Second Harmonic Generation in Microwave Regime

Abstract

Second harmonic generation (SHG), a fundamental and widelystudied phenomenon in nonlinear optics, has attracted significant attention for its ability to convert fundamental frequencies into their second harmonics. While the dominant SHG research has been focused on the optical and infrared regimes, its investigation in the microwave range presents challenges due to the requirements of materials with higher nonlinear coefficients and highpower microwave sources. Here, we provide an overview of methods together with underlying mechanisms for SHG in microwave frequencies, and discuss prospects and insights into the future developments of SHGbased technologies. The discussions on both numerical analyses and experimental studies will offer guidance for further SHG research and communication advancements in microwave regime.

Keywords

second harmonic generation (SHG) / microwave / nonlinear / methods

1 Introduction

Nonlinear optics investigate the behavior of light in nonlinear media, in which the optical response is not proportional to the amplitude of the applied electromagnetic (EM) field \(\left\lbrack {1,2}\right\rbrack\) . Second harmonic generation (SHG), a fundamental process in nonlinear optics, occurs when an incident wave interacts with a nonlinear material, producing a signal at twice the frequency of the incident wave [ 3 - 5 ]. The SHG phenomenon was discovered by Franken et al. in 1961 [ 6 ], and then has been extensively studied and utilized in various research fields, including laser technology [ 7 - 10 ], optical communication [ 11 - 13 ], and material science [ 14 - 17 ]. While SHG is well-documented in the optical and infrared regimes, its exploration in the microwave range is relatively slow, primarily attributed to diminished nonlinear responses. In this review, various SHG methods in microwave regime are summarized and listed as much as possible, laying the foundation for further research into the microwave SHG process.

Traditional SHG equipment in microwave regime employs an SHG device and two antennas, one for receiving the incident fundamental frequency signal, while the other one for radiating the generated second harmonic(SH)signal. However, these traditional systems for microwave SHG usually occupy a larger volume, have low generating efficiency, and present difficulties in their integration with the circuit. More importantly, the traditional microwave antenna is typically connected to the microwave channel by feed sources and feeder lines. The introduction of the feeders will cause phase shift and energy dissipation to the antenna system, thus adding difficulties to achieving phase matching. In this case, the methods in optical waves for generating \(\mathrm{{SH}}\) signal with higher generation efficiency and smaller device size can be borrowed and utilized in microwave range.

Typically, the generation of SH involves interactions between EM waves and nonlinear media. In optical frequencies, the nonlinear media include two dimensional (2D) materials [ 16 ], nanotubes [ 18 ], ferroic materials [ 19 - 22 ], spoof surface plasmonics [ 23 ] and metamaterials [ 24 , 25 ]. The methods of utilizing these nonlinear materials or effective media in SHG can be adapted to microwaves [ 26 - 33 ]. The materials and effective media [ 34 ] that exhibit nonlinear polarization or susceptibility responses under the illumination of strong EM fields lead to the generation of harmonic frequencies. In addition to the above methods applicable to both optical and microwave ranges, there are specific techniques tailored for microwave SHG, such as gas plasma [ 35 - 38 ], diode elements [ 39 - 42 ], time-varying system [ 43 - 47 ], diode-integrated metamaterials [ 48 , 49 ], and memristive system [ 50 , 51 ]. These specialized approaches expand the toolkit available for researchers and engineers working in the microwave regime. The diagram in Fig. 1 shows the main methods for SHG both in optical and microwave bands.

In the microwave range, early in 1969, Janes B. Beyer and colleagues demonstrated the generation of SH signal in parallel-plate capacitors filled with gas plasma [ 52 ]. Fifty-five years later, Nikolaev et al. realized phase-matched spin wave SHG in nanoscale ferromagnetic Yttrium Iron Garnet (YIG) waveguides [ 26 ]. The ongoing interest in microwave SHG is driven by its promising applications in frequency conversion [ 48 , 50 ] , signal processing [ 53 , 54 ] , and the development of advanced communication systems [ 40 , 55 ]. Efficient frequency doubling, for instance, enables the creation of higher-frequency signals from existing microwave sources, driving innovations in high-frequency technologies. In addition, the exploration of SHG in microwaves simultaneously offers numerous exciting opportunities. The development of novel nonlinear materials and metamaterials holds great potential to significantly enhance SHG efficiency. Integrating SHG devices with existing microwave technologies could lead to groundbreaking applications in telecommunications, radar systems, and signal processing.

Despite its implementation potential, SHG in microwave range faces several significant challenges, such as the design and optimization of materials with higher nonlinear coefficients, the requirement for high-power microwave sources to induce substantial nonlinear effects, and the increase of overall conversion efficiency [ 56 ]. Additionally, artificial structures and resonances in nonlinear metamaterials can amplify structural dispersion effects, making it critical to precisely control dispersion for phase-match between propagating waves. So the dispersion and absorption properties of metamaterials at microwave frequencies must be carefully controlled to achieve phase-match for efficient coupling of wave energy, maximizing SHG efficiency [ 27 , 57 ]. Considering the above, understanding the mechanism of SHG methods in the microwave domain benefits breaking these barriers and contributes to the broader field of nonlinear electromagnetics, offering valuable insights into wave interactions across various media and conditions.

This review intends to present a comprehensive summary of methods and underlying mechanisms for SHG in microwave frequencies. The article is organized as follows. Section 1 provides an introduction. Section 2 is dedicated to SHG methods that are suitable in both optical and microwave ranges, including ferroic materials, metamaterials and surface plasmon polaritons. Section 3 focuses on the SHG methods that are unique in microwave frequencies, summarized as gas plasma, diode and diode-bridge elements, diode element-integrated metamaterials, memris-tive systems, and time-varying systems. Finally, we provide the conclusion and outlook in Section 4.

2 SHG Methods Applicable in Both Microwave and Optical Bands

We first briefly discuss the methods of realizing SHG in microwave frequencies that are similar to those in optical ranges, as there are many publications on such topic in optics [ 19 , 21 - 23 , 57 ]. Those include the SHG with ferroic materials, with metamaterial possessing negative-index (NI) and epsilon-near-zero (ENZ) properties, and via surface plasmon polaritons (SPPs).

2.1 SHG in Ferroic Materials

SHG phenomena in ferroic materials predominantly arise from the nonlinear polarization or magnetization responses under EM incident fields. In the 1990s, Liu et al. demonstrated SHG in a ferroelectric liquid crystal (FLC) material, i.e., o-nitroalkoxyphenyl biphenylcarboxylate 1 [ 19 ]. This ferroelectric material is specifically synthesized for large nonlinear coefficients, exhibiting pronounced birefringence property and great flexibility of FLC molecular synthesis, resulting in thermodynamically stable materials. In the experiment, various interaction lengths can be obtained for a fixed material orientation by moving the sample relative to the incident beam. The SHG is an oscillating sine function whose period is proportional to the birefringence for a specific propagation direction.

From the perspective of ferroelectric materials’ centrosymmetry in paraelectric phase, a phase transition to the ferroelectric state will break the symmetry, resulting in an efficient SHG. An approach for generating a phase transition is applying an electric field to the crystal and inducing a polarization. This field-induced SHG can be called spontaneous SHG. \(\mathrm{{AgNa}}{\left({\mathrm{{NO}}}_{2}\right)}_{2}\) crystal emerged as a promising ferroelectric nonlinear medium with high second order nonlinear susceptibility for spontaneous SHG. In 2010, Kityk’s group investigated the \(\mathrm{{AgNa}}{\left({\mathrm{{NO}}}_{2}\right)}_{2}\) crystal and observed \(\mathrm{{SH}}\) signals originating from the spontaneous electric field-induced polarization [ 20 ]. The SHG phenomenon is explained through the free energy associated with the interaction between spontaneous electric field-induced polarizations and spatially inhomogeneous electric polarizations generated by propagating optical waves. The relatively high second-order nonlinear optical susceptibilities, combined with various phase-matching geometries, promote these materials as high-performance candidates for nonlinear optical applications. A decade later, the ferroelectric material \({\mathrm{{NbOI}}}_{2}\) nanosheets were employed for \(\mathrm{{SHG}}\) , achieving a conversion efficiency higher than \({0.2}\%\left\lbrack {22}\right\rbrack\) . In this context, the SHG originates from the interaction between anisotropic polarization and the excitonic resonances in \({\mathrm{{NbOI}}}_{2}\) , with strain- and electric-tunable nonlinear responses.

Besides the above ferroelectric materials, ferromagnetic materials can also be employed to obtain efficient SHG. For example, engineering the dispersion spectrum of spin waves in nanoscale Yttrium Iron Garnet (YIG) waveguides enables the generation of highly efficient resonant SH signals [ 26 ]. In the designed YIG nano-waveguides, the elliptical precession of the magnetization leads to a significant double-frequency component in dynamic magnetization, resulting in the second-harmonic spin wave generation. While Refs.[ 19 ] and [ 20 ] involve traditional angular phase matching, Ref.[ 26 ] achieves phase matching by adjusting the thickness, because the nano-waveguide introduces mode dispersion as a new dimension for matching the phase velocity. By tailoring the thickness and dispersion properties, the interaction strength and the SHG efficiency can be enhanced, paving the way for advanced applications in spintronics and nonlinear optics.

2.2 SHG in Epsilon-Near-Zero and Negative- Index Metamaterials

Metamaterials, artificially-engineered periodic or non-periodic structures, exhibit unique SHG and high harmonic generation abilities that are controllable by unit cell design and low-power external stimuli [ 29 , 31 ] . Notably, well-designed meta-materials with adaptive dispersion properties facilitate both phase-matching and backward wave propagation. Then, phase matching, essentially energy conservation, makes it possible to manipulate EM waves at will.

The ENZ metamaterial [ 57 ] is characterized by an infinite phase velocity, indicating that the wave number is nearly zero and there is negligible phase change during wave propagation. This property enables the achievement of ideal phase-matching conditions. The NI metamaterial [ 49 ], where the power flows opposite to the phase velocity, can achieve a nonlinear mirror that will generate \(\mathrm{{SH}}\) signals in backward direction, responding to the fundamental incident wave [ 58 - 60 ].

These ENZ and NI metamaterials can be achieved with varactor-diode-loaded unit cells, and a waveguide is usually used to support the SHG modes [ 61 - 63 ]. Alec et al. arranged varactor-diode loaded split-ring-resonator (SRR) into a periodic array in an aluminum waveguide to realize a strongly enhanced SHG, as shown in Fig. 2 (a)[ 49 ]. The right bottom shows the unit cell of varactor-diode loaded SRR. The structural parameters are well-designed to ensure that the resonant frequency of the loaded SRR is below the waveguide’s cut-off frequency so that the resonant wave can propagate in the waveguide. The SH spectra are supported by nonlinear transfer matrix method calculations. Through transfer matrix method, the second-order nonlinear susceptibility is extracted from the SH spectra. A backward SHG phenomenon is achieved through nonlinear-optical mirror effect in a nonlinear NI metamaterial.

To improve the SHG efficiency, in 2014, Alù’s group used double ENZ cross-slit nonlinear channels with anomalous tunneling effect to achieve SHG in both forward and backward directions [ 57 ], as shown in Fig. 2 (b). The metamaterial consists of cross-slit channels loaded with nonlinear dielectric and perfect electric conductor (PEC) slabs. Fabry-Perot (FP) transmission resonance [ 64 , 65 ] and ENZ tunneling resonance will be excited simultaneously in this configuration, strengthening the field amplitude interacting with nonlinear dielectric slab and enhancing the nonlinear permittivity in the dielectric material. One of the FP transmission resonance is the SH mode, while the other one is the fundamental frequency (FF) mode. In this ENZ medium, the refractive indices at both frequencies are effectively equal to zero and phase matching is automatically achieved, with an infinite coherence length if losses are negligible. This provides a new means of enhancing SHG efficiency.

2.3 SHG via Surface Plasmon Polaritons

Moreover, metamaterials that support SPPs are also capable of generating SH SPPs in both optical and microwave domains. In optical frequencies, Giuseppe et al. studied hyperbolic plas-monic nanorod metamaterial slabs which perform free-electron nonlinear responses [ 23 ]. The nonlinear polarization of free-electrons can be described by the hydrodynamic mode. Due to rich modes supported by the hyperbolic anisotropic slab waveguide, the enhanced SHG can be observed in a broad spectral range. In microwaves, surface plasmon polaritons make it possible to realize SHG with ultrathin geometry and ultralow loss.

As a new trend, Liu et al. obtained backward SHG through NI conformal surface plas-monic metamaterials in 2018 [ 27 ]. The NI meta-materials consist of ultrathin mirror-symmetrically corrugated metallic strips, as illustrated in Fig. 2 (c). The nonlinearity was introduced by the nonlinear current-voltage relationships of the diode SMV-1231-079 LF [ 66 ]. By well designing the metamaterial structure, the SHG occurred in the negative-index dispersion region with a peak efficiency of 0.067%.

While the SHG phenomena in above two references [ 23 ] and [ 27 ] are fixed to one operating frequency, in Ref.[ 30 ], Cui’s group make the operating frequency of SHG adjustable. This dynamically controllable SHG is achieved with varactor diodes, whose capacitance can be manipulated through applied voltages. In fact, they achieved both forward and backward phase-matching SHGs in the reconfigurable plasmonic meta-waveguide with engineered capacitance, as shown in Fig. 2 (d)[ 30 ]. Maximum conversion efficiencies of \({0.80}\%\) and \({0.14}\%\) are obtained for backward and forward SHGs, respectively. This voltage-control method opens new techniques for improving the SHG efficiency and enabling the flexible applications of tunable SHG devices.

3 SHG Methods Tailored for Microwave Frequencies

There are some unique methods of obtaining SHG in the microwave frequencies, such as using gas plasma, diode elements, diode-integrated metamaterials, memristive system, time-varying systems, etc. In the following, we will discuss these methods in detail.

3.1 SHG in Gas Plasma

Harmonic generation in gas plasma has been a key area of research in the microwave range, with its origin tracing back to the 1940s [ 67 ]. This process involves the conversion of microwave radiation into its higher-order harmonics upon interaction with the gas plasma, presenting important implications for various applications.

In 1965, J. H. Krenz and Kino from Stanford University obtained an SHG of \({25}\%\) conversion efficiency in gas plasma [ 68 ]. Four years later, J. Asmussen et al. from Michigan State University found that microwave SHG can be realized in a plasma capacitor [ 52 ], of which the structure is shown in Fig. 3 (a). They developed a nonlinear one-dimensional mathematical model which contains the reactive nonlinearities from spatial variation of electric field. They found that harmonic generation is dependent on the working gas and gas pressure. Since then, to generate high-pressure and microscale plasmas, significant efforts have been undertaken [ 36 , 37 , 69 ] . A major advancement in micro-plasma generation techniques involves the use of resonant structures, such as split ring resonators (SRRs), dielectric resonators, and photonic crystal cavities. For example, Jeffrey’s team [ 36 ] obtained radiated and conducted harmonics in micropla-sma, which was generated in a \({150}- \mu \mathrm{m}\) discharge gap of an SRR.

The physical origin or mechanism responsible for harmonic generation in plasmas can be attributed to several factors. When microwaves propagate through a gas plasma, as the gas molecules or particles become ionized, the interaction between EM waves and the charged ions or electrons can induce nonlinear effects [ 70 ], such as harmonic generation. This nonlinearity arises primarily from the ponderomotive force [ 69 ], which acts on the plasma particles within the sheath region. The key mechanism behind harmonic generation is the nonlinear motion of electrons in the oscillating electric field of microwaves, often referred to as spatial sources [ 37 ]. In addition, the modulation of electron density in the plasma can also contribute to harmonic generation. Variations in electron density influence the refractive index of the plasma, causing generation of harmonics by incident microwaves.

As noted by Krenz and Kino [ 68 ], the motion equation of plasma electrons, which indicates the relationship between current and velocity, plays a crucial role in the nonlinear harmonic generation process. A theory of plasma harmonic generation is developed, and the velocity of the electrons and the electric field can be written as the superposition of static and oscillating terms, in the form of

(1) \[\mathbf{v}= {\mathbf{v}}_{0}+ \frac{1}{2}\mathop{\sum }\limits_{{p = 1}}^{\infty }\left({{\mathbf{v}}_{p}{\mathrm{e}}^{\mathrm{j}{p\omega t}}+ {\mathbf{v}}_{p}{}^{* }{\mathrm{e}}^{-\mathrm{j}{p\omega t}}}\right)\]

(2) \[\mathbf{E}= {\mathbf{E}}_{0}+ \frac{1}{2}\mathop{\sum }\limits_{{p = 1}}^{\infty }\left({{\mathbf{E}}_{p}{\mathrm{e}}^{\mathrm{j}{p\omega t}}+ {\mathbf{E}}_{p}{}^{* }{\mathrm{e}}^{-\mathrm{j}{p\omega t}}}\right)\]

where \({\mathbf{v}}_{p}\) and \({\mathbf{E}}_{p}\) denote the electron velocity and the electric field of the \(p\) -th harmonic, respectively. Both the external electric field excitation and electron density play important roles in SHG. Each factor influences second-harmonic electron density, which in turn affects the electron’s velocity. The frequency conversion efficiency of \({25}\%\) is consistent with the results reported by Pantell [ 71 ], who has pointed out that the maximum achievable conversion efficiency to the \(m\) -th harmonics is \(1/{m}^{2}\) .

In line with the explanation provided in Krenz’s work, J. Asmussen et al. found that it is the spatial variation of electron density and radiofrequency electric field that dominate the SHG process in 1969 [ 52 ]. In their experiment, the electron temperature \({T}_{\mathrm{e}}\) does not vary significantly from a constant value. Then, considering the one-dimensional condition, there are equations describing the plasma system

(3) \[\frac{\partial {n}_{\mathrm{e}}}{\partial t}+ \frac{\partial }{\partial x}\left({{n}_{\mathrm{e}}{v}_{x}}\right)= 0 \]

(4) \[\frac{\partial {v}_{x}}{\partial t}+ {v}_{x}\frac{\partial {v}_{x}}{\partial x}= -\frac{e}{m}{E}_{x}- \frac{{3k}{T}_{\mathrm{e}}}{m{n}_{\mathrm{e}}}\frac{\partial {n}_{\mathrm{e}}}{\partial x}\]

(5) \[\frac{\partial {E}_{x}}{\partial x}= \frac{e}{{\epsilon }_{0}}\left({N -{n}_{\mathrm{e}}}\right)\]

where \({n}_{\mathrm{e}}\) is electron number density, \({v}_{x}\) denotes electron fluid velocity along \(x\) direction, \({E}_{x}\) is electric field, \(N\) represents ion number density, \(k\) and \({\epsilon }_{0}\) denote the Boltzmann constant and permittivity in vacuum, \(e\) and \(m\) are the charge and mass of electrons. Under microwave excitation, the velocity \({v}_{x}\) and electric field \({E}_{x}\) can be written in the similar form of Eqs.(1) and (2)[ 68 ]. Then, given the fact that there is no static contribution under equilibrium condition, i.e., the direct current (DC) electron velocity \({v}_{0}= 0\) and static electric field \({E}_{0}= 0\) , Eqs.(3)-(5) can be written as follows

(6) \[\mathrm{j}{p\omega }{v}_{p}+ \frac{1}{2}{v}_{p - 1}\frac{\mathrm{d}{v}_{p - 1}}{\mathrm{\;d}x}= -\frac{e}{m}{E}_{p}- \frac{{3k}{T}_{\mathrm{e}}}{m{n}_{0}}\frac{\mathrm{d}{n}_{p}}{\mathrm{\;d}x}\]

(7) \[\mathrm{j}{p\omega }{n}_{p}+ {n}_{0}\frac{\mathrm{d}{v}_{p}}{\mathrm{\;d}x}= 0 \]

(8) \[\frac{\mathrm{d}{E}_{p}}{\mathrm{\;d}x}= -\frac{e}{{\epsilon }_{0}}{n}_{p}\]

where \({n}_{p}\) represents the electron number density of the \(p\) -th harmonics. From the above equations, it is evident that the \(\mathrm{{SH}}\) velocity \({v}_{2}\) (with \(p = 2\) ) is influenced not only by the variation of \(\mathrm{{SH}}\) electric field \({E}_{2}\) and electron number density \({n}_{2}\) , but also by the motion of electrons \({v}_{1}\) at the FF. This means that, under microwave excitation, the oscillation of electrons will generate the SHs.

Regarding the conversion efficiency, it was observed that a second radiofrequency (RF) field within the ultra-high frequency (UHF) range improved the impedance matching at the S-band. The introduction of a modest UHF power increases the discharge stability and sustains the discharge process even in the absence of S-band input. However, an excessive UHF power exceeding \({200}\mathrm{\;{mW}}\) leads to a reduction in conversion efficiency due to the thermal loss.

In the aforementioned studies, SHG has been explored primarily from the perspective of high-frequency oscillations. However, the contributions of ion motions, which are seen as the resistive sources [ 37 ] of nonlinearity in plasma, have been overlooked. In 1990, Brunel developed a model that incorporates multiphoton ionization to predict the generation of odd harmonics [ 2 ]. Multiphoton ionization takes place around the peaks of electric field, leading to the production of plasma current from the ionization process, which subsequently drives harmonic generation.

In their theoretical analysis, the following model is used, shown as

(9) \[\frac{1}{2}n\frac{\partial }{\partial t}\left({{m}_{\mathrm{e}}{\mathbf{v}}_{\mathrm{f}}^{2}}\right)+ \frac{1}{2}{m}_{\mathrm{e}}{\mathbf{v}}_{\mathrm{f}}^{2}\frac{\partial n}{\partial t}= \mathbf{J}\cdot \mathbf{E}\]

(10) \[{\mathbf{v}}_{\mathrm{f}}= \frac{1}{N}\mathop{\sum }\limits_{{\alpha = 1}}^{N}{\mathbf{v}}_{\alpha }\]

where \({\mathbf{v}}_{\mathrm{f}}\) is the fluid velocity, \({\mathbf{v}}_{a}\) is the \(\alpha\) -th electron velocity, \(\mathbf{J}\) represents the electric current density, \(n\) is the electron number density, \({m}_{\mathrm{e}}\) is mass of electron, and \(N\) represents the total electron number in a small volume \({\Delta V}\) . The first term on the left in Eq.(9) denotes the change of plasma kinetic energy from the coherent motion, while the second term represents the heat dissipation during the ionization process. In the latter, heat energy will be consumed into kinetic energy for electrons. In their experiment, the laser pulses propagate halfway through the gas slab, with plasma emerging from the right side of the gas slab. Fig. 3 (b) presents a step-like plasma density profile, which serves as the source of harmonic generation. The spectrum of the emerging EM field, which was collected from the right side of the gas slab, is illustrated in Fig. 3 (c).

Furthermore, S. K. Sinha and A. C. Sinha studied and optimized the SHG output in parallel plate waveguides, highlighting the significant influence of waveguide geometry on the output modes [ 38 ]. In a plasma waveguide, the second-and third-harmonic signals are attributed to the ponderomotive force, as proposed by Sharma [ 72 ]. In 2007, Fu’s team promotes SHG in plasma-filled waveguide in getting high-power and high-frequency RF sources with tunable properties [ 35 ]. For the SH, Helmholtz’s equation can be written as

(11) \[{c}^{2}{\nabla }^{2}{\mathbf{E}}_{2}+ 4{\omega }^{2}{\epsilon }_{p}\left({2\omega }\right){\mathbf{E}}_{2}= \\- \frac{e}{4{m}_{e}}\frac{{\omega }_{p}^{2}}{{\omega }^{2}}\nabla \left({{\mathbf{E}}_{1}\cdot {\mathbf{E}}_{1}}\right)- \frac{e}{m}{\mathbf{E}}_{1}\nabla \cdot {\mathbf{E}}_{1}\]

where \(c\) is the light speed in vacuum, \({\epsilon }_{p}\) is the plasma permittivity, \({\omega }_{p}\) represents the plasma frequency, and \({\mathbf{E}}_{1}\left({\mathbf{E}}_{2}\right)\) is the electric field of FF (second harmonic). We consider that both \({\mathrm{{TE}}}_{0n}\) (transverse electric) and \({\mathrm{{TM}}}_{0n}\) (transverse magnetic) fundamental harmonic mode excitations in cylindrical coordinate system can be written in the form of

(12) \[{\mathrm{{TE}}}_{0n}\left\{\begin{matrix}{E}_{r}= 0 \\{E}_{\theta }= A{J}_{1}\left({{k}_{\mathrm{c}}r}\right){\mathrm{e}}^{-\mathrm{j}{k}_{z}z}\\{E}_{z}= 0 \end{matrix}\right.\]

(13) \[{\mathrm{{TM}}}_{0n}\left\{\begin{matrix}{E}_{r}= \mathrm{j}{k}_{\mathrm{c}}A{J}_{1}\left({{k}_{\mathrm{c}}r}\right){\mathrm{e}}^{-\mathrm{j}{k}_{z}z}\\{E}_{\theta }= 0 \\{E}_{z}= {k}_{z}A{J}_{0}\left({{k}_{\mathrm{c}}r}\right){\mathrm{e}}^{-\mathrm{j}{k}_{z}z}\end{matrix}\right.\]

where \({J}_{1}\left(\cdot \right)\) and \({J}_{0}\left(\cdot \right)\) are the first and zeroth order Bessel function of the first kind, \(A\) is the amplitude of the field, \(\mathrm{j}\) is the imaginary unit, \({k}_{\mathrm{c}}\) is the cut-off wavenumber, and \({k}_{z}\) is the wavenumber in \(z\) direction. And by solving Eq.(9) in the cylindrical coordinate system, it shows that the generated \(\mathrm{{SH}}\) modes in cylinder waveguide are both transverse magnetic (TM)- like waves. The ratios of amplitude of the generated \(\mathrm{{SH}}\) to the fundamental modes are shown in Fig. 4 (a) and Fig. 4 (b). This phenomenon arises from the reverse relationship between the nonlinear term and frequency.

In 2016, Hopwood’s team experimentally excited nonlinearities in micro-plasmas using microwaves and provided a theoretical explanation of this phenomenon with a 2D fluid model [ 36 ]. The harmonic generation in this work originates from the sheath oscillation. As shown in Fig. 5 (a), the micro-plasma generator is formed by a split ring resonator on a copper-coated RogersTMM \({}^{\circledR }{10}\) substrate. The1-mm wide SRR is connected to the substrate edge by a copper strip, which serves as the input port. To generate stronger harmonics, plasma electrodes with unequal surface areas are usually employed. A dielectric plasma limiter is placed on the top of SRR to flexibly control the plasma volume and electrode symmetry. As microwave propagates through the micro-plasmas, the electrons oscillate and interact with the EM field, leading to the generation of harmonics at multiples of FF. The sheath width of micro-plasmas can be modulated by the electrode potential, which will be further amplified by electrode asymmetry. As shown in Fig. 5 (b), the presence of micro-plasmas surely enhances the harmonic generation, especially for the harmonics beyond the second order.

Then, in 2018, D. Pederson et al. investigated the nonlinear behaviors of plasmas by coupling the electron momentum equations to EM wave simulation within a resonator system [ 37 ]. Dense and subwavelength plasmas were generated in the resonator, with electron dynamics exhibiting nonlinearity due to three key physical factors: charge density, magnetic field, and inertial term. The authors developed a theoretical framework based on the perturbation equation to analyze these effects. Their findings indicated that the harmonic power is primarily produced by the inertial factor in the electron equation, agreeing with S. K. Sinha and A. C. Sinha’s conclusion [ 38 ]. In other words, the inertial contribution plays a dominant role in harmonic generation, followed by the influence of the magnetic field.

In summary, the inertial term plays a crucial role in plasma SHG, followed by the influence from spatial variation of electron density and EM fields. Additionally, the resistive ionization process also affects the SHG mechanism. Reversely, harmonic generation can be used as a diagnostic tool to probe the properties of plasma, such as electron density and temperature. By analyzing the harmonics, researchers can infer some important plasma parameters, such as refractive index and plasma density. Furthermore, higher-frequency wave generation from the source has potential applications in high-frequency communication and radar technologies. Despite the promising applications, challenges exist in efficiently generating and controlling harmonics. Either plasma’s condition, such as density and temperature, or the power of the external microwave source, should be optimized to open up new possibilities for more practical applications.

3.2 SHG with Diode Elements

Diodes, such as varactor diodes and Schottky diodes, play a critical role in microwave circuits. Their nonlinear capacitance-voltage (C-V) characteristics make them suitable components for harmonic generation. Varactor-diodes, in particular, are highly efficient for this purpose due to their voltage-dependent capacitance \(\lbrack {42},{55}\) , 73-76]. When driven by a microwave signal, the capacitance variation induces a nonlinear response, resulting in the generation of harmonics at multiples of the input frequency. Schottky diodes also exhibit pronounced nonlinear behaviors, owing to the rapid switching capability and low forward voltage drop. When subjected to high-frequency signals, they generate harmonics as a result of the junction properties in electric circuits [ 77 , 78 ] .

Schottky diodes available by sputtering process are in common use for harmonics generation. For example, in 2011, Huang’s group fabricated a right/left-handed coplanar waveguide-transmission line (RLH CPW-TL) composite on the gallium arsenide (GaAs)-based Schottky diode to generate harmonics, which is shown in Fig. 6 (a)[ 79 ]. This is the first time that a monolithic circuit was integrated into the CPW-TL structure in microwaves and generated an SH signal in K-band frequencies. Maximum conversion efficiency is \({4.7}\%\) and the harmonic strength is easily influenced by the frequency variation because RLH CPW-TL has anomalous dispersion. Three years later, Huang et al. applied this design and fabrication technology to W-band frequencies, and an \(\mathrm{{SH}}\) of \({77.6}\mathrm{{GHz}}\) was realized with a \({38.8}\mathrm{{GHz}}\) signal as input [ 80 ]. This SHG technology can be expanded to a higher-frequency range, such as terahertz (THz) frequencies, by changing and optimizing the structural parameters.

Besides Schottky diodes, the commercially available varactor diodes also exhibite non-neligi-ble nonlinear responses in SHG process. By positioning two varactor-diodes in the gap of two coupled SRRs and arranging them in opposite directions, as shown in Fig. 6 (b), researchers are able to generate SHs [ 48 ]. In this operation mechanism, FF resonance is seen as the symmetric mode, while the SH resonance is the asymmetric mode. The two oppositely arranged diodes ensure that the induced SH current flows in the asymmetric mode while the fundamental current flows in the fundamental symmetric mode. Otherwise, both fundamental and SH currents flow in the symmetric mode, i.e., resonance at fundamental EM wave only. The coupling of the two resonance modes through the complementary split rings improves the SHG conversion.

However, GaAs Schottky diodes [ 80 - 82 ], varactor-diodes [ 83 ] and n-metal-oxide-semiconductor (nMOS)[ 54 ] doublers all have conversion loss, which diminishes the overall conversion efficiency. In addition, the conversion loss depends on the bandwidth and frequencies. To deal with the loss problem, transistor multipliers that require lower input power and provide conversion gain can be employed for SHG and higher harmonic signal generation. Timothy et al. designed two gallium nitride (GaN) doublers with conversion gain at a \({10}\mathrm{{dBm}}\) input power and a tripler with \({8.6}- {11.5}\mathrm{{dBm}}\) output power at \({19}\mathrm{{dBm}}\) input across the W-band [ 53 ]. The cascading of a post-multiplication amplifier stage facilitates the conversion gain. This is the first time that GaN monolithic microwave-integrated circuit doublers with conversion gain in W-band have been demonstrated.

Another approach to achieve SHG is introducing nonlinear active chips into the waveguide or transmission line systems. In 2016, Cui’s group members generated SH SPPs by inserting a nonlinear active chip into plasmonic waveguides [ 28 ]. As is shown in Fig. 6 (c), the commercial nonlinear chip, AMMC-6120, functions as a frequency multiplier. Thin metallic corrugated strip structures were fabricated asymmetrically on the bottom and top of a dielectric slab. A significant gain, around 10 from 6 to \({10}\mathrm{{GHz}}\) , was obtained for SH SPP conversion. This active SHG technology can be further applied to controlling high-order harmonics generation in microwaves and SPP mixers at \(\mathrm{{THz}}\) frequencies.

3.3 SHG with Diode-Integrated Metamaterials

As is known, field localization and enhancement can strengthen the interaction between EM waves and nonlinear elements, thus enhancing the nonlinear responses and improving the conversion efficiency of SHG. Considering this, nonlinear components are typically positioned at the capacitive regions of metamaterials to boost the SHG. Then, through the effective medium theory, the effective nonlinear susceptibilities of the diode-integrated structure can be obtained to describe the EM responses of the designed device when nonlinear diodes are integrated into meta-materials or metasurfaces [ 48 , 49 , 84 - 87 ].

Using an analytical effective medium theory, David R. Smith et al. analyzed the magnetic response of nonlinear component integrated SRRs and obtained the effective magnetic susceptibility [ 86 ]. Taking the SRRs unit cell in Fig. 7 (a), which exhibits magnetic response to EM fields, as an example, the magnetization \(\widetilde{\mathbf{M}}\left( t\right)\) is the superposition of all the dipole moments \(\widetilde{\mathbf{m}}\left( t\right)\)

(14) \[\widetilde{\mathbf{M}}\left( t\right)= N\widetilde{\mathbf{m}}\left( t\right)= N{\mathbf{I}}^{\left( 1\right)}A = N\frac{\mathrm{d}\widetilde{\mathbf{Q}}\left( t\right)}{\mathrm{d}t}A =\\{NA}{C}_{0}\frac{\mathrm{d}\widetilde{\mathbf{q}}\left( t\right)}{\mathrm{d}t}\]

where \(N\) is the magnetic dipole moments’ volume density, \(A\) is the effective area of the equivalent magnetic dipole circuit. The equivalent circuit is shown in Fig. 7 (b), where the diode can be integrated in the capacitive region.

Following the perturbative expansion for \(\widetilde{\mathbf{q}}\left( t\right)\) , the magnetization \({\widetilde{M}}_{y}\left( t\right)\) can be written as

(15) \[{\widetilde{M}}_{y}\left( t\right)= -\mathrm{j}{NA}{C}_{0}\left\lbrack {\mathop{\sum }\limits_{{n =- \Lambda }}^{\Lambda }{\omega }_{n}{q}^{\left( 1\right)}\left({\omega }_{n}\right){\mathrm{e}}^{\left(-\mathrm{j}{\omega }_{n}t\right)} +}\right.\\\mathop{\sum }\limits_{r}{\omega }_{r}{q}^{\left( 2\right)}\left({\omega }_{r}\right){\mathrm{e}}^{\left(-\mathrm{j}{\omega }_{r}t\right)} +\\\left.{\mathop{\sum }\limits_{s}{\omega }_{s}{q}^{\left( 3\right)}\left({\omega }_{s}\right){\mathrm{e}}^{\left(-\mathrm{j}{\omega }_{s}t\right)}}\right\rbrack \\{M}_{y}^{\left( 1\right)}\left({\omega }_{n}\right)= {\chi }_{yy}^{\left( 1\right)}\left({\omega }_{n}\right){H}_{y}\left({\omega }_{n}\right)\\{M}_{y}^{\left( 2\right)}\left({\omega }_{r}\right)= \mathop{\sum }\limits_{\left( nm\right)}{\chi }_{yyy}^{\left( 2\right)}\left({{\omega }_{r};{\omega }_{n},{\omega }_{m}}\right){H}_{y}\left({\omega }_{n}\right){H}_{y}\left({\omega }_{m}\right)\]

(18) \[{M}_{y}^{\left( 3\right)}\left({\omega }_{s}\right)= \mathop{\sum }\limits_{\left( nmp\right)}{\chi }_{yyyy}^{\left( 3\right)}\left({{\omega }_{s};{\omega }_{n},{\omega }_{m},{\omega }_{p}}\right){H}_{y}\left({\omega }_{n}\right)\\{H}_{y}\left({\omega }_{m}\right){H}_{y}\left({\omega }_{p}\right)\\{\chi }_{yy}^{\left( 1\right)}\left({\omega }_{n}\right)= \frac{{\omega }_{0}^{2}{\omega }_{n}^{2}{\mu }_{0}{A}^{2}V{C}_{0}}{D\left({\omega }_{n}\right)} \\{\chi }_{yyy}^{\left( 2\right)}\left({{\omega }_{r};{\omega }_{n},{\omega }_{m}}\right)= -\mathrm{j}a\frac{{\omega }_{0}^{6}\left({{\omega }_{n}+ {\omega }_{m}}\right){\omega }_{n}{\omega }_{m}{\mu }_{0}^{2}{A}^{3}V{C}_{0}}{D\left({\omega }_{n}\right) D\left({\omega }_{m}\right) D\left({{\omega }_{n}+ {\omega }_{m}}\right)} \]

(21) \[{\chi }_{yyyy}^{\left( 3\right)}\left({{\omega }_{s};{\omega }_{n},{\omega }_{m},{\omega }_{p}}\right)= \\- \frac{2}{3}{a}^{2}\frac{{\omega }_{0}^{10}\left({{\omega }_{n}+ {\omega }_{m}+ {\omega }_{p}}\right){\omega }_{n}{\omega }_{m}{\omega }_{p}{\mu }_{0}^{3}{A}^{4}V{C}_{0}}{D\left({\omega }_{n}\right) D\left({\omega }_{m}\right) D\left({\omega }_{p}\right) D\left({{\omega }_{n}+ {\omega }_{m}+ {\omega }_{p}}\right)} \\\left\lbrack {\frac{1}{D\left({{\omega }_{n}+ {\omega }_{m}}\right)} +\frac{1}{D\left({{\omega }_{n}+ {\omega }_{p}}\right)} +\frac{1}{D\left({{\omega }_{m}+ {\omega }_{p}}\right)}}\right\rbrack +\\ b\frac{{\omega }_{0}^{8}\left({{\omega }_{n}+ {\omega }_{m}+ {\omega }_{p}}\right){\omega }_{n}{\omega }_{m}{\omega }_{p}{\mu }_{0}^{3}{A}^{4}V{C}_{0}}{D\left({\omega }_{n}\right) D\left({\omega }_{m}\right) D\left({\omega }_{p}\right) D\left({{\omega }_{n}+ {\omega }_{m}+ {\omega }_{p}}\right)} \]

where \(D\left({\omega }_{n}\right)= {\omega }_{0}^{2}- {\omega }_{n}^{2}- \mathrm{j}\gamma {\omega }_{n}\) with \(\gamma\) being the damping parameter. Thus, nonlinear component-or material-integrated metamaterials can be described as a continuous homogeneous medium with effective susceptibility and polarizability. Alec et al. in Smith’s group experimentally demonstrated the magneto-electric coupling in a varactor-diode-loaded metamaterial in microwave frequency. This nonlinear coupling takes the form of SHG via \({\chi }_{emm}^{\left( 2\right)}\left({{2\omega };\omega ,\omega }\right)\left\lbrack {87}\right\rbrack\) .

In 2008, Shadrivov and his group members fabricated nonlinear metamaterials formed by double SRRs and wires with varactor diodes integrated into the capacitive region. The magnetic resonance can be modulated dynamically via input power [ 84 ]. The nonlinear structure is illustrated in Fig. 8 (a). The varactor-diode (Sky-works SMV-1405) introduces nonlinear magnetic dipole in the proposed structure, causing an effective nonlinear magnetization and permeability. As illustrated in Fig. 8 (b) and Fig. 8 (c), this device presents power-dependent transmission properties and selective orders of harmonic generation. In 2010, Poutrina et al. applied varactors to SRR structure, realizing the third- and fifth-order harmonic generation [ 86 ]. The designed SRR structure containing two gaps that are both integrated with varactors is shown in Fig. 8 (d).

The authors determined the effective constitutive parameters and effective susceptibilities that are formed by the metacrystals via the effective medium theory, and then explained the generation of the third- and fifth-order harmonics, as shown in Fig. 8 (e). Moreover, varactor-diode-loaded SRR was utilized by Huang et al. to obtain sum frequency generation in 2011 [ 85 ], which was verified by experiments and explained through effective medium theory. The above works are all based on the effective medium theory after the introduction of nonlinear electric elements, providing a potential tool for high-harmonic generation and wave mixing in metacrys-tals.

For diode-integrated harmonic generation methods, there exist some challenges. On one hand, managing the heat that is generated in high-power harmonic generation is crucial for reliable operation. Advanced cooling techniques and thermal management strategies are being developed to handle this problem. On the other hand, with the trend towards miniaturization in electronic systems, integrating harmonic generation circuits into compact modules is a key focus. This involves developing integration leveraging advancements in microfabrication.

3.4 SHG in Memristive System

Memristors and diode bridges, as common circuit elements, play significant roles in the realization of SHG devices. It is well known that, a diode bridge is the straightforward way to realize a frequency doubler due to the full-wave rectified output. In this case, the SHG efficiency is \({4.5}\%\) and the diode bridge can transform about \({18.9}\%\) of the input power into all higher harmonics [ 88 ]. To suppress the odd harmonics and obtain only the even harmonics, the output signal is usually adjusted symmetrically [ 88 ]. Mem-ristors [ 89 ] are resistive circuit elements with memory, which comprise the memcapacitive and meminductive [ 86 ] systems, so-called capacitors and inductors with memory (depending on the history).

In 2012, G. Cohen et al. used a memristive bridge circuit, illustrated in Fig. 9 (a), to actively generate second and higher harmonics, more efficiently compared to the standard diode bridge [ 51 ]. The SHG in this memristive system can be explained via a resistance switching model. To obtain the considerable generation of higher harmonics, the voltage source should be increased and the frequency ought to be decreased. Advantageously, the operation voltage for memristive circuits is lower than that in diode circuits, in which there exists a \({0.7}\mathrm{\;V}\) barrier voltage for the p-n junction [ 88 ]. This work adopts a voltage source as the excitation, generating heat losses during the operation. So, in order to deal with this problem, Toshihiro’s team studied passive or non-volatile conductive bridge memory that allows higher harmonic generation [ 50 ]. In this system, the resistance switching characteristics are modulated by the RF input. The designed structure is displayed in Fig. 9 (b). They found that the transmission was independent of the resistance state for RF frequencies from \({500}\mathrm{{MHz}}\) to \({6.5}\mathrm{{GHz}}\) . Notably, the direct-current resistance is irreversible during the heat cycling while the frequency multiplication is reversible, indicating that the nonlinear resistance is attributed to a process that is reversible for heat cycling.

The above works all show incident dependence of second or higher harmonic signals. What’s more, Ref.[ 50 ] experimentally demonstrated the input power-dependent properties, while there’s only theoretical analysis in [ 51 ]. The conductance of a memristor shows nonlinear relation with current or voltage, and also depends on its internal state variables, such as the resistance value or ion distribution of the channel. This dynamic nonlinearity makes it easier to generate higher harmonics in nonlinear processes. The memory effect can accumulate or amplify specific components of a harmonic signal.

3.5 SHG with Time-Varying Systems

Besides the above SHG phenomena induced by nonlinear properties, time can serve as a new degree of freedom to realize and control SHG in microwave. A time-varying system is a system where parameters, such as permittivity, conductivity and boundary condition change dynamically over time. Photon energy conservation constraints will be broken in time-varying systems. Compared with traditional nonlinear systems, time-modulated systems can achieve efficient harmonic generation in a wider frequency range and precisely control the output spectrum through modulation parameters, providing new solutions for frequency conversion and wavefront manipulation.

The temporal discontinuities in element phase introduce a new dimension to manipulate the momentum component at the interface of time-varying metasurface, enabling the Lorentz nonreciprocity in light-matter interactions in the absence of nonlinear materials. In 2015, V. Sha-laev’s team expanded classical Snell’s law to a more general form by introducing time-gradient phase discontinuity. The developed law is applicable to situations that are either reciprocal or non-reciprocal, making it possible to build magnetic-free optical isolators with harmonic signals [ 43 ]. In the same year, an anomalous nonreciprocal electromagnetically induced transparency effect was achieved by a surface impedance meta-surface. The metasurface consists of an array of ultrathin SRR geometry loaded with varactors, as shown in Fig. 10 (a), whose capacitance is changed with a time-modulated input signal and is analyzed via full-wave finite-element simulation [ 44 ]. These works lay foundation for harmonics control in communication.

With the rapid progress of information and communication technologies, time-modulated antenna systems that can manipulate wave behaviors are desired to generate harmonics for multichannel communication. For example, in 2007, M. Secmen et al. used frequency diverse transmitting array antennas to continuously scan the range and angle without any phase shifter [ 45 ]. The antenna array is made of spatially periodically arranged radiating elements which are time-modulated periodically with a progressive incremental frequency shift. The structure is illustrated in Fig. 10 (b) and this array shows property of range dependent antenna patterns. Four years later, multiple harmonic beams steering at different sidebands with various shapes are realized through a time-modulated linear array by L. Poli et al.[ 46 ] The structural parameters are optimized using Particle Swarm algorithm and the beam shape after optimization is shown in Fig. 10 (c).

However, the above antennas cannot dynamically control the incident EM waves with efficient harmonic manipulation. To address this problem, the time-domain digital coding metasur-face could be employed, realizing individual modulation of harmonics’ strength and phase. In 2018, Prof. Cui’s group designed and experimentally realized the independent control of harmonic amplitudes and phases by a reflective time-domain digital coding metasurface, as illustrated in Fig. 10 (d)[ 47 ]. As the bias voltage of the varactor diodes can be actively modulated by a predefined field programmable gate array (FPGA) chip, the harmonic amplitudes and phases are accurately engineered. In addition, beam scanning for multiple harmonics can also be implemented by adjusting digital code sequences. This time-modulated method can be extended into the terahertz and optical range combined with advanced modulation technologies.

4 Conclusion and Outlook

This paper provides a review on SHG mechanisms that are applicable for microwave frequencies, and they are summarized in Tab. 1 , with representative example publications. In those methods, some of them are similar to those used in optical bands, including ferroic materials, metamaterials, and surface plasmon polaritons. In ferroic materials, the SHG effect is rooted in the nonlinear polarization or magnetization responses, while metamaterials and SPPs can be seen as effective media that possess nonlinear permittivity or permeability, which leads to SHG phenomenon. In addition, NI and ENZ metama-terials with nearly ideal phase-match are typically employed to generate \(\mathrm{{SH}}\) signals efficiently in forward and backward directions.

Other than these adapted from optical SHG, there are also unique methods for SHG in microwave frequencies, such as gas plasma, diode elements, diode-integrated metamaterials, mem-ristive systems, and time-varying systems. The SHG phenomena in gas plasma originate from spatial factors and resistive ionization. As for diodes, the inherent nonlinear capacitance-voltage relation in equivalent electric circuits results in harmonic generation. Memristive systems generate \(\mathrm{{SH}}\) signals more efficiently than traditional diode bridges. It is expected that the memcapaci-tive and meminductive systems can be used for passive or low-loss harmonic generation. Finally, time-varying components provide a new dimension to manipulate harmonics and wavefront.

Microwave SHG phenomenon can be used to achieve frequency/wavelength conversion and switching in communications. Furthermore, SHG responses serve as sensors to characterize and demarcate the spectral and structural properties of new materials. The ability of using SHG responses as precise sensors offers a powerful tool for material characterization, providing insights into the spectral and structural properties of emerging materials. Overall, nonlinear optics expands the range of interactions and phenomena beyond what is possible with linear optics, enabling advanced technologies and scientific discoveries. Microwave SHG will advance the development in communication systems, sensors, and radar technology. As research progresses, the understanding and implementation of SHG in microwaves will undoubtedly contribute to the development of next-generation microwave technologies.

References

[1] N. Bloembergen . Nonlinear optics: Past, present, and future [J]. IEEE Journal of Selected Topics in Quantum Electronics , 2000 . 6 ( 6 ): 876 - 880 .

[2] F. Brunel . Harmonic generation due to plasma effects in a gas undergoing multiphoton ionization in the high-intensity limit [J]. Journal of the Optical Society of America B , 1990 . 7 ( 4 ): 521 - 526 .

[3] S. Tang , T. Krasieva , Z. Chen , G. Tempea , B. Tromberg . Effect of pulse duration on two-photon excited fluorescence and second harmonic generation in nonlinear optical microscopy [J]. Journal of Biomedical Optics , 2006 . 11 ( 2 ): 20501 -

[4] P. Campagnola , M. Wei , A. Lewis , L. Loew . High-resolution nonlinear optical imaging of live cells by second harmonic generation [J]. Biophysical Journal , 1999 . 77 ( 6 ): 3341 - 3349 .

[5] S. Kruk , R. Camacho-Morales , L. Xu , M. Rahmani , D. Smirnova , L. Wang , H. Tan , C. Jagadish , D. Neshev , Y. Kivshar . Nonlinear optical magnetism revealed by second-harmonic generation in nanoantennas [J]. Nano Letters , 2017 . 17 ( 6 ): 3914 - 3918 .

[6] P. Franken , A. Hill , C. Peters , G. Weinreich . Generation of optical harmonics [J]. Physical Review Letters , 1961 . 7 ( 4 ): 118 - 119 .

[7] R. Gauderon , P. Lukins , C. Sheppard . Three-dimensional second-harmonic generation imaging with femtosecond laser pulses [J]. Optics Letters , 1998 . 23 ( 15 ): 1209 - 1211 .

[8] S. Chu , T. Liu , C. Sun , C. Lin , H. Tsai . Real-time second-harmonic-generation microscopy based on a 2-GHz repetition rate Ti: Sapphire laser [J]. Optics Express , 2003 . 11 ( 8 ): 933 - 938 .

[9] S. Jun , S. Jeon , J. Kwon , J. Lee , C. Kim , S. Hong . Full-color laser displays based on optical second-harmonic generation from the thin film arrays of selenium nanowires [J]. ACS Photonics , 2021 . 9 ( 2 ): 368 - 377 .

[10] Z. Zhang , Y. Gao , X. Li , X. Wang , S. Zhao , Q. Liu , C. Zhao . Second harmonic generation of laser beams in transverse mode locking states [J]. Advanced Photonics , 2022 . 4 ( 2 ): 26002 -

[11] K. DeLong , R. Trebino , J. Hunter , W. White . Frequency-resolved optical gating with the use of second-harmonic generation [J]. Journal of the Optical Society of America B , 1994 . 11 ( 11 ): 2206 - 2215 .

[12] K. Gallo G. Assanto . All-optical diode based on second-harmonic generation in an asymmetric waveguide [J]. Journal of the Optical Society of America B , 1999 . 16 ( 2 ): 267 - 269 .

[13] J. Berthelot , G. Bachelier , M. Song , P. Rai , G. Francs , A. Dereux , A. Bouhelier . Silencing and enhancement of second-harmonic generation in optical gap antennas [J]. Optics Express , 2012 . 20 ( 10 ): 10498 - 10508 .

[14] A. Chowdhury , D. Ye , Z. Song , S. Zhang , H. Hedderich , B. Mallick , S. Thirunahari , S. Ramakrishnan , A. Sengupta , E. Gualtieri , C. Bouman , G. Simpson . Second harmonic generation guided Raman spectroscopy for sensitive detection of polymorph transitions [J]. Analytical Chemistry , 2017 . 89 ( 11 ): 5958 - 5965 .

[15] C. Yang , C. Guo , J. Xu , H. Zhang . Device design for multitask graphene electromagnetic detection based on second harmonic generation [J]. IEEE Transactions on Microwave Theory and Techniques , 2024 . 72 ( 7 ): 4174 - 4182 .

[16] A. Khan , L. Zhang , K. Ishfaq , A. Ikram , T. Yildrim , B. Liu , S. Rahman , Y. Lu . Optical harmonic generation in 2D materials [J]. Advanced Functional Materials , 2022 . 32 ( 3 ): 2105259 -

[17] Y. Wang , J. Xiao , S. Yang , Y. Wang , X. Zhang . Second harmonic generation spectroscopy on two-dimensional materials [J]. Optical Materials Express , 2019 . 9 ( 3 ): 1136 - 1149 .

[18] J. Zhang , W. Zhao , P. Yu , G. Yang , Z. Liu . Second harmonic generation in 2D layered materials [J]. 2D Materials , 2020 . 7 ( 4 ): 42002 -

[19] J. Liu , M. Robinson , K. Johnson , D. Walba , M. Ros , N. Clark , R. Shao , D. Doroski . The measurement of second-harmonic generation in novel ferroelectric liquid crystal materials [J]. Journal of Applied Physics , 1991 . 70 ( 7 ): 3426 - 3430 .

[20] B. Sahraoui , R. Czaplicki , A. Klöpperpieper , A. Andrushchak , A. Kityk . Ferroelectric AgNa (NO₂), crystals as novel highly efficient nonlinear optical material: Phase matched second harmonic generation driven by a spontaneous and electric field induced polarizations [J]. Journal of Applied Physics , 2010 . 107 ( 11 ): 113526 -

[21] N. Sherstyuk , E. Mishina , S. Lavrov , A. Buryakov , M. Marchenkova , A. Elshin , A. Sigov . Optical second harmonic generation microscopy for ferroic materials [J]. Ferroelectrics , 2015 . 477 ( 1 ): 29 - 46 .

[22] I. Abdelwahab , B. Tilmann , Y. Wu , D. Giovanni , I. Verzhbitskiy , M. Zhu , R. Berté , F. Xuan , L. Menezes , G. Eda , T. Sum , S. Quek , S. Maier , K. Loh . Giant second-harmonic generation in ferroelectric NbOI2 [J]. Nature Photonics , 2022 . 16 ( 9 ): 644 - 650 .

[23] G. Marino , P. Segovia , A. Krasavin , P. Ginzburg , N. Olivier , G. Wurtz , A. Zayats . Second-harmonic generation from hyperbolic plasmonic nanorod metamaterial slab [J]. Laser & Photonics Reviews , 2018 . 12 ( 2 ): 1700189 -

[24] P. Vabishchevich , S. Liu , M. Sinclair , G. Keeler , G. Peake , I. Brener . Enhanced second-harmonic generation using broken symmetry III-V semiconductor Fano metasurfaces [J]. ACS Photonics , 2018 . 5 ( 5 ): 1685 - 1690 .

[25] K. Kim J. Kim . Dielectric chiral metasurfaces for second-harmonic generation with strong circular dichroism [J]. Annalen der Physik , 2020 . 532 ( 8 ): 2000078 -

[26] K. Nikolaev , S. Lake , G. Schmidt , S. Demokritov , V. Demidov . Resonant generation of propagating second-harmonic spin waves in nano-waveguides [J]. Nature Communications , 2024 . 15 . 1827 -

[27] L. Liu , L. Wu , J. Zhang , Z. Li , B. Zhang , Y. Luo . Backward phase matching for second harmonic generation in negative-index conformal surface plasmonic metamaterials [J]. Advanced Science , 2018 . 5 ( 11 ): 1800661 -

[28] H. Zhang , Y. Fan , J. Guo , X. Fu , T. Cui . Second-harmonic generation of spoof surface plasmon polaritons using nonlinear plasmonic metamaterials [J]. ACS Photonics , 2015 . 3 ( 1 ): 139 - 146 .

[29] Y. Chou Chau . Boosting second harmonic generation efficiency and nonlinear susceptibility via metasurfaces featuring split-ring resonators and bowtie nanoantennas [J]. Nanomaterials , 2024 . 14 ( 8 ): 664 -

[30] X. Gao , J. Zhang , H. Zhang , L. Liu , Q. Ma , P. Xu , T. Cui . Dynamic controls of second-harmonic generations in both forward and backward modes using reconfigurable plasmonic metawaveguide [J]. Advanced Optical Materials , 2020 . 8 ( 8 ): 1902058 -

[31] M. Chen , X. Fang , M. Li , D. Ramaccia , A. Toscano , F. Bilotti , D. Ding . Design of switchable SHG and THG using metasurface-based frequency mixing system [J]. IEEE Antennas and Wireless Propagation Letters , 2024 . 23 ( 2 ): 598 - 602 .

[32] J. Wu , J. Liu , Z. Ren , M. Leung , W. Leung , K. Ho , X. Wang , Q. Shao , S. Yang . Wideband coherent microwave conversion via magnon nonlinearity in a hybrid quantum system [J]. npj Spintronics , 2024 . 2 . 30 -

[33] Y. Hu , C. Xue , Y. Sun . Unidirectional second harmonic generation based on electromagnetic induced transparency-like phenomenon derived from standing waves [J]. Physics Letters A , 2018 . 382 ( 27 ): 1818 - 1822 .

[34] W. Choe , P. Banerjee , F. Caimi . Second-harmonic generation in an optical medium with secondand third-order nonlinear susceptibilities [J]. Journal of the Optical Society of America B , 1991 . 8 ( 5 ): 1013 - 1022 .

[35] W. Fu Y. Yan . Harmonic generation of high-power microwave in plasma filled waveguide [J]. International Journal of Infrared and Millimeter Waves , 2007 . 29 ( 1 ): 43 - 50 .

[36] J. Gregório , S. Parsons , J. Hopwood . Microwave harmonic generation and nonlinearity in microplasmas [J]. Plasma Sources Science and Technology , 2016 . 25 ( 3 ): 35018 -

[37] D. Pederson , K. Kourtzanidis , L. Raja . Nonlinear hydrodynamic effects in dense microplasmas interacting with microwaves [J]. Journal of Applied Physics , 2018 . 124 ( 4 ): 43303 -

[38] S. Sinha A. Sinha . Second harmonic generation in collisionless plasma in parallel plate guide [J]. Physics of Plasmas , 1996 . 3 ( 1 ): 59 - 63 .

[39] D. Pardo , J. Grajal , B. Mencía , S. Pérez , J. Mateos , T. González . Harmonic generation and noise in GaAs and GaN schottky diodes [C]// 21st International Symposium on Space Terahertz Technology , 2010 . 23 - 25 .

[40] X. Gu , J. De Almeida , S. Hemour , R. Khazaka , K. Wu . Nonlinearity and parameterization of Schottky diodes-based battery-free harmonic transponder for millimeter-wave 5G applications [J]. Applied Physics Letters , 2021 . 119 ( 6 ): 63504 -

[41] D. Shim , S. Sankaran , K. O . Complementary antiparallel schottky barrier diode pair in a 0.13-µm logic CMOS technology [J]. IEEE Electron Device Letters , 2008 . 29 ( 6 ): 606 - 608 .

[42] Z. Chen , Z. Chen , W. Choi , K. O . 610-GHz fourth harmonic signal reactively generated in a CMOS voltage controlled oscillator using differentially pumped varactors [J]. IEEE Solid-State Circuits Letters , 2020 . 3 . 46 - 49 .

[43] A. Shaltout , A. Kildishev , V. Shalaev . Time-varying metasurfaces and Lorentz non-reciprocity [J]. Optical Materials Express , 2015 . 5 ( 11 ): 2459 - 2467 .

[44] Y. Hadad , D. Sounas , A. Alù . Space-time gradient metasurfaces [J]. Physical Review B , 2015 . 92 ( 10 ): 100304 -

[45] M. Secmen , S. Demir , A. Hizal , T. Eker . Frequency diverse array antenna with periodic time-modulated pattern in range and angle [C]// 2007 IEEE Radar Conference , 2007 . 427 - 430 .

[46] L. Poli , P. Rocca , G. Oliveri , A. Massa . Harmonic beamforming in time-modulated linear arrays [J]. IEEE Transactions on Antennas and Propagation , 2011 . 59 ( 7 ): 2538 - 2545 .

[47] J. Dai , J. Zhao , Q. Cheng , T. Cui . Independent control of harmonic amplitudes and phases via a time-domain digital coding metasurface [J]. Light: Science & Applications , 2018 . 7 . 90 -

[48] T. Nakanishi , Y. Tamayama , M. Kitano . Efficient second harmonic generation in a metamaterial with two resonant modes coupled through two varactor diodes [J]. Applied Physics Letters , 2012 . 100 ( 4 ): 44103 -

[49] A. Rose , D. Huang , D. Smith . Controlling the second harmonic in a phase-matched negative-index metamaterial [J]. Physical Review Letters , 2011 . 107 ( 6 ): 063902 -

[50] Y. Yin , C. Uchida , K. Tsukamoto , H. Hayashi , T. Nakaoka . Modulated conductive bridge memory characteristics by radio frequency input and non-volatile switching of frequency multiplication [J]. Electronics Letters , 2022 . 58 ( 21 ): 804 - 806 .

[51] G. Cohen , Y. Pershin , M. Di Ventra . Second and higher harmonics generation with memristive systems [J]. Applied Physics Letters , 2012 . 100 ( 13 ): 133109 -

[52] J. Asmussen J. Beyer . Microwave harmonic generation in a plasma capacitor [J]. IEEE Transactions on Electron Devices , 1969 . 16 ( 1 ): 18 - 29 .

[53] T. Sonnenberg , S. Verploegh , M. Pinto , Z. Popovic . W-band GaN HEMT frequency multipliers [J]. IEEE Transactions on Microwave Theory and Techniques , 2023 . 71 ( 10 ): 4327 - 4336 .

[54] N. Zhang , L. Belostotski , J. Haslett . D-band broadband passive frequency tripler using antiparallel diode-connected nMOS transistor pair in 22-nm CMOS SOI [J]. IEEE Microwave and Wireless Components Letters , 2020 . 30 ( 7 ): 689 - 692 .

[55] N. Zhang , L. Belostotski , J. Haslett . Relations of time-varying circuit parameters and idlerless parametric harmonic generation for reconfigurable frequency multipliers [J]. IEEE Transactions on Microwave Theory and Techniques , 2021 . 69 ( 5 ): 2554 - 2568 .

[56] J. Chen , C. Hu , F. Kong , J. Mao . High-performance second-harmonic-generation (SHG) materials: New developments and new strategies [J]. Accounts of Chemical Research , 2021 . 54 ( 12 ): 2775 - 2783 .

[57] C. Argyropoulos , G. D'Aguanno , A. Alù . Giant second-harmonic generation efficiency and ideal phase matching with a double ɛ-near-zero cross-slit metamaterial [J]. Physical Review B , 2014 . 89 ( 23 ): 235401 -

[58] A. Popov V. Shalaev . Negative-index metamaterials: Second-harmonic generation, Manley-Rowe relations and parametric amplification [J]. Applied Physics B , 2006 . 84 ( 1-2 ): 131 - 137 .

[59] V. Agranovich , Y. Shen , R. Baughman , A. Zakhidov . Linear and nonlinear wave propagation in negative refraction metamaterials [J]. Physical Review B , 2004 . 69 ( 16 ): 165112 -

[60] I. Shadrivov , A. Zharov , Y. Kivshar . Second-harmonic generation in nonlinear left-handed metamaterials [J]. Journal of the Optical Society of America B , 2006 . 23 ( 3 ): 529 - 534 .

[61] R. Marqués , J. Martel , F. Mesa , F. Medina . Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides [J]. Physical Review Letters , 2002 . 89 ( 18 ): 183901 -

[62] D. Smith , W. Padilla , D. Vier , S. Nemat-Nasser , S. Schultz . Composite medium with simultaneously negative permeability and permittivity [J]. Physical Review Letters , 2000 . 84 ( 18 ): 4184 - 4187 .

[63] V. Veselago . The electrodynamics of substances with simultaneously negative values of ɛ and μ [J]. Soviet Physics Uspekhi , 1964 . 10 ( 4 ): 517 - 526 .

[64] J. Porto , L. Martín-Moreno , F. García-Vidal . Optical bistability in subwavelength slit apertures containing nonlinear media [J]. Physical Review B , 2004 . 70 ( 8 ): 81402 -

[65] J. Porto , F. García-Vidal , J. Pendry . Transmission resonances on metallic gratings with very narrow slits [J]. Physical Review Letters , 1999 . 83 ( 14 ): 2845 - 2848 .

[66] SMV123x series: hyperabrupt junction tuning varactors [EB/OL]. 2016 -

[67] H. Margenau L. Hartman . Theory of high frequency gas discharges. II. Harmonic components of the distribution function [J]. Physical Review , 1948 . 73 ( 4 ): 309 - 315 .

[68] J. Krenz G. Kino . Harmonic generation and parametric oscillations in a plasma [J]. Journal of Applied Physics , 1965 . 36 ( 8 ): 2387 - 2395 .

[69] H. Sobhani , M. Vaziri , H. Rooholamininejad , A. Bahrampour . Impact of nonlinear absorption on propagation of microwave in a plasma filled rectangular waveguide [J]. Waves in Random and Complex Media , 2016 . 26 ( 3 ): 272 - 283 .

[70] M. Kamal-Al-Hassan , H. Ito , N. Yugami , Y. Nishida . Dynamic of ion density perturbations observed in a microwave-plasma interaction [J]. Physics of Plasmas , 2005 . 12 ( 11 ): 112307 -

[71] P. Tien H. Suhl . A traveling-wave ferromagnetic amplifier [J]. Proceedings of the Institute of Radio Engineers , 1958 . 46 ( 4 ): 700 - 706 .

[72] R. Sharma R. Sharma . Microwave harmonic generation in a plasma-filled waveguide [J]. Journal of Applied Physics , 1979 . 50 ( 9 ): 5655 - 5660 .

[73] F. Yu , K. Lyon , E. Kan . Harmonic generation from integrated nonlinear transmission lines for RFID applications [C]// 2010 IEEE MTT-S International Microwave Symposium , Anaheim, CA, USA : 2010 . 844 - 847 .

[74] Z. Wang , Y. Luo , L. Peng , J. Huangfu , T. Jiang , D. Wang , H. Chen , L. Ran . Second-harmonic generation and spectrum modulation by an active nonlinear metamaterial [J]. Applied Physics Letters , 2009 . 94 ( 113 ): 134102 -

[75] D. Filonov , Y. Kramer , V. Kozlov , B. Malomed , P. Ginzburg . Resonant meta-atoms with nonlinearities on demand [J]. Applied Physics Letters , 2016 . 109 ( 11 ): 111904 -

[76] H. Kim , A. Kozyrev , A. Karbassi , D. van der Weide . Compact left-handed transmission line as a linear phase-voltage modulator and efficient harmonic generator [J]. IEEE Transactions on Microwave Theory and Techniques , 2007 . 55 ( 3 ): 571 - 578 .

[77] J. Grajal , V. Krozer , E. Gonzalez , F. Maldonado , J. Gismero . Modeling and design aspects of millimeter-wave and submillimeter-wave Schottky diode varactor frequency multipliers [J]. IEEE Transactions on Microwave Theory and Techniques , 2000 . 48 ( 4 ): 700 - 711 .

[78] K. Rasilainen , J. Ilvonen , A. Lehtovuori , J. Hannula , V. Viikari . On design and evaluation of harmonic transponders [J]. IEEE Transactions on Antennas and Propagation , 2015 . 63 ( 1 ): 15 - 23 .

[79] J. Huang , J. Dong , H. Yang , H. Zhang , C. Tian , T. Guo . Design and fabrication of frequency multiplier based on composite right/left-handed nonlinear transmission lines [J]. IEEE Transactions on Microwave Theory and Techniques , 2011 . 59 ( 10 ): 2486 - 2493 .

[80] J. Huang Q. Zhao . A W-band monolithic broadwidth frequency multiplier on millimeter Schottky diode and metamaterial [J]. Microwave and Optical Technology Letters , 2015 . 57 ( 3 ): 525 - 530 .

[81] D. Porterfield , T. Crowe , R. Bradley , N. Erickson . A high-power fixed-tuned millimeter-wave balanced frequency doubler [J]. IEEE Transactions on Microwave Theory and Techniques , 1999 . 47 ( 4 ): 419 - 425 .

[82] R. Camarero , J. Marin , O. Menéndez , P. de Paco . On designing a W-band active coherent array [C]// 2011 IEEE MTT-S International Microwave Workshop Series on Millimeter Wave Integration Technologies , Sitges, Spain : 2011 . 176 - 179 .

[83] G. Hegazi , A. Ezzeddine , F. Phelleps , P. Mcnally , K. Pande , P. Rice , P. Pages . W-band monolithic frequency doubler using vertical GaAs varactor diode with n⁺ buried layer [J]. Electronics Letters , 1991 . 27 ( 3 ): 213 - 214 .

[84] I. Shadrivov , A. Kozyrev , D. van der Weide , Y. Kivshar . Tunable transmission and harmonic generation in nonlinear metamaterials [J]. Applied Physics Letters , 2008 . 93 ( 16 ): 161903 -

[85] D. Huang , A. Rose , E. Poutrina , S. Larouche , D. Smith . Wave mixing in nonlinear magnetic metacrystal [J]. Applied Physics Letters , 2011 . 98 ( 20 ): 204102 -

[86] E. Poutrina , D. Huang , D. Smith . Analysis of nonlinear electromagnetic metamaterials [J]. New Journal of Physics , 2010 . 12 ( 9 ): 93010 -

[87] A. Rose , D. Huang , D. Smith . Demonstration of nonlinear magnetoelectric coupling in metamaterials [J]. Applied Physics Letters , 2012 . 101 ( 5 ): 51103 -

[88] S. Makarov , R. Ludwig , S. Bitar . Practical electrical engineering [M]. Switzerland : Springer Cham , 2019 . 795 - 849 .

[89] M. Di Ventra , Y. Pershin , L. Chua . Circuit elements with memory: Memristors, memcapacitors, and meminductors [J]. Proceedings of the IEEE , 2009 . 97 ( 10 ): 1717 - 1724 .

Funding

National Natural Science Foundation of China(12274339)