## Abstract

The ever increasing demand for bandwidth in wireless communication systems will inevitably lead to the extension of operating frequencies toward the terahertz (THz) band known as the ‘THz gap’. Towards closing this gap, we present a multi-level amplitude shift keying (ASK) terahertz wireless communication system using terahertz spatial light modulators (SLM) instead of traditional voltage mode modulation, achieving higher spectral efficiency for high speed communication. The fundamental principle behind this higher efficiency is the conversion of a noisy voltage domain signal to a noise-free binary spatial pattern for effective amplitude modulation of a free-space THz carrier wave. Spatial modulation is achieved using an an active metamaterial array embedded with pseudomorphic high-electron mobility (pHEMT) designed in a consumer-grade galium-arsenide (GaAs) integrated circuit process which enables electronic control of its THz transmissivity. Each array is assembled as individually controllable tiles for transmissive terahertz spatial modulation. Using the experimental data from our metamaterial based modulator, we show that a four-level ASK digital communication system has two orders of magnitude improvement in symbol error rate (SER) for a degradation of 20 dB in transmit signal-to-noise ratio (SNR) using spatial light modulation compared to voltage controlled modulation.

© 2016 Optical Society of America

## 1. Introduction

The demand for high speed wireless access is increasing due to large amounts of data needed for new emerging applications such as the consumer market that is already demanding 20, 40 and 100 Gbit/s wireless technologies for Super Hi-Vision (SHV) and Ultra High-Def (UHD) TV data [1]. Terahertz (THz) carrier frequencies will offer the advantage of higher data speed, sub-millimeter antenna size and short range security especially suitable for portable devices. Although they are susceptible to atmospheric loss, THz digital communication systems have been demonstrated near certain windows, especially around the 300–400 GHz range that has shown promise for high bit-rate data transmission [1–3]. In addition to THz sources and detectors, one of the key components of a THz communication system is a modulator that is used to modulate the carrier waves with data streams. Designing high-speed modulators for the frequency regime (0.3 – 10 THz, *λ* = 1*mm* – 30*μm*), so called ’terahertz gap’ [4], has been a difficult primarily due to lack of suitable materials for constructing electronic or quasi-optical devices. In recent years, the research in metamaterial [5] has slowly narrowed this gap by showing promise in applications ranging from bio-detection to security screening [6–9]. But most metamaterial based terahertz modulators have been limited to high voltage and low speed devices [10,11] that makes it unsuitable for the applications that are demanding high-speed and low-voltage operations. We use a metamaterial based THz modulator with embedded HEMT that has been demonstrated to operate at high speed (∼ 10*MHz*) and operating at as low as 1V makes it suitable for high-speed and low-power communication systems [12]. With a crowded RF spectrum, high data rate and spectral efficiency are top priority in wireless communication systems [13]. The next step towards that goal is to create spectral efficient multi-level amplitude shift keying (ASK) modulators for terahertz frequencies. A simple block diagram of a terahertz communication system based on multi-level ASK is shown in Fig. 1. In this system, voltage controlled terahertz modulators [11, 12] can be used for multi-level terahertz modulation by applying different control voltages for their respective transmit symbols. In addition to poor terahertz sources and detectors, noisy modulator electronics result in low SNR which inherently limits the number of levels for amplitude modulation.

We present a novel modulation technique that breaks this barrier from low SNR by using a spatial light modulator consisting of electronically controlled metamaterial tiles in array, where each tile is controlled by an equivalent binary signal, ‘high’ or ‘low’, and the multilevel ASK modulation is effectively and indirectly achieved by choosing the number of tiles based on the transmit symbol. In principle, this converts a multi-level voltage domain to multi-element spatial domain making it immune to voltage noise in the transmit control circuitry since there is no such thing as ‘spatial noise’. This results in a higher spectrally efficient terahertz modulator since you can transmit more symbols for a given voltage noise in the transmit circuitry when using a spatial light modulation compared to voltage controlled modulation. The idea is extensible to visible light communication and RF communication as well. Using an active metamaterial based terahertz modulator, we demonstrate two orders of magnitude improvement in symbol error rate (SER) for 20 dB degradation in signal-to-noise ratio (SNR) in the transmit circuitry. In the next section we detail the principle behind the terahertz modulator and it’s design fabrication techniques. We then show the terahertz characterization result for both voltage and spatial modulation. Finally, from the characterization results we develop a terahertz communication model and compare the SER from analysis and simulation for both the modulation schemes.

## 2. Operating principle and device fabrication details

The basic principle of our terahertz (THz) spatial light modulator (SLM) is based on a electronically controlled absorption coefficient in a metamaterial using embedded pHEMT devices as described below. For a metamaterial designed with electric-LC (ELC) resonators [14, 15] shown in Fig. 2(a), the average permittivity, without factoring spatial dispersion, can be expressed in a Drude-Lorentz form [16, 17]

*ε*

_{∞}is the permittivity of the base substrate material and

*f*is the plasma frequency, generally related to the geometry.

_{p}When the pHEMT is ”off” (*V _{GS}* = −1

*V*), the values for

*L*and

_{HEMT}*R*are negligible and the resonant frequency

_{HEMT}*f*

_{0}can be expressed in terms of the equivalent circuit parameters shown in Fig. 2(b) as

*C*is the capacitance associated with the split gap of the metamaterial as shown in the close-up drawing in Fig. 2(c) and

_{MM}*L*is the inductance associated with the current loop in each half of the metamaterial loop. The damping factor Γ

_{MM}*, related to the loss in the ELC,*

_{e}*R*, can be expressed as [16] See appendix A.1 for a brief explanation and first-order expression for the equivalent circuit.

_{LOSS}The ELC usually does not exhibit a strongly dispersive permeability (*μ̄*(*f*) ≈ *μ*_{0}). Since the dispersion relation is complex, *ε̄* (*f*) = *ε*_{1}(*f*) + *iε*_{2}(*f*), the absorbtion of the transmissive terahertz wave, is directly proportional to the imaginary part of the dispersion *ε*_{2}(*f*) and, the maximum absorption occurs at the resonant frequency *f*_{0} known as the *anomalous dispersion* [18].

When the pHEMT is “on” (*V _{GS}* = 0

*V*), a high-mobility, two-dimensional electron gas (2-DEG) channel is formed between the drain and the source of the pHEMT which can be described by the Drude conductivity [19]

*σ*

_{0}is the DC conductivity [Siemens/sq] given by the expression

*σ*

_{0}=

*e*

^{2}

*N*/

_{S}τ*m*

^{*}. Here,

*e*is the electron charge,

*N*is the two-dimensional carrier density of the 2-DEG,

_{S}*τ*is the scattering or relaxation time,

*m*

^{*}is the effective electron mass and

*ω*the angular frequency of the incident time-varying field.

The equivalent circuit for the 2-DEG can be represented as a series R-L structure shown in Fig. 2(b) where *R _{HEMT}* = 1/

*σ*

_{0}and

*L*=

_{HEMT}*τ*/

*σ*

_{0}, as derived in Eq. (12) and Eq. (13) in appendix A.2. After converting the equivalent circuit into a single parallel R-L-C circuit as explained in appendix A.2, the new resonant frequency of the metamaterial

*f′*

_{0}and the damping frequency Γ′

*are expressed in Eq. (14) and Eq. (15). As shown later in the experimental results, the new resonance is dominated by the damping factor because of the pHEMT conductance thus attenuating the transmissive terahertz wave and hence creating a terahertz modulator at the resonant frequency of the metamaterial. The principle of modulation is explained in appendix A.2 and the terahertz absorption plot (*

_{e}`Im`[

*ε̄*(

*f*)]) shown in Fig. 8 with realistic values of

*f*

_{0}and Γ

*for*

_{e}*V*= 0

_{GS}*V*and

*V*= −1

_{GS}*V*. Assembling such actively and individually controllable metamaterials as tiles in an array, one can construct an all solid-state electronically controlled THz spatial light modulator (SLM).

The metamaterial design was fabricated using a commercial GaAs process with an active device that is a planar-doped pseudomorphic HEMT (pHEMT) based on AlGaAs-InGaAs-AlGaAs quantum well heterostructure [20]. Based on the design principle explained in the previous section, the metamaterial device is constructed of a planar array of subwavelength-sized ELC resonators using the top metal layer, 2.1*μm* thick gold (Fig. 2(a)). Dimension of each element is 42 *μ*m wide by 30 *μ*m in height and they are repeated with a period of 55 *μ*m × 40*μ*m. The line width of the metamaterial is 4*μ*m and the split gap is 3 *μ*m. The metamaterial was designed to be resonant at 0.45 THz using a commercial finite difference time domain (FDTD) solver. A pHEMT is embedded in the split gap of each ELC element [12] [Fig. 2(c)] which is a standard device offered in the process, that is constructed using pseudomorphic undoped InGaAs and a lightly doped Schottky layer creating a heterojunction. The gate voltage with respect to the source (or drain), *V _{GS}*, controls the charge density in the 2-DEG layer between the split-gap and thus changing the resonant frequency as explained in the previous section. When the metamaterial is ”on” (

*V*= −1

_{GS}*V*), the charge carriers in the 2-DEG is completely depleted resulting in the metamaterial functioning at it’s designed resonant frequency(

*f*

_{0}). And when it is ”off” (

*V*= 0

_{GS}*V*), a high-mobility (∼ 3000

*cm*

^{2}/

*V*·

*s*) channel with a computed charge density of

*N*= 7.37 × 10

_{S}^{12}

*cm*

^{−2}is formed between the split gap thus shifting the resonant frequency to

*f′*

_{0}as explained in the previous section. Because of the localized pHEMT in the split-gap, the device has also been demonstrated for fast modulation [12](∼10

*MHz*) that is capable of much higher speed with proper design. Compared to other electrically controlled metamaterial [10, 11], where the whole substrate is used to control the resonance, this device offers element-level control that offers higher switching speed which one of the key performance metrics for high speed terahertz communication systems.

To demonstrate our terahertz SM, we constructed a 2×2 element device, where each element is 1.0 × 1.3*mm*^{2} with 551 elements with an active area of 0.88 × 1.12*mm*^{2} for each element as shown in Fig. 2(d). The 2×2 tile array is die-attached (only at the corners) to a high-resistive silicon (*ρ* − *Si*) substrate that is mounted on a FR-4 based PCB with a hole underneath the *ρ* − *Si* to allow THz transmission. All the gates of the HEMTs for each tile are connected to a 100 *μm X* 100 *μm* bond pad and all the metamaterials for each tile are connected to a separate bond pad which provides the DC bias for the drain/source connection. These bond pads are bonded out to the test circuit board such that each tile can be biased (*V _{GS}*) independently thus enabling a 2×2 electrically controlled terahertz SLM.

## 3. Results

#### 3.1. Measurement setup

A schematic of the THz imaging and characterization system is shown in Fig. 3. Our terahertz spatial light modulator (SLM) is characterized in transmission mode using a commercial continuous-wave (cw) THz spectroscopy system, TeraScan 1550 by Toptica Photonics [21]. The cw THz spectrometer generates linearly polarized THz frequency from 60 GHz to 1.2 THz using a pair of tuned lasers (1546 and 1550nm). The temperature controlled beat frequency is fiber-coupled to a InGaAs photo-diode with a bow-tie antenna which is bias modulated (±1.2*V*) at 7.629 kHz. The THz focusing optics consist of two 76.2 mm diameter 90° off-axis parabolic mirrors (OAPMs) each with an effective focal length of *f _{L}* = 152.2

*mm*. The source-side OAPM,

*f*from the source, collimates the THz beam to the second OAPM that focuses the radiation on the SLM that is

_{L}*f*away. The THz detector (fiber-coupled InGaAs photo-mixer) is placed right behind the SLM to measure the aggregate THz radiation power passing through the SLM. Since the diameter of receiver’s semi-hemispherical silicon lens is approximately the size of the SLM, we avoid the second set of optics to give us better fidelity i.e. signal-to-noise ratio (SNR). The SLM is orientated such that, the linearly polarized electric field of the THz wave is across the split gaps of the metamaterial elements.

_{L}The detected photo-current is pre-amplified using a programmable gain amplifier (PGA) and then lock-in detected (*τ* = 620 *ms*) using Toptica’s proprietary FPGA module. The detected photocurrent *I _{ph}* can be expressed as

*I*∝

_{ph}*E*(Δ

_{THz}cos*ϕ*) [22] where,

*E*is the amplitude of the terahertz electric field and Δ

_{THz}*ϕ*is the phase difference between the terahertz wave and the laser beat signal at the detector. Therefore, the detected photocurrent

*I*oscillates with the THz frequency with the period set by the length of the terahertz beam. The frequency response of a sample is calculated by measuring the envelope of the oscillating

_{ph}*I*. The oscillating period limits the frequency resolution, which was ≈ 0.2 GHz for our setup. Please note that, in the rest of manuscript we will refer to the electric field of the incident terahertz wave on the detector as

_{ph}*I*in nano amperes that is proportional to the strength of the incident terahertz electric field.

_{ph}This setup was used to characterize both the voltage and spatial modulation and the measured data is used to model the THz communication system described in a later section.

#### 3.2. Voltage modulation characterization

We characterized the voltage controlled modulation and spatial modulation in a transmission geometry using the experimental apparatus shown in Fig. 3. The transmission spectra for each modulation configuration is result of a frequency scan from 447 GHz to 455 GHz with a step size of 0.005 GHz and lock-in time-constant of 620 ms. For voltage controlled modulation, the gate-to-source voltage (*V _{GS}*) of all the embedded HEMTs was swept from 0V to −1V with a step size −0.125V and the transmission spectra for each

*V*(

_{GS}*I*

_{ph,VGS0}(

*f*) =

*I*(

_{ph}*f*)[

*V*

_{GS0}]) was measured as shown in Fig. 4(a). In order to emphasize the modulation depth near the modulation frequency, the differential transmission spectra for each

*V*(Δ

_{GS}*I*

_{ph,VGS0}(

*f*)) was calculated using the data from Fig. 4(a) as shown in Fig. 4(b) which can be expressed as Δ

*I*

_{ph,VGS0}(

*f*) =

*I*

_{ph,VGS0}(

*f*)−

*I*(

_{ph_ref}*f*) where,

*I*(

_{ph_ref}*f*) =

*I*

_{ph,0V}(

*f*) i.e. the transmission spectra when all the metamaterial is ”off”. A total of 36% modulation is observed at 450.2 GHz. In order to demonstrate the 4-level ASK THz communication system,

*V*values were calculated for equal discrete steps of |Δ

_{GS}*I*| ≈ 0.4

_{ph}*nA*using the function Δ

*I*(

_{ph}*V*) as shown in Fig. 4(c). Δ

_{GS}*I*(

_{ph}*V*) is expressed by fitting data points of

_{GS}*I*

_{ph,VGS0}(

*f*

_{0}) at 450.2 GHz in Fig. 4(b) using a 3

*−*

^{rd}*order*polynomial fitting function. This enables us to achieve uniform spacing of modulated THz wave in spite of the inherent nonlinearity in our modulator. Let Φ

*(*

_{I}*V*) denote the function Δ

*I*(

_{ph}*V*) that can be expressed as

_{GS}#### 3.3. Spatial modulation characterization

For characterizing spatial modulation, each of the four spatial tiles were turned “on”(*V _{GS}* = −1

*V*) and “off”(

*V*= 0

_{GS}*V*) sequentially in a clockwise or anti-clockwise direction while the transmission spectra using the envelope of the detected current was measured as shown in Fig. 5(a). As in the case of voltage modulation, in order to emphasize the modulation depth near the resonant frequency, the differential transmission spectra for each incremental element (Δ

*I*(

_{phN}*f*)) was calculated using the data from Fig. 5(a) as shown in Fig. 5(b) which can be expressed as Δ

*I*(

_{phN}*f*) =

*I*(

_{phN}*f*) −

*I*(

_{phN_ref}*f*) where,

*I*(

_{phN_ref}*f*) =

*I*

_{ph,0V}(

*f*) i.e. the transmission spectra when all the elements are “off”. A total of 36% modulation is observed at 450.2 GHz with an average of 9% modulation for each element. Figure 5(c) shows the envelope photocurrent of the differential transmission spectra at 450.2 GHz in Fig. 5(b), Δ

*I*(

_{phN}*f*

_{0}) is plotted as function of number of spatial elements ”on”. As evident from these plots, binary switching the tiles on and off has an equivalent effect of modulating the terahertz wave. These Δ

*I*values are used in the system simulation model where we lay the foundation for multilevel THz amplitude modulation using effective spatial light modulation.

_{phN}#### 3.4. Analysis and simulation of symbol error rate (SER) for a multilevel amplitude modulation THz communication system

An important performance metric almost universally used for a digital communication system that is corrupted by noise, is the probability of error in the output signal commonly measured as bit error rate (BER) for a binary signaling system. For a multilevel signaling system like ours, the same metric is measured as symbol error rate (SER). For a binary signaling system (voltage levels ±*V _{P}*) with zero mean additive white Gaussian noise (AWGN) and mid-point threshold for signal detection, the BER (

**P**(

*error*) =

**P**

*) can be expressed as [23]*

_{e}The system model used for our analysis and simulation of symbol error rate (SER) is shown in Fig. 6. For the purpose of demonstration, a 2×2 spatial light modulator was characterized (Fig. 5) and is modeled in this system as a 4-level terahertz amplitude modulator that can transmit 2-bit (log_{2}(4)) symbols. In order to create a platform for comparison, the voltage modulator was characterized (Fig. 4) and 4-level modulation model was extracted from it which is also implemented in the system as shown in Fig. 6.

For voltage modulation, the bit stream is converted to 2-bit symbols that is a gate-to-source voltage (*V _{GS}*) chosen from 4 values within the range (0,−1V) as shown in the inset table in Fig. 6.

*Additive white Gaussian noise*(

*AWGN*) is added to this

*V*symbol (

_{GS}*σ*in Fig. 6) and applied to the terahertz modulator model which represents all the electronic noise referred at the output of the

_{N,TX}*V*control circuitry. The output of the terahertz modulator model is a differential photocurrent (Δ

_{GS}*I*) based on the

_{Ph}*V*− Δ

_{GS}*I*relation derived from voltage modulation characterization as shown in Fig. 4(c). Noise (AWGN) is added to this detected differential photocurrent (

_{Ph}*σ*in Fig. 6) which represents noise in the terahertz channel and the electronic noise in the demodulator referred at the input of the demodulator. The resultant differential photocurrent is demodulated using corresponding thresholds to slice the input into one of the four symbols and the transmitted bit stream is reproduced from it.

_{N,RX}For spatial modulation, the bit stream is also converted to 2-bit symbols that represents the number of spatial tiles that are ”on” based on the inset table in Fig. 6. Since this mapping from bit stream to the spatial modulation inherently remains binary in nature, the voltage noise in the transmit circuitry does not affect the noise performance of the spatial modulation system. The rest of the model is exactly same as described in the voltage modulation system. The additive noise after the terahertz detector affects the spatial modulation the same way as the voltage modulation case.

In order to demonstrate the noise advantage of spatial modulation over voltage modulation, we analyze the SER of the system for a zero mean AWGN and a variance of
${\sigma}_{N,\hspace{0.17em}TX}^{2}$ and
${\sigma}_{N,\hspace{0.17em}RX}^{2}$ for the transmit and receive noise respectively. Based on the same principle of derivation as for BER in Eq. (6), we can express the symbol error rate (SER = **P**(*error*) = **P*** _{e}*) of our system as

*(*

_{I}*V*) − Δ

_{GSk}*I*| is the signal strength of the

_{phk,th}*K*symbol with respect it’s threshold, Φ

^{th}*(*

_{I}*V*) expresses

_{GSk}*K*the symbol voltage referred at the input of the demodulator using the function Φ

^{th}*(*

_{I}*V*) in Eq. (5) with reference to their corresponding threshold current (Δ

*I*) that is used to slice a received signal for detecting the transmitted symbol. Similarly, the transmit voltage noise is referred at the input of the demodulator using the function in Eq. (5) as ${\mathrm{\Phi}}_{I}^{2}\left({\sigma}_{N,TX}\right)$ which is added to the receive noise power ${\sigma}_{N,RX}^{2}$ to account for the total noise power in the system.

_{phk,th}*γ*is the fitting co-efficient to account for the non-linear function Φ

*(*

_{I}*V*) and the unequal magnitude of the

*V*symbols.

_{GSk}In order to verify our analysis, the digital communication system was simulated using a common computer-aided technique know as the Monte Carlo method [24]. It is in essence a time-domain technique that sequentially simulates a deterministic or random bit-stream with added noise. If *N* symbols are processed through the system, out of which *n* are observed to be in error, the SER can be expressed as the sample mean: *p̂* = *n*/*N*. In the limit *N* → ∞ the estimate *p̂* will converge to the true value *p*. For finite *N*, it has been shown that *N* should be in the order of 10/**P*** _{e}* [24]. We simulated with

*N*= 10

^{7}bits for a high certainty SER output from our simulation. The simulation was done for each value of the signal-to-noise ratio (SNR) for both the voltage and spatial modulation.

Figure 7(a) shows the calculated and simulated SER for voltage and spatial modulation as function of the transmit SNR which can be expressed as
$\u3008{V}_{\mathit{GSk}}-{V}_{\mathit{GSk},\mathit{th}}\u3009/\sqrt{2}{\sigma}_{N,TX}$ where, the numerator is average of the *V _{GS}* symbols with reference to the their corresponding thresholds and the denominator is standard deviation of the added transmit noise. The plots show a good match of the simulated SER with the calculated values, for

*γ*= 0.75. It can be clearly observed from the voltage modulation plot, the SER increases by almost two orders of magnitude with a 20 dB decrease in SNR, where as the SER for the spatial light modulation remains unchanged. This confirms our design objective which is predicted by our analysis that spatial light modulation provides immunity to voltage noise in the transmit electronics compared to voltage controlled modulation. The SER for voltage modulation is asymptotic at ≈ 10

^{−3}due to the constant receive noise of

*σ*= 0.05 nA. It is observed that the SER of the spatial light modulation is better at higher SNR due to an artifact related to the non-linear relation of

_{N,RX}*V*− Δ

_{GS}*I*as seen in Fig. 4(c) and Eq. (5).

_{ph}Figure 7(b) shows the calculated and simulated SER for voltage and spatial light modulation as function of the receive SNR which can be expressed as
$\u3008{\mathrm{\Phi}}_{I}{V}_{\mathit{GSk}}\u3009-\mathrm{\Delta}{I}_{\mathit{phk},\mathit{th}}/\sqrt{2}{\sigma}_{N,RX}$ where, the numerator is average of the receive symbols with reference to the their corresponding thresholds and the denominator is standard deviation of the added receive noise. The plots show a good match of the simulated SER with the calculated, for *γ* = 1.0. It can be observed that both modulation schemes have decreasing SER with decreasing SNR. This is expected from analysis as the added receive noise affects both the schemes equally. Again, the spatial light modulation still shows better SER due to an artifact related to the non-linear relation of *V _{GS}* − Δ

*I*as seen in Fig. 4(c) and Eq. (5).

_{ph}## 4. Conclusion

In conclusion, we have demonstrated a terahertz communication system based on active metamaterial-based spatial light modulators that is immune to voltage noise in the transmit electronics compared to voltage controlled modulation. This enables multi-level amplitude modulation which otherwise would not have been possible due to low SNR. We show experimental results on both voltage and spatial light modulation. And using models created from such experimental measurements, we show excellent immunity to transmit electronic noise as predicted by our analysis. Monte Carlo simulations confer with our analysis showing two orders of magnitude improvement in symbol error rate (SER) for 20 dB SNR degradation due to transmit voltage noise. The result of this work is a very important step towards realizing higher spectral efficient modulation technique eg. multi-level ASK, for high speed wireless terahertz digital communication systems.

## A. Appendices

## A.1. Circuit model for the electric-coupled LC (ELC) resonator

When *V _{GS}* = −1

*V*, ie. the pHEMT is ”off”, the ELC resonator can be represented by an equivalent RLC circuit as shown in Fig. 2(b) constituting

*L*,

_{MM}*C*,

_{MM}*R*, where in its simplest form,

_{LOSS}*C*is the capacitor associated with the split gap of the ELC,

_{MM}*L*is the inductor associated with the circulating current in the two symmetric loops. And

_{MM}*R*is the resistive loss of the metal (gold) for the circulating inductor current. The effective permeability of such a metamaterial can be expressed in a Drude-Lorentz form [16, 17] as expressed in Eq. (1). The capacitor

_{LOSS}*C*can be expressed as

_{MM}*C*=

_{MM}*K*

_{SiN3}

*ε*

_{0}

*A*/

*s*, but a more accurate expression can be used from [16]

*H*and

*W*are the total height and width of the split gap and

*α*is a geometry dependent constant. The inductor can be expressed in its first-order form as [16]

*l*is the length of the inductive loop and

*b*is the width of the conducting strip.

*R*can be approximated as [31] where

_{LOSS}*l*is the length of the conductor forming the inductive loop,

*σ*is conductivity of conductor,

*P*is the perimeter of the cross-section of the conductor and

*δ*is the skin depth of conductor at

*f*

_{0}.

The resonant frequency for the dispersion Eq. (1) can be calculated using Eq. (2). The damping factor Γ* _{e}* in Eq. (1) can be calculated as [16]

## A.2. Principle of voltage controlled terahertz wave modulator

When the gate-to-source voltage (*V _{GS}*) of the pHEMT is increased towards

*V*= 0

_{GS}*V*, a two-dimensional electron charge (2-DEG) channel layer between the source and drain is formed, which can be described by the Drude conductivity [19, 30], as expressed in Eq. (4). See reference [20] for a relation between carrier concentration and

*V*. This conductance can be represented as a series R-L structure (

_{GS}*L*,

_{HEMT}*R*) as shown in Fig. 2(b), that can be expressed as

_{HEMT}*L*is typically referred to as the

_{HEMT}*kinetic*inductance [30] and the

*R*is the DC resistance of the 2-DEG channel. It should be noted that the resistance of the ohmic layer connecting the pHEMT to the split also appears in series with

_{HEMT}*L*−

_{HEMT}*R*which can be combined into one series R-L element. It can be seen from Fig. 2(a), that the source and drain of the pHEMT is connected directly to the split gap, therefore, the pHEMT circuit model,

_{HEMT}*L*−

_{HEMT}*R*, is connected across the metamaterials capacitor

_{HEMT}*C*in the equivalent circuit shown in Fig. 2(b).

_{MM}In order to find the new resonant frequency *f′*_{0} and the damping frequency Γ′* _{e}* when the pHEMT is “on” (eg.

*V*= 0

_{GS}*V*), the circuit in Fig. 2(b) can be converted to a single, parallel R-L-C circuit with elements

*R*,

_{E}*L*,

_{E}*C*using series-parallel circuit transformation [31]. After the transformation, the resonant and damping frequency can be expressed as

_{MM}*f′*

_{0}≈

*f*

_{0}and Γ′

*is dominated by the pHEMT conductance and therefore, Γ′*

_{e}*≫ Γ*

_{e}*.*

_{e}The principle of our modulation can be demonstrated by plotting the imaginary part of the dispersion Eq. (1) which is directly proportional to the absorbtion of the electromagnetic wave transmitting through the metamaterial sample. For the purpose of demonstration, we choose a some realistic values for *V _{GS}* = 0

*V: f*

_{0}= 0.55

*THz*, Γ

*= 0.15*

_{e}*THz*and

*V*= −1

_{GS}*V: f*

_{0}= 0.5

*THz*, Γ

*= 0.025*

_{e}*THz*. Figure 8 shows the absorbtion plot for our drude-lorentz model dispersion of Eq. (1). As seen from the plot, the primary reason for the modulation is due to the increase in the damping factor or loss in the ELC resonator due to the conductance in the pHEMT that shunts the metamaterial capacitor

*C*.

_{MM}## Acknowledgments

This research was funded in part by the Office of Naval Research under U.S. Navy Contracts N00014-09-1-1075 and the National Science Foundation Awards No. ECCS-1002152. We also acknowledge the support of DoD Defense University Research Infrastructure Program (DURIP) under grant N00014-12-1-0888 equipment purchase used for test and characterization. We acknowledge Willie Padilla and David Shrekenhamer at Boston College for their collaboration on the initial effort for design of the ELC resonator and discussion and comments regarding our experimental setup. We thank Jessie Tovera at Qualtre Inc. for his valuable support in the SLM assembly.

## References and links

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