杜遇林，謝欣榮，陳紅勝，高飛

（1 浙江大學(xué) 杭州國際科創(chuàng)中心 極端光學(xué)技術(shù)與儀器全國重點(diǎn)實(shí)驗(yàn)室 量子信息交叉中心，杭州 310027）

（2 浙江大學(xué) 國際電磁科學(xué)院浙江大學(xué)分院 國際聯(lián)合創(chuàng)新中心，海寧 314400）

0 Introduction

Terahertz（THz）waves have attracted extensive interest for their distinguished features in wave-matter interactions and have achieved two important applications，i.e.，THz Time-Domain Spectroscopy（THz-TDS）and imaging technique［1-6］.However，the low output power of the THz radiation sources has put limitations on the signal-to-noise ratio and sensitivity of these two applications［7-8］.THz signals are mainly produced by optoelectronic means，and recently spintronic method has attracted great research efforts［9-10］.Photoconductive Antenna（PCA）is one of the most popular optoelectronic THz sources，and the typical pattern of which is a pair of electrodes deposited on a semiconductor substrate［11］.The general principle of PCAs is that femtosecond-laser-excited photocarriers are driven by a biasing voltages to generate picosecond pulse currents and emit THz waves.In such PCAs，the low-output issue is attributed to two aspects，i.e.，the low photon-electron conversion efficiency，and low vertical directivity resulting from propagating Transmission Line（TL）modes along the horizontal electrodes.

To date，much effort has been devoted to addressing the low photon-electron conversion efficiency and low vertical directivity of PCAs，while being limited to solving these two problems separately.To improve the photon-electron conversion efficiency，two approaches have been proposed：adopting semiconductors with shorter carrier lifetime and higher carrier mobility［12-15］and integrating nanoscale plasmonic structures on the semiconductor［16-27］.Microscale metamaterial resonators have been integrated with PCAs to suppress the TL electrode modes at specific frequencies，thus increasing the vertical directivities of PCAs in narrow bands［28-31］.WANG Chi et al.promoted this scheme by employing a microscale Spoof Surface Plasmon Polariton（SSPP）structure and realized broadband suppression［32］.However，the global output efficiency can only be increased sufficiently when the low photon-electron conversion efficiency and low vertical directivity are addressed simultaneously.

Here we propose hierarchical spoof plasmonic structures to increase the output efficiency of PCAs.Hierarchical structures combine nanoscale plasmonic structure and microscale SSPP［33-39］structure to address low-output issues from two aspects simultaneously.In detail，the metallic nanoblock array raises energy conversion efficiency by Spoof Localized Surface Plasmon Resonance（SLSPR），and periodic microscale gratings suppress horizontal radiation based on the forbidden band of the fundamental SSPP mode and the orthogonality between the electric dipole source and the higher-order SSPP mode.With the cooperation of hierarchical structures，the radiation power density in the vertical direction increases in a broad band from 0.86 THz to 1.51 THz.

1 Theory and simulation

Fig.1 shows the schematic diagrams of PCAs without and with hierarchical spoof plasmonic structures.Both PCAs adopt the Bottom-Located Thin-Film（BLTF）structure proposed by Burford N and El-Shenawee M［20］，where golden electrodes are sandwiched between a Low-Temperature-grown Gallium Arsenide（LTGaAs）substrate and a Si lens for shortening the carrier's average transport distance.The general principle of PCA follows that with an incident femtosecond laser beam，photocarriers are excited inside the semiconductor substrate，and the carrier population is propelled by biasing voltages applied on electrodes to form a picosecond current between central teeth pair，i.e.，the anode and cathode（see Fig.1）.The photocurrent then generates-z-direction THz radiation converged by a Silicon（Si）collimating lens.Fig.1（b）shows the hierarchical spoof plasmonic structures we proposed to address these two problems simultaneously，which includes a golden nanoblock array set above the substrate to elevate energy conversion efficiency and periodic teeth pairs integrated between electrodes along thex-axis to suppress horizontal radiation.

Fig.1 The schematic diagrams of PCAs without and with hierarchical plasmonic structures.Both PCAs share a substrateelectrode-lens structure along the-z-direction

Firstly，we will analyze the influence of nanoscale plasmonic structure.Fig.2（a）illustrates the anode-gapcathode region，and the zoom-in view shows the details of the nanoblock array.The nanoblock with a square cross-section ofb×b，b=200 nm，and a thickness oft1=75 nm，constructs a two-dimensional array with a center-to-center spacing ofs=520 nm.Due to the nonlinear increase in the biasing electric field near the electrodes，it is beneficial for the electrodes to collect carriers when the laser is focused onto the gap asymmetrically.And as electrons mobility is significantly higher than that of holes，the laser which is focused onto the anode can further reduce average carrier transport path lengths［17］.The array is thus set only on the anode side according to the laser irradiation area.The laser is a Gaussian beam spatially，and a Gaussian pulse temporally with a wavelength ofλ=800 nm.Since the Fr?hlich condition（b＜s＜λ）is satisfied，and the permittivity of goldεAuis negative according to the Drude model，SLSPR is excited under laser irradiation，leading to electric field enhancement around the structure，which corresponds to the conversion efficiency improvement.Here the nanoblock arrays can be treated as ensembles of interacting electric dipoles where the near-field coupling results in the suppression of scattering into the far field and the confinement of energy in the interstitial sites，presenting as resonant enhancement of the electric field.

Fig.2 The principle of improving conversion efficiency with nanoblock array

For quantitative analysis，we develop a three-dimensional simulation in COMSOL Multiphysics.Theypolarization laser beam is defined by［20］

Dx=Dy=2 μm is Half Power Beam Width（HPBW），（0，y0）is the center point of the nanoblock array，and peak electric fieldE0is calculated by

Table 1 Laser parameters

Naturally，a stronger electric field will induce more photocarriers and generate a larger photocurrent.In COMSOL Multiphysics，we can use the electric field distribution results of the LT-GaAs substrate from the frequency domain study to calculate the transient photocarrier generation rate，which can be written as［20］

wherehis Planck constant，cis the light speed in free space，kpc=0.086 is the photoconductor extinction coefficient of LT-GaAs（i.e.，the imaginary part of the refractive index of LT-GaAs），t0=2 ps is the pulse center location，and the exponential term comes from temporal Gaussian property of laser as mentioned before.The time-average power flow of electric fieldPoavestablishes connections between the electromagnetic and semiconductor fields and is written as

Besides，we can get the electrostatic field distribution results from the stationary study（the applied biasing DC voltage isVbias=30 V）.The electrostatic field distribution results combined with the photocarrier generation rate enable us to get the terminal current between the anode and cathode（the gap distance isg=10 μm）by conducting a time-dependent study，in which the evolution process of carriers in LT-GaAs substrate is described by coupled Poisson's equation and drift-diffusion equation，and carriers recombination conforms Schottky-Read-Hall（SRH）recombination model and Auger recombination model.Related parameters of LTGaAs are listed in Table 2.Simulation results are presented in Fig.2（c）as the comparison of time-variant photocurrents.In both cases，photocurrents rise and fall drastically within 0.5 ps around 2 ps.With a nanoblock array，the peak value grows from 3.561 9×10-4A to 5.245 7×10-4A，and the increasing rate reaches around 47%，proving that the conversion efficiency has been increased.However，the horizontal radiation leakage is still to be solved to avoid its negative impact on global output efficiency.

Table 2 LT-GaAs parameters

The horizontal leakage is caused by TL modes propagating along the electrodes，which can be suppressed by optimizing the structure of the electrodes.Different from metamaterial resonator schemes［28-31］，a broader bandwidth is attained from the bandgap of the SSPP structure.As Fig.3（a）shows，the width of the electrode and tooth isw=5 μm anda=5 μm，respectively.The distances between the electrode pair ared=40 μm.Teeth are decorated periodically on both infinitely long electrodes，and the period length isp=50 μm.Periodic subwavelength gratings are a kind of classical spoof plasmonic structure that support the propagation of surface electromagnetic modes，but here it is used to restrict electromagnetic field from transversal propagating.We then conduct simulations using the eigenmode solver in CST Studio for further research.Fig.3（b）and Fig.3（c）are dispersion relations and electric field patterns of the unit cell circled by a dotted box in Fig.3（a），respectively.The inset in Fig.3（b）is the three-dimensional simulation model，where the thickness of electrodes ist3=200 nm（see Fig.2（b）），and the permittivity of LT-GaAs and Si areεGaAs=12.9 andεSi=11.9，respectively.The boundaries along thex-direction are periodic boundaries，and along they-andz-directions are electric boundaries.Observing Fig.3（b），the two dispersion curves are lying below the light line，corresponding to two eigenmodes respectively.When the frequency is higher than the asymptomatic frequency（0.795 THz），the first eigenmode cannot be excited.Meanwhile，in terms of Fig.3（c），the electric field pattern of the second eigenmode demonstratesy-even parity，which is orthogonal to they-odd parity of the practical current source.Thus，the second eigenmode also cannot be excited efficiently even though the permissible band of higher-order eigenmode partially overlaps with the forbidden band of the first eigenmode.Both propagating modes arising from the SSPP structure are unexcited as we expected，thus suppressing the horizontal radiation leakage in the operating frequency range.

Fig.3 The principle of improving vertical directivity with SSPP structure only

Next，we conduct simulations by utilizing the time domain solver of CST Studio to validate the vertical directivity enhancement brought by the periodic gratings.The simulation model contains the whole antenna except for the nanoscale plasmonic structure.The radius and focus length of the Si collimating lens isr=550 μm andF=224.5 μm，respectively.All the geometry parameters of the microscale and nanoscale structures are listed in Table 3.And the electrodes with a finite length totally include 21 teeth.The practical photocurrent is replaced by a dipole source，representing a discrete constant unit current port connecting the anode and cathode，and lens focus coincides with it.Fig.3（d）shows excited near-field mode patterns on electrodes at 1.01 THz without and with periodic gratings，respectively.It appears that the electromagnetic field propagates freely to the ends of bare electrodes，while on SSPP-modified electrodes，lateral propagation along thex-direction is almost blocked.The radiation power pattern shown in Fig.3（e）provides a more intuitive result，from which we can see the horizontal directivity along thex-axis decreases by 3 dB，accompanied by the increase of the vertical directivity along the-z-direction from 19.00 dBi to 20.32 dBi.

Table 3 Structures geometry parameters

We further analyze the joint effect of the hierarchical structures composed of both nanoscale and microscale structures.The difficulty of trans-dimensional and multi-physical co-simulation denies the possibility of touching results directly with simulations，but a mathematical method can be adopted to connect the results of the nanoscale and microscale structures.The electrodes and lens compose a linear system where the output is the product of the input photocurrent and its spectral response and is proportional to the input photocurrent.Here the input photocurrents refer to the ordinary and nanoblock-enhanced photocurrents as we have discussed above，and they can be converted to the frequency domain using the Fourier transform.And the spectral responses of radiation power density in the-z-direction of the regular and SSPP-integrated antennas are obtained in CST simulations above，of which the excitation source is a constant amplitude current.By multiplying Fourier coefficients by corresponding frequency responses at the same frequency，we can achieve the vertical radiation power density spectra of PCAs under different circumstances（Fig.4（a））.In the experiment，the reflection of THz wave at the Si-air interface because of the impedance mismatching will attenuate since material loss exists；while in the simulation，the material is lossless in ideal scenarios，and thus the resonance forming inside the Si lens will cause spectral oscillations，so here only the envelopes of THz signals are shown.The PCA with the hierarchical structure（red solid line）shows a huge increasement in vertical radiation power density compared with others in a broad band（0.86～1.51 THz，blue region）because it combines the advantages of both the nanoscale and microscale structures simultaneously.The necessity of these two structures is further analyzed.The comparison between the vertical radiation power density of the PCA with microscale structure only（orange dash-dotted line）and with hierarchical plasmonic structure（red solid line）confirms that the introduction of nanoscale structure raises output efficiency over the whole simulation frequency range.On the other hand，the introduction of the microscale structure can be found to bring great improvement in downward radiation with a comparison between the vertical radiation power density of the PCA with the nanoscale structure only（green dotted line）and with hierarchical plasmonic structure.Here the working frequency range of the PCA with the microscale structure is 1.03～1.47 THz（green region），where the start frequency is slightly higher than the theoretical value because of the requirement of the sufficiently short evanescent length.The far-field power patterns shown in Fig.4（b）further illustrate the functions of the microscale structure.We can see that the PCA with the nanoscale structure only owns the largest horizontal radiation since the increasement of the photon-electron conversion efficiency enlarges radiation power isotropically，but it’s decreased with the introduction of the microscale structures to be even smaller than that with no structure（blue dashed line）.Besides，it is surprising to find that in the-z-direction，the vertical radiation with microscale structure only is even stronger than that with the nanoscale structure only（see Fig.4（b））.Quantitatively，at 1.27 THz，in the circumstance of the same input laser，radiation power density experiences a growth of 4.77 dB from-35.32 dBW/m2（2.94×10-4W/m2）to-30.55 dBW/m2（8.81×10-4W/m2）.

Fig.4 The comparison of output efficiencies of PCAs

Preparing for the experiments，the effects of the accuracies of the fabrication and operation are also studied.Regarding fabrication accuracy，for the metallic nanoblock array，localized surface plasmon resonance can be excited in arranged nanoscale metallic structures with arbitrary shapes under laser irradiation［40］.For the periodic microscale gratings，simulation results show that a simulated error of 100 nm for main geometry parameters affecting the start frequency of the forbidden band（i.e.，the period length of the grating p，the width of the tooth a，the distance between the electrode pair d and the gap distance g，see Fig.3（a））causes a small fluctuation from-0.22% to 1.45%，indicating a negligible influence to the horizontal suppression.And concerning the experimental operation accuracy，for the typical PCA，the laser focus can only be at the edge of the anode at most because the metallic tooth is set above the semiconductor substrate；even if the laser is focused to a diffraction-limited spot size，the average electron transport path lengths are still relative long，only a minority of electrons can reach the anode in a sub-picosecond-scale time，while most remaining carriers will be recombined before being collected.In our work，however，profiting from the BLTF structure，photocarriers will generate when the laser focus is located inside the anode region.In other words，compared to the typical one，the PCA we proposed has a lower requirement for focus accuracy，and the spot size is unnecessary to down to the diffraction limit.

2 Conclusion

In this paper，hierarchical spoof plasmonic structures are introduced to improve the radiation power and directivity of THz-PCA.The fundamentals are SLSPR enhancement of electric field and horizontal leakage suppression using the forbidden band of SSPP structure，respectively.We numerically demonstrate that the proposed PCA achieves higher output power density in a broad band from 0.86 THz to 1.51 THz.Using advanced micro/nano fabrication technology，PCA with hierarchical structures can be fabricated for further experimental measurement.With the explorations on semiconductor material engineering and the developments of higher efficiency THz collimating components，our proposal can enable more applications as a broadband THz source.