Design and analysis of a high-performance terahertz photoconductive modulator enhanced by photonic crystal cavity

We have proposed and designed a high-speed and high-performance optically controlled terahertz (THz) intensity modulator based on the free carrier modulation of a GaAs semiconductor. The device comprises a photonic crystal cavity–waveguide coupling structure for operation in the THz region. This modulator benefits from the strong interaction between the THz wave and the photoconductive substance to obtain a deep modulation with GHz speed, even with a low external optical power. The finite element method was used to calculate the most important properties of the modulator, such as the modulation depth, insertion loss, and modulation rate. The proposed modulator also demonstrates external optical power-dependent characteristics. The results indicate that the THz intensity can be modulated at a switching frequency of 1 GHz with high modulation depths of 83 and 90.3% under the continuous wave laser pumping of 50 W/cm2 and 80 W/cm2, respectively. In addition, this modulator exhibits efficient performance under the same pumping power with a switching frequency of up to 3 GHz. The device exhibits higher modulation depths with higher laser power intensities. The outstanding properties of the proposed structure are promising for the development of modulators and switches in THz communication systems.


Introduction
Terahertz (THz) science and technology have been widely adopted in communication systems over the past several years [1][2][3]. In this area, THz modulators have emerged as one of the most functional devices under intensive research and development. To efficiently modulate a THz wave, different research groups have used various techniques and materials, including metasurfaces [4][5][6][7][8][9], two-dimensional materials [10][11][12][13][14], liquid crystals [15,16], nanoparticles [17], two-dimensional electron gas [18], and photonic crystals [19][20][21]. Most of them have tried to improve the modulator characteristics in terms of modulation speed, modulation depth, bandwidth, and energy efficiency. For example, Deng et al. demonstrated a modulator based on InSb gratings on GaAs substrate with a modulation speed of 1.2 GHz and a 46.7% transmittance modulation with the carrier frequency of 1.5 THz [22]. Another efficient THz modulator based on a composite metamaterial with double-channel heterostructure has been proposed demonstrating 1 GHz modulation speed and 85% modulation depth [23]. Although modulation depths up to 99.9% have been achieved by modulators composed of two-dimensional materials and semiconductor heterostructures, but simultaneous modulation rates were limited to the order of MHz [24][25][26].
A particular category of THz modulators based on controlling the conductivity of a material with the modulating energy is one of the most desirable schemes which has received considerable attention [27][28][29][30]. In this mechanism, modulating free carrier density affects the conductivity of the substance. The amplitude of the passing wave interacting with this substance attenuates proportionately to its conductivity. Optical modulation is preferable to electrical modulation due to its superior potential characteristics of higher bandwidth, faster speed, and higher modulation depth [31][32][33]. However, for optically controlled modulators, the achievement of large modulation depths requires considerable optical power consumption. Therefore, it is desirable to enhance the light-matter interaction by trapping the THz wave within a cavity. Better confinement of the electromagnetic wave leads to higher modulation depths, although the modulation rate reduces at high resonances; naturally this trade-off is typical for these modulators. As a result, it is necessary for a particular application to find an acceptable compromise between the modulation speed and depth.
In this study, we propose an all-optical tunable THz intensity modulator based on the photoconductive properties of semi-insulating (SI) GaAs. Exercising a THz wave through a photonic crystal waveguide and trapping it in a cavity, make a strong interaction environment with materials within. This creates a high modulating effect that could work with even low power optical pumps. The proposed THz photoconductive modulator provides adjustable modulation depths up to 90.3%, and the modulation speed, by several GHz, under applicable optical power intensities. This tunability in a wide range of THz spectrum, along with high-performance figures of merit paves the way for the development of THz modulator and switch applications for telecommunication services.

Theory and structure design
The schematic of the proposed optically tunable THz modulator is shown in Fig. 1(a). This structure comprises a 2D square array of silicon rods, in which, the lattice constant and radius of rods are assumed to be a and r, respectively. As shown in Fig. 1(b), by making a pair of linear defects (waveguides) along the x-axis (left and right) and removing three rods in the middle (cavity) along the y-axis, which is the connection between the two waveguides, we have designed a waveguide-cavity coupling mechanism. The device is designed based on a Photonic Band Gap (PBG) of a Photonic Crystal (PC), transporting electromagnetic waves of a particular bandwidth along the waveguide while forbidding them from transferring in any other direction. The included cavity is beneficial for controlling light in a narrow frequency band and/or confining the electromagnetic energy for an enough time [34][35][36][37]. Here, we locate two exterior rods for the cavity with radius of r c . A THz wave having the same frequency as the cavity's main mode is launched into the input port of the waveguide, coupled to and gone through the resonant cavity, and then extracted from the output waveguide.
Taking advantage of this condition, an SI-GaAs wafer with the thickness of h (higher than the penetration depth) and width of w is located within the cavity.
If an optical pump containing photons with higher energies than the electronic bandgap of GaAs is irradiated, there will be a very high probability that the electrons absorb the photon energy and get excited to higher states. At this point, the conductivity of GaAs, which is related to its free carrier density, increases. The time dependence of the free carrier density within the material is given by [38]: where η e is the photoconductor external quantum efficiency (multiple exciton production is neglected), α is the absorption coefficient, R is the reflectance power at the surface of GaAs, hv is the photon energy, τ is the carrier lifetime, and P is the incident optical intensity. According to Beer-Lambert law, when an electromagnetic wave passes through a substance, its intensity attenuates exponentially as a function of distance from the surface of the material via P(d) = P 0 exp(− ⍺d). Therefore, the carrier density varies with the illumination time and depth of the substance, considering the modulating effect. To calculate the spatially varying free carrier density, GaAs layer is divided into 50 nm thick sublayers with an approximately constant power for  each sublayer. The basic optical parameters of GaAs are given in Table 1.
Manipulating the free carrier density of GaAs via the optical pumping results in a variation of GaAs permittivity and its conductivity. The permittivity of GaAs at THz frequencies is calculated by the classic Drude model through the following expression [39,40].
Here, ε ∞ is the background dielectric constant, γ = 1/τ s = e/µm* is the electron scattering rate, e is the electron charge, m* is the electron effective mass, µ is the electron mobility, and the plasma frequency ( p ) is written as [39]: which is apparently dependent on the concentration of free carriers in GaAs. Eventually, the conductivity of GaAs can be found by the imaginary part of the permittivity and is expressed as σ(ω)=ωε 0 ε imag (ω) [41].

Results and discussion
Initially, we are going to investigate the characteristics of the device without optical pumping. In this configuration, the structural parameters are designed in such a way that the resonant frequency occurs around 1 THz which is our frequency of interest as a carrier wave. For this purpose, the square lattice constant and radius of rods are chosen as 96 μm and r = 0.2a = 19.2 μm, respectively. It is assumed that Si is lossless and non-dispersive with a refractive index of n = 3.4 in the THz region [44]. The simulation area includes 15 × 11 silicon rods, which are surrounded by scattering boundary conditions (SBC). A normal layer of GaAs with a thickness of 3 µm and a width of 29 µm as a photoconductive substance, is inserted. The radius of two exterior rods r c of the cavity can strongly affect the device performance; hence, we explored the transmittance spectra for three different values of r c = 13, 14, and 15 µm under the normal incidence of TE polarization using the 2D finite element numerical method. Illustrated in Fig. 2(a), when the radius increases, the cavity demonstrates stronger effects, resulting in narrower bandwidth and redshift of the resonance frequency. The insertion loss is determined to be 2.2 dB at 1 THz for r c = 14 µm, based on the equation of IL = − 10 × log (I out /I in ), where I out and I in are the transmitted and input THz power, respectively. Figure 2(b) illustrates normalized electric field (|E|⁄|E inc |) distribution at 1 THz excitation for r c = 14 µm.
Here, |E| and |E inc | represent the local electric field and the incident electric field, respectively. Inside the cavity, the THz field enhancement reaches up to 16, which is beneficial to the photoconductive-based modulators. Strong confinement of the THz wave in the vertical (z-axis) direction, the aspect ratio of the rods needs to be sufficiently large, which has a feasible value in our design based on some reported fabrication process research [45][46][47].
Tuning width and thickness of the photoconductive material, gives extra degrees of freedom to smoothly control the properties of the cavity. We evaluated the quality factor (Q) of the resonator from the transmittance spectrum. Figure 3 shows the tunability of the resonance frequency and quality factor in terms of GaAs width for three selected thicknesses of h = 3, 3.2, and 3.5 µm. It is apparent that wider GaAs provides higher Q factors and shifts the resonant frequency toward lower values. A greater Q leads to longer trapping time of the incident THz wave (i.e., Q = ω 0 τ), which increases the interaction with the photoconductor material. Thus, the modulator can operate with lower external optical powers, forgoing the modulation speed. As also observed from this figure, increasing the thickness of GaAs, the quality factor rises, and the resonant frequency experiences a redshift. A comparison of plots for h = 3 and 3.5 μm at w = 29 μm shows that the Q value enhances from 430 to Now, we can evaluate the performance of this modulator under the excitation of an optical pump. The most common way to excite massive free carriers is using a femtosecond laser as an optical pumping source. However, instead, it is more practical to use a continuous wave (CW) laser for modulation at GHz rate. We considered a CW laser with wavelength of 800 nm (higher photon energy with respect to GaAs bandgap: E g = 1.4 eV) for irradiation of GaAs. Based on the mechanism explained in the previous section, one can calculate the conductivity of GaAs for a given optical intensity. Figure 4(a) illustrates the carrier density and surface conductivity of GaAs for a 1 THz wave as a function of pump intensity. Observed from this figure, higher intensities generate greater free carrier densities, and the conductivity will increase. These results indicate that a low pump intensity in a given range generates a carrier density of about 10 14 cm −3 , and then even with this low concentrations, efficient modulation of THz wave can be constructed. The calculated permittivity (real and imaginary parts) of n-doped GaAs for 1 THz wave is depicted in the inset of Fig. 4(a). In the high optical power region, the real part decreases rapidly, while the imaginary part increases quickly, and GaAs becomes a lossy material. Figure 4(b) illustrates GaAs conductivity under applying a pump intensity of 40 W/cm 2 as a function of distance from the surface of GaAs and illumination time. The conductivity near the surface, dramatically enhances when the illumination time is more than the carrier lifetime. Figure 4(c) shows the surface conductivity of GaAs overtime for three pump intensities of 60, 40, and 20 W/cm 2 . As expected, the extra generated carriers over the illumination time enhance the conductivity, and finally, it reaches a steady state. The conductivity versus distance from GaAs surface under different pump intensities has been plotted in Fig. 4(d). The conductivity increases with optical intensity, meanwhile decreasing with the distance from the surface. As observed from this figure, under the pump intensity of 40 W/cm 2 , the surface conductivity at the steady state is   Figure 5 shows the transmittance spectra in the steady state under different pump intensities. Without pumping power, the THz wave is guided through the waveguide with a transmission peak of around 60% at 1 THz. Increasing the pumping power, the transmission of the THz wave will be gradually reduced, and finally, the transparency of this frequency window will be completely suppressed. Applying pumping power intensities of 10, 30, and 50 W/cm 2 , the transmission peak drops to 36, 17, and 10% respectively. This behavior is due to the photoresponse of GaAs. Higher optical powers enhance the conductivity, resulting in greater absorptance and reflectance of the THz wave while transmitted through the waveguide. Due to the strong interaction of the THz wave with GaAs in the resonant cavity, the transmitted THz wave suffers a significant attenuation.
Moreover, after illumination, there is almost no frequency shift of the transmittance spectra that would adversely affect the properties of the modulator. Because of the appropriate narrow band transmittance, the proposed modulator is especially suitable for THz CW signals.
In the absence of optical excitation, GaAs is nearly transparent for the THz wave propagation, and there is no absorption loss as shown in Fig. 6(a). However, under the continuous wave laser pumping of 50 W/cm 2 , THz power dissipation in GaAs reaches 36% at 1 THz, and the absorptance of GaAs is calculated as follows [48]: with ω being the angular frequency, imag(ε) the imaginary part of the GaAs permittivity, and |E| the electric field amplitude. The integral is applied over the active absorption region V abs . Figure 6(b) shows the variations of reflectance caused by application of the same pumping. The reflectance has a minimum value of about 3% at 1 THz and rises to about 39%, applying a pump intensity of 50 W/cm 2 . Figure 7 demonstrates the modulation depth under pumping up to 120 W/cm 2 for several values of r c at their corresponding resonance frequency. Based on the modulator behavior, higher pump intensities reduce the transmitted THz power, which results in greater modulation depth. This figure also shows that under a relatively small intensity of 40 W/cm 2 , the modulation depths enhance rapidly and reach the high values of around 89.5, 79.1, and 62.2% for r c = 15, 14, and 13 µm, respectively. Subsequently, by further increasing the optical intensity, the modulation depth tends to be saturated at higher values, which is high enough for all-optical THz modulators. Furthermore, for a given intensity, a higher modulation depth can be achieved for the larger exterior rods. This is due to the strong interaction of THz wave and photoconductive material for a greater r c as mentioned before.
Modulation rate is another key parameter of the THz modulator that indicates the ability to follow a high-speed modulating signal. The modulation rate of the proposed device depends on the lifetime of free carriers and is also influenced strongly by the trapping time of the THz wave in the cavity. Since the free carrier lifetime is greater than the cavity lifetime, the response time of the modulator is controlled by the carrier lifetime. In this modulator, high-speed modulation in GHz frequency along with high modulation depth is possible even with low external power. We evaluate the dynamic response of the proposed modulator under the modulating laser pulses with a duty cycle of 50%, as shown in Fig. 8(a). The rise and fall times of the THz intensity are attained by solving Eq. 1, which is related to the excess carrier concentration and the carrier lifetime. When the illumination laser is off, the intensity of the transmitted THz wave increases and reaches a steady state. Afterward, during the illumination time of the laser, the carrier concentration increases in time, and therefore the THz intensity drops to lower values which is dependent on the pump power intensity. The intensity of the transmitted THz wave under laser power of 30, 50, and 80 W/cm 2 at modulation frequencies of 0.5, 1, and 2 GHz are illustrated in Fig. 8 (b-d). One can notice that a lower modulation depth is achieved when the modulation speed increase from 0.5 to 2 GHz. For the modulation rate of 0.5 GHz, the transmitted terahertz intensity is a square wave. As the modulation rate increases to 2 GHz, the modulated terahertz waveform becomes a semi-triangle wave.
Based on the analysis mentioned above, this modulator demonstrates that its modulation properties depend strongly on pump power intensity. Figure 9 shows the device performance characteristics at 1 THz under various modulation rates up to 3 GHz. This figure indicates that by increasing the modulation rate up to 1 GHz, the modulator exhibits no drop in modulation depth, but at higher rates, modulation depth decreases significantly. It is due to the insufficient absorbed energy in the interval of laser irradiation at higher modulation rates, and therefore much fewer free carriers are to be generated. Figure 9 also indicates that the pumping  power also has a considerable effect on modulation. Higher modulation depths appear as the pumping power rises to higher levels. Therefore, according to the requirement of considerable modulation depth, one should involve a laser with a proper optical power. As a result, enhancing the laser intensity from 30 W/cm 2 to 80 W/cm 2 , modulation depths from 72 to 91% were observed at a modulation rate of 0.5 GHz. Increasing the speed to 3 GHz and under the same laser intensities, modulation depths from 56 to 77% were achieved.

Conclusion
In summary, we designed and optimized a high-speed and highly efficient THz intensity modulator with external power-dependent characteristics. The proposed structure is composed of a photonic crystal cavity that makes a strong interaction between THz waves and included photoconductor GaAs. We have demonstrated that variation of GaAs conductivity by laser illumination, causes the strong modulation of THz waves. Engineering the structural parameters of this device provides tunability for the desired function. Furthermore, numerical investigation at 1 THz exhibited excellent performance, with the ability to follow a high-speed modulating signal up to 3 GHz. With the carrier wave of 1 THz, for the modulating rate of 1 GHz, and under the optical power intensity of 80 W/cm 2 , high modulation depths more than 90% were observed. In a similar situation at the modulating rate of 3 GHz, the modulation depth of 76% was found. Also, higher modulation depths of up to 99% could be achieved by increasing the optical power intensity further. These unique characteristics of the modulator make it a good choice for developing future high-speed and energy-efficient THz communication systems. Fig. 9 Dependence of modulation depth of a 1 THz carrier wave on the rate of the modulating optical pump driven with a squared shape optical wave with three various intensities