\hslash }v\to A^{\ast }\to A^{+}+e$$E16

After an atom undergoes ionization, resulting in the emission of electrons from an inner shell, a vacancy is created in that shell. A higher-energy electron from an outer shell can then transition to fill this vacancy, releasing energy. Subsequently, another electron can absorb this energy and be emitted into the continuum, a process referred to as the Auger effect. The quantum interference between the direct ionization process of the inner-shell electrons and the self-ionization process of the two excited electrons leads to the emergence of asymmetric linear resonances. Fano explained the origin of this resonance phenomenon using perturbation theory and proposed a formula to describe the resonant line shape in scattering cross-sections [25]:

$$\sigma =\frac{{\left(\epsilon +q\right)}^2}{{\epsilon }^2+1}$$E17

where q is the shape parameter and the energy ε is defined as 2(E-E_F)/Γ, where E_F represents the resonance energy and Γ corresponds to the width of the self-ionized state.

In 1961, Fano introduced the asymmetric parameter q, which represents the ratio of the probabilities for atoms to transition to a discrete or continuous state, as shown in Figure 7. When the asymmetric parameter q is equal to 0, the resonance line shape is symmetrical, displaying a dip in the scattering spectrum, which is often referred to as an anti-resonance. As the asymmetric parameter |q| tends to infinity, the likelihood of atoms transitioning to a continuous state becomes negligible, and the resonant line shape is predominantly determined by the discrete state, characterized by a typical Lorentzian line pattern. When the asymmetric parameter q equals 1, the probabilities of transitioning to the discrete and continuous states are equal, giving rise to the appearance of asymmetric resonance lines.

Figure 7.

Schematic diagram of Fano resonance based on the superposition of a discrete state and a continuum state.

2.3.2 Classical analogy of Fano resonance

Resonance is a phenomenon commonly observed in systems that exhibit self-enhancement in response to external excitations at specific frequencies, known as the resonant frequency or natural frequency of the system. In classical physics, resonance is often introduced by applying a periodically changing force to a resonator. When the frequency of this force matches or is very close to the natural frequency of the system, the amplitude of the oscillations reaches its maximum value. However, there are cases where the opposite occurs, and the amplitude of the oscillations reaches its minimum value. For instance, in 2006, Young S Joe et al. proposed the classic analogy of Fano resonance using a dual oscillator model [26]. Consider two weakly coupled double oscillators with natural frequencies ω₁ and ω₂, respectively, connected by a light spring. Applying a periodically changing force to the first oscillator, the equations of motion for the two oscillators are:

$${\ddot{x}}_1+{\gamma }_1{\dot{x}}_1+{\omega }_1^2{x}_1+{\upsilon }_{12}{x}_2=a_1{e}^{\mathit{i\omega t}}$$E18

$${\ddot{x}}_2+{\gamma }_2{\dot{x}}_2+{\omega }_2^2{x}_2+{\upsilon }_{12}{x}_1=0$$E19

where

$$x_1=c_1{e}^{\mathit{i\omega t}},x_2=c_2{e}^{\mathit{i\omega t}}$$E20

By combining Eqs. (18)-(20), we can derive:

$$c_1=\frac{{\omega }_2^2-{\omega }^2+i{\gamma }_2\omega }{\left({\omega }_1^2-{\omega }^2+i{\gamma }_1\omega \right)\left({\omega }_2^2-{\omega }^2+i{\gamma }_2\omega \right)-{\upsilon }_{12}^2}{a}_1$$E21

$$c_2=\frac{{\upsilon }_{12}}{\left({\omega }_1^2-{\omega }^2+i{\gamma }_1\omega \right)\left({\omega }_2^2-{\omega }^2+i{\gamma }_2\omega \right)-{\upsilon }_{12}^2}{a}_1$$E22

The phase difference between the two oscillators is given by ${\phi }_2-{\phi }_1=\pi -\theta$. The additional phase difference θ can be obtained from Eq. (21):

$$\theta ={tan}^{-1}\left(\frac{{\gamma }_2\omega }{{\omega }_2^2-{\omega }^2}\right)$$E23

For the first oscillator, which is driven by the periodically changing force, its amplitude varies with frequency as illustrated in Figure 8(a). It exhibits two resonance peaks, with the lower frequency resonance displaying a symmetric line shape, while the higher frequency resonance exhibits an asymmetric resonant line shape. Moreover, for the higher frequency resonance, the frequencies at which interference cancelation and interference phase occur are extremely close, resulting in an amplitude of zero during interference cancelation. This arises from the interference between the periodic changing force and the second oscillator. From the oscillator vs. frequency curve of the first harmonic oscillator, as depicted in Figure 8(c), we observe a sudden phase change from π to 0 to π around the second resonant frequency. Similarly, the second oscillator also exhibits two resonances, as shown in Figure 8(b), but with symmetric resonant line shapes. The phase vs. frequency curve for the second oscillator, illustrated in Figure 8(d), reveals a phase change of π at the resonance.

Figure 8.

Oscillators ((a) and (b)) and their phases as functions of frequency ((c) and (d)) in a dual oscillator model [26].

3. Applications of 2D PhC meta-systems

A metasurface is an ultrathin and integrable 2D planar artificial material. It is composed of sub-wavelength metal or dielectric unit structures that possess remarkable capabilities to control various properties of light, including amplitude, phase, polarization, and more. These capabilities enable metasurfaces to achieve optical functionalities such as ultralensing, holography, and generation of vortex light. Metasurfaces represent a revolutionary advancement in the realm of 2D planar materials, offering a solution to the size constraints of traditional optical devices. They hold immense promise for replacing conventional optical components, thereby significantly enhancing the performance of modern optical systems. Due to their exceptional ability to manipulate light amplitude, 2D PhC meta-systems have achieved significant breakthroughs in the fields of structural color and polarization manipulation.

3.1 Generating structural colors

3.1.1 Structural colors based on surface plasmon polaritons (SPPs)

In addition to utilizing metasurfaces’ phase modulation function for applications such as light refraction, holographic projection, and superlensing, an important and captivating application of metasurfaces involves using amplitude modulation to achieve structural color displays. Inspired by the vibrant hues found in nature, the mesmerizing colors exhibited by plasma structures have fascinated humanity since ancient times. Glassmakers have successfully created vivid and remarkable colors by incorporating metal nanoparticles into glass, as exemplified by the Roman Lycurgus cup (Figure 9) and stained glass [27, 28].

Figure 9.

The color of the Lycurgus cup in reflected and transmitted lights.

The Roman Lycurgus cup possesses a remarkable characteristic known as dichroism. When illuminated by an internal white light source, it appears a radiant ruby red. Conversely, when illuminated by the same light source from outside, it transforms into a captivating emerald green. Unlike the dynamic colors generated by light sources with finite bandwidth, these breathtaking color filtering effects arise from the resonance coupling between light and metal nanoparticles, specifically through a phenomenon known as SPPs [29, 30, 31, 32, 33]. This resonance phenomenon involves the interaction of free electrons within the nanoparticles with the incident light, resulting in the distinct and enchanting color transformations observed in the Roman Lycurgus cup.

Surface plasmons refer to the oscillation of free electrons at metal interfaces [34, 35, 36, 37, 38, 39, 40]. When light with a specific wavelength interacts with a metal nanoparticle, the electric field of the light causes the free electrons within the particle to resonate, leading to strong optical absorption. This resonance is known as Localized Surface Plasmon Resonance (LSPR). The LSPR mode exhibits remarkable capability in confining light to subwavelength volumes. Consequently, LSPR-based color filters have garnered significant attention due to their exceptional spatial resolution, long-lasting durability, straightforward manufacturing process, and wide viewing angle visibility. These attributes offer immense potential for plasma-color-based optics in applications such as high-resolution color printing, high-density data storage, and color displays. The advancement of nanofabrication technology has enabled precise design of nanoparticle size and structure, effectively transforming them into “optical nanoantennas” and allowing unprecedented control over optical responses. Consequently, research on structural color based on metal plasmas has seen extensive development in recent years [41, 42, 43, 44, 45].

Grating structures that possess the ability to disperse different wavelengths of light are highly desirable for plasma color filters [46, 47, 48, 49, 50, 51]. By ensuring that the grating period is smaller than the incident wavelength, the undesirable effects of diffraction can be minimized. This allows for efficient achievement of transmission or reflection that remains insensitive to the angle of incidence. In 2010, Xu et al. introduced a grating structure composed of aluminum-aluminum selenide-aluminum, which was periodically arranged on magnesium fluoride [46]. By carefully designing the period and width of the grating, they were able to precisely control the transmission spectrum through these arrays, resulting in a highly efficient color filter capable of transmitting any color. The transmittance of light near the resonant wavelength exceeded 50%, and the full width at half maximum (FWHM) was approximately 100 nm, enabling the production of more vibrant colors, including red, green, and blue.

The plasma nanostructure mentioned above consists of three layers of material: the top layer is composed of a metal nanostructure array, a dielectric spacer layer, and an underlying metal reflective layer. These structures, known as gap plasma structures or metal-insulator–metal (MIM) structures, exhibit a resonant pattern characterized by the magnetic field within the dielectric gap confined between the metal nanostructure and the reflector [52, 53, 54, 55, 56]. However, the excitation of surface plasmons in a one-dimensional grating is limited to incident light perpendicular to the grating direction. This limitation can be overcome by employing 2D nanostructured arrays, such as high-density silver nanorod arrays and aluminum nanoclusters [47, 57]. These arrays allow for the excitation of surface plasmons and can efficiently absorb light, enabling the production of vibrant colors.

In 2016, Masashi designed a metal-insulator-metal nanoscale disk structure consisting of a 40 nm thick aluminum disk on a 30 nm alumina film, which was placed on top of an aluminum film, as shown in Figure 10 [58]. By modifying the diameter and periodicity of the structural unit, a color display with a wide range of colors within the visible spectrum was achieved. Unlike traditional plasmon structures that rely on periodic arrangements, this structure’s optical properties simultaneously reflect the characteristics of a single unit. This implies that the optical interaction between structures is minimal, enabling the realization of color displays using individual nanodisks. An optical microscope image of the processed letter “Nano” demonstrates the formation of colors solely by a single disk, without the presence of other neighboring units.

Figure 10.

Schematic of aluminum nanodisks and optical microscopy images of the nanodisk arrays [58].

2D grating nanostructures can selectively excite SPPs, LSPRs, or a combination of both. This excitation leads to the formation of distinct peaks or dips in the transmission or reflection spectrum. Similarly, subwavelength nanopore arrays exhibit comparable optical responses. The interaction of incident light with surface plasmons gives rise to an extraordinary phenomenon known as anomalous light transmission, characterized by an abnormal zero-order transmission. This abnormal transmission arises from the interaction of SPPs among neighboring nanopores, resulting in enhanced transmission at specific resonant positions in the spectrum. By adjusting the periodicity of the nanopore array, this effect can be harnessed to produce vivid and diverse colors [57, 59, 60, 61, 62, 63, 64, 65, 66, 67].

In 2015, Yang’s research group introduced a polarization-insensitive color filter consisting of a periodic perforated silver film [68]. The three functional layers of this filter include a 25 nm-thick silver film with perforations on top and a 100 nm-thick silver mirror at the bottom. These layers are separated by a 45 nm silica spacer medium. The combination of the periodically perforated silver film and the silver mirror enables the realization of bright, highly saturated, and uniformly distributed yellow, magenta, and cyan colors, thereby enhancing overall color quality.

Color filtering can be achieved by utilizing periodic hole structures, but this approach often requires multiple elemental structures to induce SPP interference between adjacent elements, resulting in larger pixel sizes. On the other hand, the excitation of LSPRs through periodic or isolated metal nanodisks enables color generation, allowing for pixel sizes to be scaled down to the wavelength scale. The near-field interaction between nanodisks and nanopores presents an opportunity to produce pixels smaller than half a wavelength.

In 2012, Kumar et al. presented a method to achieve full-color printing at the optical diffraction limit by designing a nanodisk structure based on a metal mirror [69]. The size of the nanodisks is optimized to encode color information onto the silver/gold nanodisks situated above the pore reflector. By varying the size and spacing of the nanodisks, different colors can be produced, visible directly under a brightfield light microscope. The authors demonstrated this by creating the well-known Lena pattern with structural colors. Notably, the smallest pixel in the image measures 2 × 2, yet clear and distinguishable colors are visible under bright field illumination. This pixel size of 250 × 250 nm reaches the theoretical diffraction limit of optical microscopy. Additionally, the authors prepared alternating color checkerboard patterns with 3 × 3 and 2 × 2 arrays of nanodisks per square. The successful rendering of distinguishable colors in such small pixels validates the capability of this approach, achieving a color printing resolution of 100,000 dots per inch (dpi).

3.1.2 Structural colors based on dielectric materials

Devices employing plasmonic metal nanostructures to achieve structural color offer precise control over light absorption and scattering beyond the optical diffraction limit. These devices possess the advantages of compact size and high resolution. Similarly, metasurface structural colors based on dielectric materials have been extensively investigated, addressing the limitations associated with metals, such as high losses that result in broadening of spectral peaks. Dielectric metasurface structural colors exploit the resonance of Mie scattering, which relies on the geometry and size of the particles [70, 71, 72, 73, 74, 75, 76, 77, 78]. In principle, dielectric nanoparticles with a high refractive index (n) can manipulate electric and magnetic dipoles. The electric dipole is primarily governed by the plasmonic metasurface while Mie scattering occurs when the incident light wavelength and nanoparticle size are comparable ($2R\approx \lambda /n$). This mechanism also offers the opportunity to design the structural color of dielectric metasurfaces by leveraging higher-order multipoles introduced through Mie resonance [79].

Resonant magnetic responses to light have been observed in particles of various shapes and materials, including rings [80], ellipsoids [81], discs and cylinders [82], and spherical bodies [83]. This diverse array of shapes presents an opportunity to tailor both magnetic and electric resonances by designing the geometric parameters of nanostructures made from different materials. Furthermore, when designing a structure, careful consideration must be given to selecting the appropriate structural material, as it significantly impacts the optical properties of light, including transmission, reflection, optical loss, and efficiency. A wide range of media materials are available, each with its own distinct advantages and disadvantages. Currently, silicon (Si) and titanium dioxide (TiO₂) are among the most commonly employed materials in this field.

Si is widely utilized in metasurface structural colors due to its cost-effectiveness, high reliability, and compatibility with optoelectronic devices [84, 85]. Si possesses a high refractive index, which enables precise control and manipulation of light [86, 87]. Significantly, when subwavelength Si particles interact with visible wavelengths, they exhibit strong electromagnetic resonances. This optical property is harnessed in the design of silicon nanowires, which serve as effective color filters. Numerous studies have explored various geometries for silicon, such as nanopillars [88] and crosses [89], resulting in high-quality resonances within the visible light range. These advancements have facilitated the achievement of vibrant colors with high purity and a wide color gamut.

In 2016, Proust et al. fabricated amorphous nano-silicon cylinders on silicon oxide substrates, with a fixed period of 1 μm between the nanocolumns [90]. This period is significantly larger than the wavelength of visible light, ensuring minimal coupling between the silicon cylinders. By leveraging resonant light scattering from individual silicon particles, they achieved a structural color display. The relative intensity of the electrical and magnetic resonances could be easily adjusted based on the length-to-diameter ratio of the cylinders. This approach enabled the realization of a color printing technology using silicon nanopillars with low-order electrical and magnetic Mie resonances. The interaction between these resonances produced darkfield scattered colors within the visible spectral range, resulting in vibrant colors spanning the entire visible spectrum.

In 2017, Brugger and colleagues conducted a study where they designed, fabricated, and measured the structural colors of amorphous silicon, aluminum, and silver nanodisks, comparing their color properties [91]. The Si structures exhibit a gradient intensity modulation corresponding to changes in nanodisk size, while the Al and Ag structures exhibit abrupt tonal and intensity variations. Moreover, the Si structure displays a broader range of colors compared to metals, thanks to its superior confinement of electromagnetic fields and lower losses. By adjusting the diameter and gap of the nanodisks, the color and intensity produced by the medium’s metasurface can be finely tuned. Additionally, pixel patterns consisting of different colors were created to showcase the color capabilities of high-resolution, isolated nanostructures. The pixel cells have pitches of 500 nm and a resolution exceeding 50,000 dpi, provide significant optical contrast, enabling the clear visualization of distinguishable patterns under brightfield microscopy. These patterns rival the color richness of plasmonic structural hues. The brightfield structural colors achieved in this study encompass a continuous range of hues and saturations and are observable through conventional light microscopy, photography, and even with the naked eye under white illumination.

Apart from amorphous silicon, monocrystalline silicon offers lower losses in the visible light range. When the wavelength is fixed at 410 nm, the extinction coefficients of amorphous silicon, polysilicon, and monocrystalline silicon are measured to be 2.02, 0.752, and 0.269, respectively. These values indicate that monocrystalline silicon serves as an ideal material for filters. In 2017, Takahara and colleagues prepared silicon nanoblocks using 150 nm monocrystalline silicon [92]. They achieved a diverse range of vibrant structural colors by adjusting the geometric parameters. Furthermore, owing to the high refractive index of monocrystalline silicon, pixels composed of individual nanoblocks can exhibit high-quality tertiary colors. The resolution of the metasurface based on this medium reaches approximately 85,000 dpi.

When it comes to amorphous silicon filters, the saturation and color gamut of structural colors are often limited by two factors: high losses and substrate effects. Substrates can have a significant impact on the performance of nanostructures, leading to broader peak shapes due to forward scattering in free space. However, by employing nanostructures without substrates, it is possible to suppress forward scattering and achieve narrower peak shapes. Therefore, Si color filters have the potential to enhance the saturation of structural colors by minimizing substrate effects and reducing material loss.

In 2017, Dong and colleagues proposed a novel structural color design that addresses these challenges while maintaining high resolution. They introduced silicon nanocylindrical structures [93]. A 70 nm thick Si₃N₄ film was employed as an anti-reflective layer, covering the Si substrate and effectively suppressing substrate-induced influences. This approach leverages the Kerker effect, resulting in sharpened reflection peaks and more saturated colors. The structural color achieved by this design covers the visible range and can reach 120% of the sRGB color gamut, thus offering improved color saturation and an expanded color range.

When it comes to amorphous silicon filters, the saturation and color gamut of structural colors are often limited by two factors: high losses and substrate effects. Substrates can have a significant impact on the performance of nanostructures, leading to broader peak shapes due to forward scattering in free space. However, by employing nanostructures without substrates, it is possible to suppress forward scattering and achieve narrower peak shapes. Therefore, Si color filters have the potential to enhance the saturation of structural colors by minimizing substrate effects and reducing material loss.

In 2017, Dong and colleagues proposed a novel structural color design that addresses these challenges while maintaining high resolution. They introduced silicon nanocylindrical structures [94]. A 70 nm thick Si₃N₄ film was employed as an anti-reflective layer, covering the Si substrate and effectively suppressing substrate-induced influences. This approach leverages the Kerker effect, resulting in sharpened reflection peaks and more saturated colors. The structural color achieved by this design covers the visible range and can reach 120% of the sRGB color gamut, thus offering improved color saturation and an expanded color range.

In 2019, Yang and colleagues introduced a metasurface structure color based on a multilayer configuration [95]. The design consists of three layers of nanoblocks on a silicon oxide substrate. From top to bottom, the layers include SiO₂, TiO₂, and Si₃N₄. Remarkably, approximately 85% of the reflected light energy is confined within the desired reflected color. The use of refractive index matching in the multilayer structure effectively suppresses high-order modes associated with short-wavelengths, leading to improved monochromaticity in the reflection spectrum. As a result, higher color gamut structural colors can be achieved.

Experimental results demonstrate color gamuts of 128% sRGB, 95% Adobe RGB, and 68% Rec. 2020. The metasurface design achieves these broad color ranges, while maintaining a spatial resolution of 18,000 dpi.

3.2 Polarization manipulation

Polarization is an intrinsic property of electromagnetic waves, and the manipulation of polarization states has significant applications in modern electromagnetism. In communication and sensing, the conversion of linearly polarized light to circularly polarized light can enhance the beam’s resistance to external environmental changes, scattering, and diffraction, thereby improving the stability and fidelity of signal transmission. Traditionally, polarization control devices rely on birefringence crystals to modulate the polarization state. However, these devices are often bulky, have a narrow operating bandwidth, and offer limited choices of crystal materials. In contrast, metasurface-based polarization conversion is gaining attention due to its flexible design and compact structure.

Highly symmetrical microstructures, such as circularly symmetrical ones, do not alter the polarization state of light. Thus, breaking this symmetry provides additional degrees of freedom for polarization control. Anisotropic micro-nanostructures, such as rectangular [96, 97], elliptical [98], and asymmetric double structures [99, 100, 101], play a key role in achieving polarization control using metasurfaces. Reference [102] illustrates a quarter-sheet metasurface composed of two substructural units (highlighted in red and green). When illuminated with linearly polarized light, these substructural elements generate orthogonal linearly polarized light with equal amplitude and a phase difference of π/2, resulting in circularly polarized light. This metasurface exhibits a high degree of circular polarization (>0.97) across a wide spectral range of lambda = 5 μm–12 μm, although the polarization conversion efficiency is low. By increasing the number of metasurface layers to two or three, the conversion efficiency and operational bandwidth of linearly polarized light to circularly polarized light can be significantly improved. Reference [103] demonstrates a three-layer ABA-type anisotropic metasurface that efficiently converts linearly polarized light to circularly polarized light in the microwave frequency range. As processing technology advances, polarization control based on metasurfaces has been extensively studied [97, 99, 100, 101, 102, 104]. For instance, a TiO₂ metasurface can generate linearly polarized light with varying polarization angles at different propagation distances according to specific requirements [105].

The key to polarization control is breaking the symmetry using anisotropic microstructures. By independently adjusting the eigenmodes associated with orthogonal polarizations, the metasurface can achieve equal transmission or reflection efficiency at a specific frequency while introducing a phase delay of Δφ. Single-layer and multi-layer metasurfaces have been developed to regulate the polarization of linearly polarized light to circularly polarized light, as well as to control the rotation of linearly polarized light [105, 106, 107, 108, 109, 110].

The transmitted electric field component through the metasurface can be described using the Jones matrix. For incident circularly polarized light, the transmitted electric field can be expressed as:

$$\left(\begin{array}{c}{E}_x^t\\ E_y^t\end{array}\right)=\left(\begin{array}{cc}{T}_{\mathit{xx}}& T_{\mathit{xy}}\\ T_{\mathit{yx}}& T_{\mathit{yy}}\end{array}\right)\left(\begin{array}{c}{E}_x^i\\ E_y^i\end{array}\right)=T_{\mathrm{lin}}\left(\begin{array}{c}{E}_x^i\\ E_y^i\end{array}\right)$$E24

For the incident circularly polarized electric field component, the transmitted electric field of the metasurface can be expressed as:

$$\left(\begin{array}{c}{E}_{+}^t\\ E_{-}^t\end{array}\right)=\left(\begin{array}{cc}{T}_{++}& T_{+-}\\ T_{-+}& T_{--}\end{array}\right)\left(\begin{array}{c}{E}_{+}^i\\ E_{-}^i\end{array}\right)=T_{\text{circ}}\left(\begin{array}{c}{E}_{+}^i\\ E_{-}^i\end{array}\right)$$E25

where $T_{++}=\frac{1}{2}\left(T_{\mathit{xx}}+T_{\mathit{yy}}\right)\pm \frac{i}{2}\left(T_{\mathit{xy}}+T_{\mathit{yx}}\right)$ and $T_{\pm \mp }=\frac{1}{2}\left(T_{\mathit{xx}}-T_{\mathit{yy}}\right)\mp \frac{i}{2}\left(T_{\mathit{xy}}+T_{\mathit{yx}}\right)$. In the case of positive incidence, the x and y directions may not align with the primary axes of the metasurface structure, but certain properties of the Jones matrix related to the symmetry of the metasurface structures can be generalized as follows:

1. If the metasurface lacks reflection or rotational symmetry, all components in the Jones matrix may differ.

2. If the metasurface structure possesses mirror symmetry, we have $T_{\mathit{xy}}=T_{\mathit{yx}}$ and $T_{++}=T_{--}$. Furthermore, if the incident light is polarized parallel or perpendicular to the symmetry plane, we have $T_{\mathit{xy}}=T_{\mathit{yx}}=0$.

3. If the metasurface structure has C₄ or C₃ rotational symmetry, we have $T_{\mathit{xx}}=T_{\mathit{yy}}$, $T_{\mathit{yx}}=-T_{\mathit{xy}}$ and $T_{+-}=T_{-+}$.

To achieve orthogonal polarization transformations (x- and y-linearly polarized light, left and right circularly polarized light), the metasurface needs to maximize the off-diagonal components of the Jones matrix. To achieve the transition between linearly and circularly polarized light, the metasurface needs to introduce a phase difference of π/2 between the orthogonal components.

The anisotropic optical response of asymmetric bistructured nanoantennas offers a promising approach to realize metasurface quarter waveplates. Zhao et al. demonstrated this concept by utilizing the detuned plasma resonance of two orthogonal nanoblocks with different sizes, enabling the conversion of circularly polarized light to linearly polarized light in the near-mid-infrared band, as shown in Figure 11 [99]. In their study, when the metasurface was illuminated by horizontally and vertically polarized light, distinct resonance decline regions were observed in the transmission efficiency. These regions corresponded to the dipole resonances in the x and y directions of the nanoblocks, respectively. In the non-resonant region between the dipole resonances, the transmitted light exhibited a phase difference close to π/2 for x- and y-polarized light, with a wavelength-dependent amplitude ratio. As a result, the metasurface effectively converted incident circularly polarized light into transmissively polarized light with a linear polarization exceeding 80% and a polarization angle dependent on the wavelength. To achieve the reverse conversion from linearly polarized light to circularly polarized light, the polarization angle of the incident light needed to be adjusted according to the operating wavelength. This ensured that the transmitted orthogonal components had equal amplitudes along the main orthogonal axis of the nanostructure.

Figure 11.

The transformation from line polarization to circular polarization [99]. (a) the schematic of a metasurface enables the transformation of polarization. (b) Transmitted spectrum of the x and y polarizations.

Another type of metasurface used for polarization conversion is the half-waveplate, which is capable of changing the direction of linear polarization or achieving the mutual conversion of orthogonal circularly polarized light. One example of such a metasurface is the geometric phase nanostructure unit mentioned earlier, which functions as a typical half-waveplate. This metasurface consists of a rectangular nanoantenna array rotated at a 45-degree angle, a dielectric spacer layer, and a metal reflective layer. When x-polarized light is incident, the electric field (E) can be decomposed into orthogonal components (E₁ and E₂) along the two main axes of the nanoantenna. These two components excite the orthogonal electric dipoles of the nanoantenna simultaneously. By ensuring that the reflection coefficients of the detuned major and minor axes are equal and introducing a phase difference of π, the incident x-polarized light can be converted into reflected y-polarized light. The half-waveplate metasurface exhibits a reflectivity exceeding 90% and achieves a polarization conversion efficiency of up to 97% within a bandwidth range of 800 nm (from 900 to 1700 nm). Even at incidence angles up to 40 degrees, the metasurface maintains a long working bandwidth and high conversion efficiency.

The final type of metasurface is designed to generate both radially and angularly polarized light from orthogonal linearly polarized incident light [98]. This metasurface not only converts x-polarized Gaussian beams into radial Bessel-Gauss beams but also transforms y-polarized Gaussian beams into angular Bessel-Gauss beams. Additionally, the metasurface is capable of focusing the transmitted beams. The structural unit of this metasurface is a regular hexagon with a lattice constant of 650 nm. The nanostructure itself is an oval-shaped silicon column with a height of 715 nm, designed to operate at a wavelength of 915 nm. The intensity distribution, when measured with a polarizer, exhibits maximum values along either the diagonal or perpendicular to the diagonal direction. This observation confirms that the transmitted beams are radially polarized and angularly polarized vector beams, respectively. Furthermore, the spot size after focusing is approximately 5 μm.

3.3 Holography

3.3.1 Digital holographic technology

With the continuous advancement of computer technology, the integration of holographic technology with computing has given rise to two innovative branches: digital holographic technology and computational holographic technology. Digital holography employs CCD sensors in lieu of traditional holographic recording media. The interference fringe patterns captured by CCD sensors are then fed into computers. Subsequent to this data input, a process of diffraction reproduction takes place [111, 112, 113, 114, 115, 116]. A visual representation of this procedure can be found in Figure 12. The specific diffraction algorithms employed can be tailored to the specific scenario, utilizing techniques such as vector diffraction, Fresnel diffraction, or Fraunhofer diffraction. The employment of computer-based diffraction reproduction in digital holography offers an array of conveniences, allowing for diverse forms of processing that are otherwise challenging to achieve within the optical domain. This encompasses techniques like frequency domain filtering and other intricate operations. These advantages have propelled digital holography into numerous applications, particularly within the realm of holographic microscopy.

Figure 12.

Schematic diagram of the digital holography technology.

3.3.2 Computational holographic technology

While traditional holographic methods enable the presentation of 3D objects, they necessitate intricate processes involving interference optical paths, encompassing exposure, development, and fixing stages. These procedures are complex and demanding, inhibiting ease of production. Additionally, real-world objects require the generation of object light within the optical path to acquire interference fringes. The creation of holograms for virtual objects remains challenging, thereby limiting the widespread applicability of conventional holography. In contrast, computational holography leverages computer-based computations to generate holograms [117]. This approach obviates the need for physical interference optical paths and seamlessly interfaces with various coding and processing equipment. Figure 13 illustrates this concept. Computational holography calculates the amplitude and phase distribution of holograms corresponding to target images. These distributions are then translated into patterns of light fields using coding equipment or materials. Spatial Light Modulators (SLM), Digital Micromirror Devices (DMD), as well as structures produced through laser direct writing, Electronic Beam Lithography (EBL), and Focused Ion Beam (FIB) represent common coding methods. The perceptibility of an object is contingent upon the light field conveying its information to the human eye. Post-hologram regulation, the modified light field reaches the human eye, aligning the received information with that of the original object. Consequently, individuals perceive the holographic target object’s projection. Notably, no tangible object occupies the observed imaging position, underscoring the mechanics of virtual reality.

Figure 13.

Schematic diagram of the computer-generated hologram (CGH).

Energy efficiency, polarization response, imaging resolution, diffraction angle, and field of view of the final hologram are influenced by the chosen coding material or device. Metasurface structures, 2D artificial metamaterials, manipulate electromagnetic wave amplitude, phase, and polarization by altering structural orientation and size. Metasurface devices possess sub-wavelength thickness and adopt a 2D configuration, rendering their fabrication more convenient than the 3D techniques typical of metamaterials. Their capacity to control electromagnetic waves within sub-wavelength scales enables the surmounting of limitations associated with conventional materials. Consequently, the integration of metasurface material coding into holographic sample computation has gained significant attention. Recent strides in metasurface holography include spanning phase holography, amplitude holography, and simultaneous phase and amplitude control holography technologies.

3.3.3 Metasurface holographic technology

Since the inception of computer-generated hologram (CGH) technology in 1965, there has been a continuous exploration of the influence of coding materials or devices on the ultimate quality of holographic imaging [118]. Frequently employed devices such as Digital Micromirror Devices (DMD) and Spatial Light Modulators (SLM) are known to possess pixel sizes of considerable dimensions, inherently constraining the achievable diffraction angles and field of view of holograms [119]. To illustrate, the widely used SLM with an 8 μm cell size typically offers a field of view angle that scarcely exceeds 5 degrees. In fact, the combination of SLMs and lenses presents a potential means to enhance the field of view and diffraction angle in holographic imaging. However, this approach inadvertently results in smaller holographic image dimensions, which may prove impractical for real-world applications [120]. Enter metamaterials—an emerging class of synthetic structural materials proposed by researchers in recent years. Metamaterials exhibit the capacity to achieve amplitude or phase modulation within scales smaller than the wavelength of light, enabling novel capabilities in holography and other domains.

The ability of metasurface structures to manipulate electromagnetic waves at scales beneath the wavelength, combined with their advantageous ease of processing, positions them as an optimal material for encoding computational holograms [121, 122]. Currently, metasurface-encoded holograms encompass pure phase, pure amplitude, and complex amplitude holograms. Among these, pure phase holograms stand out for their streamlined achievement of continuous phase control. On the other hand, both pure amplitude and complex amplitude metasurface holographic structures often employ discretization techniques to simplify processing complexities. Researchers have shown a stronger affinity towards pure phase holography, largely due to its enhanced energy efficiency. The creation of phase holograms typically employs methodologies such as the Gerchberg-Saxton (GS) algorithm and the point source approach [123, 124, 125]. Phase encoding methods encompass geometric phase (PB phase) and resonant phase. Notably, discrete order amplitude holograms commonly adopt polarization response structures, which effortlessly enable polarization encryption, switching, and coding for color display. In the realm of complex amplitude hologram coding, circular and intricate phase and amplitude holograms emerge as exemplars. Capitalizing on the remarkable energy utilization offered by dielectric metasurfaces and their substantial developmental potential in metasurface holography, significant strides have been made in dielectric metasurface holography technology over recent years. Leveraging the flexibility inherent to holographic principles, the design methodologies of computational holography extend to the creation of other functional devices. Consequently, a myriad of applications stemming from generalized metasurface holography technology have also been explored and developed.

3.4 Other applications

The 2D PhC meta-system boasts a plethora of additional applications, including but not limited to daytime radiative cooling, spectroscopy leveraging meta-atoms, and the exploitation of photonic spin hall effects. Interested readers can delve into these diverse applications through the extensive literature available on the subject.

4. Outlook of 2D PhC meta-system

The future trend is expected to focus on the advancement of metasurfaces, characterized by increased degrees of freedom and enhanced adaptability. Considering the present international development landscape and drawing from relevant design and processing experiences, it is firmly believed that metasurfaces hold significant potential for development in the following four key areas:

The design and fabrication of metasurface devices using monolayer or multilayer 2D materials, such as transition metal sulfides, have recently garnered significant attention. These emerging materials possess remarkable advantages, including atomic layer thickness, and exhibit a multitude of novel photoelectric properties and quantum effects when interacting with photons. However, due to their extremely thin atomic layer scale, the interaction between light and matter is inherently weak. Therefore, addressing the urgent challenge of designing nanostructures that can enhance their interaction with light becomes imperative. Metasurfaces offer a compelling solution by generating unique electromagnetic responses tailored to specific requirements, thereby enabling the manipulation of light-matter interactions. This provides a nearly perfect platform for exploring the exotic physical properties of 2D materials. Consequently, the development of metasurfaces composed of 2D materials has become the primary focus of research. With advancements and maturation in micro-nano research, manufacturing, and processing equipment, we are better equipped to tackle the main difficulty associated with the preparation and transfer of 2D materials. This includes first-principles calculation and processing, which play a crucial role in our endeavor.

Dynamically adjustable metasurface devices represent a captivating area of research. Despite significant progress in regulating the light field of metasurfaces and their applications, the majority of current devices remain static. Once a device is fabricated, its optical response becomes fixed, limiting its adaptability. The design of dynamic metasurfaces with programmability and addressability presents both challenges and opportunities. The key to achieving dynamically adjustable metasurface devices at the sub-wavelength scale lies in independent control of each nanostructure unit within the metasurface. One method entails utilizing liquid crystals or phase change materials during metasurface fabrication. By applying distinct light or electric fields, the response of the metasurface devices can be precisely controlled, enabling dynamic adjustments. Another approach involves the integration of metasurfaces with microelectromechanical systems (MEMS). By manipulating the state of MEMS components, dynamic adjustments of the metasurface properties can be achieved. This integration offers a pathway to real-time fine-tuning and modulation of metasurfaces, thereby enhancing their versatility. Exploring these methodologies not only addresses the pressing need for dynamic metasurface designs but also opens up exciting possibilities for various applications. Leveraging liquid crystals, phase change materials, and MEMS allows researchers to unlock new frontiers in manipulating light-matter interactions, propelling advancements in the realms of optics and photonics.

Expanding the dimensionality of metasurface regulation holds great potential for advancing metasurface physics. Introducing the temporal dimension into the metasurface platform allows us to consider parameters, such as the dielectric constant (ε), varying with both space and time—expressed as ε(x, y, z, t). This interplay between space and time within the space-time metasurface enriches light-matter interactions, leading to the emergence of novel effects, including time-varying polarizability. By incorporating the temporal dimension, the space-time metasurface breaks new ground in understanding the intricate relationship between space and time in the context of metasurface physics. This augmentation paves the way for exploring the dynamic behavior and interdependencies of electromagnetic responses. The manipulation of time-varying parameters not only expands the range of possibilities but also opens avenues for harnessing unique optical phenomena and enabling advanced functionalities in metasurface devices. Investigating the space-time metasurface paradigm holds significant promise for the development of futuristic optical technologies, unlocking new realms of scientific exploration and engineering innovations.

Advanced metasurface design and machining strategies benefit from the utilization of machine learning and deep learning techniques. These branches of artificial intelligence have become powerful tools for simulating and optimizing metasurfaces, thanks to the progress in computational science. By employing these algorithms, researchers can efficiently optimize unconventional optical designs while improving calculation accuracy and saving time. Concurrently, the development of micro-nano processing technology plays a pivotal role in manufacturing unique metasurface structures. Although the testing and research of metasurfaces are predominantly limited to laboratory settings due to the challenges involved in sample processing, advanced processing strategies will drive the scalability and industrialization of metasurface devices.

5. Conclusion

This chapter provides an overview of the fundamental concepts and principles of Mie resonance, BICs, and Fano resonance. It explores how these principles are leveraged in the design and operation of 2D PhC meta-systems, which are a specific type of metasurfaces characterized by periodic structures. The chapter delves into the capabilities of 2D-PhC meta-systems in manipulating electromagnetic waves. The chapter further highlights two important applications of 2D-PhC meta-systems: generating structural colors and polarization manipulation. The ability of these meta-systems to manipulate light has led to significant advancements in the fields of structural color and polarization control. The chapter examines the breakthroughs achieved in these areas, underscoring the profound impact of 2D-PhC meta-systems in these domains.

Acknowledgments

The author would like to extend his heartfelt thanks for the support provided by the China Postdoctoral Science Foundation (No. 2023 M732734), which has been essential for the successful completion of this work.

Conflict of interest

The author declares no conflict of interest.