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Abstract
This paper demonstrates a methodology to miniaturize THz spectroscopes into a single silicon chip by eliminating traditional solid-state architectural components such as complex tunable THz and optical sources, nonlinear mixing and amplifiers. The proposed method achieves this by extracting incident THz spectral signatures from the surface of an on-chip antenna itself. The information is sensed through the spectrally-sensitive 2D distribution of the impressed current surface under the THz incident field. By converting the antenna from a single-port to a massively multi-port architecture with integrated electronics and deep subwavelength sensing, THz spectral estimation is converted into a linear estimation problem. We employ rigorous regression techniques and analysis to demonstrate a single silicon chip system operating at room temperature across 0.04–0.99 THz with 10 MHz accuracy in spectrum estimation of THz tones across the entire spectrum.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Miniaturization of complex THz technology into low-cost chip-scale platforms can enable a wide diversity of applications in sensing, imaging and spectroscopy due to its desirable properties such as penetration capability through optically opaque materials, non-ionizing photon energy, and unique spectral signatures for large bio-molecules and chemicals [1–5]. In the past decade, this effort has progressed rapidly across a wide array of chip-scale technologies including integrated circuits [6], quantum cascade lasers [7], micro-bolometers [8], nanowires [9], plasmonic structures [10], metamaterials [11], and photo-mixing in ultrafast photoconductive semiconductor materials [12]. Recently, silicon-based integrated circuit technology has been demonstrated to be capable of THz signal generation and detection including fully integrated beam-formers with mW-level output power and large-scale imagers operating at room temperature [13, 14, 16–22].
THz spectroscopy is one of the promising applications in the frequency range. Traditional time-domain terahertz spectroscopy relies on a collection of optical components including femtosecond laser, photoconductive substrates, nonlinear optical elements and mechanical delay lines which results in typically bulky and expensive system with low integration level [1]. On the other hand, finer resolution can be achieved over a much narrower frequency band in solid-state-circuit-based spectroscopy systems with coherent down-conversion receiver architectures. However, the spectrum analysis range in such systems is fundamentally limited by the tunable range of an available THz source (Fig. 1) [23, 24]. This is true because the architecture examines the various portions of the incident spectrum (after capturing with a broadband antenna) by mixing it down in frequency and digitizing it. Therefore, as shown in Fig. 1, wide band spectroscopy spanning GHz-THz requires a tunable source across GHz-THz along with GHz-THz amplifiers and mixers. The integration of these extremely wideband circuits for a single chip system, particularly the tunable GHz-THz source, is extremely challenging, inefficient and even impractical. Much narrower analysis range is also carried off with a large bank of external THz sources including frequency synthesizers and frequency multipliers [23].
Fig. 1 Proposed concept of exploiting subwavelength near-field sensing for spectral estimation of incident signal. Three examples of current distribution excited by continuous waves at three different frequencies with incident power of 1.33 nW.
In this paper, we miniaturize a THz spectroscopic receiver system into a single chip by converting the on-chip antenna into a spectrum sensing device. This eliminates the need for such wide band signal generation, mixers and amplifier circuits and the entire down-conversion architecture following the antenna. This is done by modifying the antenna from a classical single-port structure to a massively multi-port active electromagnetic surface with sub-wavelength spaced detectors that can measure the spectrally-sensitive 2D distribution of THz near-fields. We show that utilizing this spatial distribution of fields as a measure of the incident spectrum converts the spectroscopy problem into a linear estimation problem. In that regard, robust estimation methods that rely on noisy detector outputs and still perform sensitive spectral estimation is critical to the functionality of the proposed integrated architecture. Since the proposed methodology does not employ a THz source, the upper limit of spectral analysis is only set by the spectral sensitivity determined by the noise of the on-chip detectors.
The previous work demonstrated spectral estimation across 290 GHz (0.04–0.33 THz) [15], [16]. In this paper, we extend the spectrum range by a factor of 3.2× showing spectrum detection capability spanning from 0.04–0.99 THz with an accuracy of detection of upto 10 MHz for continuous-wave (C.W.) excitations. With an analysis range of 0.95 THz, this is the widest fully integrated THz spectroscope in silicon technology [23], [24]. In addition, we focus on data analysis methods to increase sensitivity of detection and reduction of spectral noise through robust estimation and regression methods. We compare techniques such as least squares, LASSO (Least Absolute Shrinkage and Selection Operator) typically applied in machine learning and statistical estimation [25–27], and non-negative estimation and analyze their performance in spectrum analysis for both narrow-band and wideband signals. As a proof-of-concept, a chip is implemented in a 0.13 µm SiGe BiCMOS technology with an area of 1.9mm × 2.6mm. Powered by a low-power battery, measurement results show that the chip has estimation capability for both narrowband and wideband cases, and achieves a resolution of prediction accuracy up to 10 MHz for C.W. excitations over 40–990 GHz. While the chip can function even at higher frequencies (due to non-requirement of an integrated source), the upper frequency of 990 GHz is currently limited by the C.W. sources in our lab comprising of frequency multipliers with Schottky diodes.
2. Spectral estimation principle
Typically, in a THz spectroscopy setup, the concerned signal is a picosecond pulse train with repetition rate of frep (Fig. 1). Due to the periodic nature of the incident field represented as , the power spectrum of the incident signal can be represented by a vector Efrep = [|a1|2, |a2|2,…, |aM|2]T ∈ RM×1 , where |am|2 represents the power of the mth harmonic. By Maxwell’s laws, the impressed current distribution on the on-chip antenna (Fig. 1) due to the incidence of the field is unique and is also periodic with the same harmonic components represented as . Figure 1 demonstrates examples of the current distribution excited by continuous waves at three different frequencies (100 GHz, 500 GHz and 1 THz). As shown in Fig. 1, due to the periodic nature of the incident current distribution, if N square-law detectors deployed over the antenna surface to measure the local average current distribution power |Js(x, y)|2, the sensor response Sfrep can be simply expressed as [16]
where (X, Y) ∈ RN×1 denotes the positions for the N sensors and |Js(X,Y)|2, |Jm(X,Y)|2 ∈ RN×1 denote the spatial sampling of |Js(x, y)|2 and |Jm(x, y)|2 over the antenna surface.Evidently, the current distribution captured by the sensor response Sfrep is related to the spectrum of the incident signal Efrep and is determined by the properties of the radiating surface and the boundary conditions. Consider that the sensor response to a single C.W. excitation at mfrep be represented as Rm ∈ RN×1. By collecting the responsivity vectors across the incident spectrum from frep to Mfrep, we can define EM responsivty matrix of the antenna represented as Rfrep = [R1, R2, …, RM] ∈ RN×M. This matrix, which is a property of the radiating surface, captures all the necessary information that converts incident spectrum to spatial field distribution. This is similar to an impulse response in a linear system. After a single-time characterization, the measured Rfrep can be used for estimation of arbitrary spectrum incidence. Therefore, when a THz pulse train with a given polarization impinges on the surface, then due to the linearity of the relationship in Eq.(1), the measured sensor response Sfrep ∈ RN×1 can be characterized as
where Vn represents the noise of the detector array. This paper proposes a method to estimate incident spectrum Efrep from the measured near-field information Sfrep. This converts the spectroscopic problem into a linear estimation problem and reduces the entire spectroscope to an on-chip radiating surface with a multitude of integrated near-field sensors.We investigate three different regression methods in the context of narrowband and wideband THz spectral estimation including least squares, LASSO (particularly ridge regression or Tikhonov regularization) and non-negative least square estimations. Given the measured sensor response Sfrep, the estimated spectrum Efrep with least square estimation is given as
This straightforward method to solve for Efrep has large spectral estimation variance under low incidence power due to the detector noise. In order to reduce the spectral noise of the THz estimator, we apply regression techniques that trades variance with bias, typically applied in estimation and machine learning for avoid over-fitting. This is done by adding an energy penalty (λ2 f (Eest)) to the minimization algorithm to eliminate large positive and negative spikes in the spectrum estimation. This can be represented by the generalized LASSO given by [26]
An example of this extra term f(Eest) is given by Ridge regression or Tikhonov regularization where Efrep is given by
where λ2 is a scaling factor which introduces a constraint on the total power of the estimated incident spectrum [25]. Considering this, the solution to the algorithm givesBoth the estimation methods search for solutions of Eest in the entire space of RN×1. Since Eest represents the power spectrum, one can ignore solutions where a single element is negative. Therefore, we can reduce the search space to where Eest ≥ 0 which leads to Eest being given by
This problem is convex, as is positive semi-definite and the non-negativity constraints form a convex set. We will investigate these regression methods for spectral estimation across 0.04-0.99 THz with the implemented chip.
3. System architecture and measured results
Figure 2 shows the architecture of the THz spectroscope implemented in a commercial silicon foundry with a 0.13 µm SiGe BiCMOS process. The chip occupies an area of 1.9mm × 2.6mm. The substrate is 250 µm thick and lossy with bulk resistivity σsub ~13.5 Ω-cm. While the lossy substrate reduces the aperture efficiency of the radiator, in this proof-of concept chip, the substrate is not thinned and no silicon lens is employed. All radiated signals are incident on the chip from the backside of the substrate. C.W. sources based on Schottky-diode based frequency multipliers are used to generate signals at these frequencies. The power is measured with Erickson power meter, which is a calibrated bolometer.
Fig. 2 Structure of the on-chip antenna, distributed sensors and the architecture of the THz spectroscope. Included is the die photo of the chip.
As shown in Fig. 2, the radiator is a log-periodic tooth shaped antenna. However, unlike a classical single-port antenna, no signal is drawn from the center port but the impressed surface current distribution is converted locally to detectable THz voltage swings which can be measured by the sensors distributed over the antenna. In total, 84 detectors are located underneath the antenna surface. As can be seen in the figure, the detectors are placed asymmetrically over the surface to ensure optimal sensing of the localized fields avoiding duplication of the responses. The number of detectors, their locations and loading effects need to be considered to ensure maximal absorption of the incident power and sensitive extraction of incident spectral information.
It can be noted that the number of detectors determine the number of unknown frequency components (not range) that can be calculated from the on-chip measurement. Increasing number of detectors can enable more accuracy, but the loading effects of these detectors on the radiator needs to be considered. This makes the design of the antenna and the massively multi-port detector configuration different from classical single-port antenna-detector topologies. As an example, single-port antenna-detector systems maximize power transfer if the antenna and the detector impedance maintains through passive matching networks over a narrow range of frequencies. For multi-port structures and antennas, maximization of power transfer is ensured by [16]. In this chip, all passive matching networks are eliminated for compactness, and the antenna and detector configurations are co-designed to ensure efficient power absorption into the detectors across the frequency range [16]. Therefore, the total collective power captured by the detectors results in an aperture area comparable to the physical area of the antenna. It can also be noted that if there is no prior information of the nature of the spectrum, then there is no basis of optimizing the locations of the detectors. Random positioning of the detectors considering the loading effects works well [16]. In a particular application if there are specific regions of the spectrum which are of interest in spectroscopy, then the detector locations can be optimized to enable optimal sensitivity in resolution and power.
Each detector consists of a hetero-junction bipolar transistor (HBT) biased in nonlinear region which can rectify the incident signal and generate a response proportional to its absorbed power. The rectified chopped signal from the detector is compared to a reference and amplified by a baseband amplifier chain with a controllable gain of 70 dB. The incident spectrum is estimated from the measured 84 sensor outputs sent out from the chip in a multiplexed fashion.
To analyze the incident spectrum, the EM responsivity matrix (Rfrep) is characterized from 40 GHz to 990 GHz with 1 GHz resolution, and the measured results are shown in Fig. 3. The spectrally-dependent distribution of the 84-sensor array response is evident from the figure. As can be seen in the figure, there are spectral regions where the responsivity drops. These are spectral regions where surface-wave excitations in the silicon chip affects the radiative modes and therefore, the overall responsivity. As can be seen in the figure, the silicon chip rests on a transparent tape (ungrounded). Taking into account these boundary conditions, the mode analysis reveals that the critically affect frequencies are where the substrate height (H) satisfies 2H ≈ (2N + 1)λsi/2, where ‘N’ is an integer. One way of interpreting this is that phase accumulation in the round trip propagation through the substrate equals π [28], [29]. This effect is stronger for higher frequencies (650GHz ~ 3.5λ/2 and 950GHz ~ 5.5λ/2), where the transistor-based detector responsivity drops as well. The surface-wave excitations can be mitigated by adding a silicon lens at the backside. But without even the external lens, the spatially distributed responses in the diodes are much higher than the noise floor to allow spectrum estimation. The relationship between the responsivity and noise is nonlinear in regression analyses and captured in Section 4.
With Rfrep being measured once, the incident spectrum Efrep can be estimated applying regression analysis. To demonstrate the functionality of the architecture, firstly C.W. excitations at unknown frequencies between 40 GHz and 990 GHz are incident on the chip and spectrum estimations are carried from the measured on-chip sensor response. Figure 4 shows some examples of spectral estimation with power levels varying between 100 nW and µ-W level. While least square estimation performs reasonably well at the lower end of the spectrum (~0.2–0.5 THz) with these power levels, at the higher portion of the spectrum (~0.6–0.99 THz), we employ LASSO and non-negative estimation for robust spectral analysis. For 1 GHz resolution, we employ 10 ms integration time and extracting signals in a parallel fashion from the six outputs simultaneously, total estimation time takes around 140 ms. This increases to 14 seconds as we narrow the resolution to 10 MHz due to the longer integration time. We will investigate, in details, their spectral estimation properties, but these examples show that collectively, these regression methods can successfully localize the frequency and incident power level across the incident spectrum simply from the sensor responses.
Fig. 4 (a) Examples of spectral estimation for single frequency excitations with least squares, LASSO and non-negative least squares regressions. (b) Procedure to successively narrow the spectral resolution down to 10 MHz for single tone excitations.
The number of frequency unknowns (bandwidth and resolution) is ultimately limited by the number of detectors. Therefore, in case of narrow-band incident spectra, the accuracy in predicting the peaks can be progressively increased by applying iteratively responsivity matrices with increased resolution on the measured sensor data. This is illustrated in the estimation examples in Fig. 4 when the chip is excited at a single tone between two characterization frequencies. The initial estimates with coarser resolution shows dual peaks around the excitation frequency which can be narrowed down with progressively increased resolution from 1 GHz to 10 MHz iteratively. Further increase in resolution is limited by sensor drift. It is to be noted however, that for the higher frequencies, non-negative least square analysis is employed to minimize output spectral variance. The architecture is versatile for multi-tone excitations as well as with wideband signals as shown in the examples in Fig. 5. It can be seen from the figure that successful spectral estimation can be achieved in both cases.
It can be noted that the surface current distribution and therefore, the estimation methodology depends on the polarization and incidence angle. Measurements shown in [16] demonstrate that the method is robust within ±10° in incidence angle and ±5° in polarization variations.
4. Regression methods for the optimization of the THz spectral noise floor and probabilistic estimation across 0.04–0.99 THz
4.1. Least-square regression
The minimal power needed for successful spectrum estimation is ultimately dependent on the noise of the detectors and the method of estimation. For least-square estimation, with no input signal (Efrep = 0), the estimated spectrum which represents the spectral noise floor is given by
where σ(Vn) is the noise r.m.s voltages of the detectors which was measured to be vn,rms=55.6 µV for a 1 second integration time. Evidently, the spectral noise floor is a function of Rfrep and it trades off with resolution. Figure 6(a) shows the measured spectrum estimation noise floor (σ(ENF)) across 40–990 GHz with frep = 1 GHz. It can be seen from the figure while sensitivity reduces at higher frequency due to the lower responsivity of the integrated detectors, the spectral noise floor varies between remains below 1 µW (10 nW−1 µW) over the spectral range. This sensitivity can be further removed with other regression techniques and also with simple modifications in packaging by thinning the substrate and employing a silicon lens to eliminate the substrate modes.Fig. 6 (a) Measured noise floor (σ(ENF)) of the estimation with least squares regression for 1 GHz resolution across 0.04–0.99 THz (b) Optimization of λopt for lowest spectral error (Δavg) for an excitation at 650 GHz for various incidence power levels. The figure also shows the comparison of spectral estimation with least-squares and LASSO with optimized regularizer. (c)–(e) Estimation quality of single tone excitations across 0.04–0.99 THz with the three regression methods. At low incidence power, LASSO and non-negative estimators can push down sensitivities to nearly −40 to −50 dBm across the spectrum, nearly 10–15 dB below the spectral noise floor achieved with least-square estimators.
4.2. LASSO (ridge regression or Tikhonov regularization)
The spectral noise analysis and sensitivity for reconstruction using generalized LASSO and non-negative least square estimation is a function of the incident spectrum itself. In other words, the estimation error (ΔLASSO) with Tikhonov regularization or Ridge regression consists of two parts: the random part ENF which, as before, is induced by noise of the detectors and the deterministic part ΔInc which is determined by the incident power Eq. (6). In LASSO, we trade-off σ(ENF) with ΔInc as shown below
To investigate the effect of the regularization parameter and the spectrum on the achievable sensitivity, we define Δavg as the average spectral error in the analysis range as
It can be seen that the average error (Δavg) is dominated by the random part σ(ENF) for low incidence powers. Therefore, it monotonically decreases with an increasing λ till ΔInc starts dominating at higher power levels. This is shown in Fig. 6(b) for an incidence tone at 650 GHz with incidence power levels varying from Pinc=−30 dBm to Pinc=−10 dBm. For λ = 0, Δavg = σ(ENF) as shown in Fig. 6(a). However, average spectral error Δavg can be significantly reduced with regularization parameter optimization for low incidence power with some pre-knowledge of the incident spectrum as shown in Fig. 6(b). The figure also shows the effect of the reduction of spectral noise floor with regularization. Figure 6(c) shows much cleaner spectral reconstruction with very low power levels (~ 1 µW at 650 GHz) compared to least-square estimators.
In order to compare the quality of spectral estimations across the three different regression methods, we define a metric called Estimation Quality (EstQ) as [30], [31].
Figure 6(c)–(e) show the comparison of the measured estimation quality across 0.04–0.99 THz for the three regression methods with optimized regularizers. It can be observed that for higher incidence powers, all the methods perform reasonably equally with high estimation qualities. This is not surprising since optimal regularization factor for high incidence power is (λopt ≈ 0), which reduces LASSO to least-squares. At low incidence power, however, the regularization methods reduce the sensitivity to nearly −40 to −50 dBm across 0.04–0.99 THz, nearly 10–15 dB below the spectral noise floor through least-square estimation.
4.3. Probabilistic estimation of THz spectrum
The accuracy of estimating single frequency C.W. excitations for frequency and power can be further increased by combining Tikhonov regularization and non-negative estimator in an iterative procedure. As an example, Fig. 7 shows successful estimation of a single tone excitation with only 20 nW of incidence power at 650 GHz (~ 15 dB below spectrum noise floor in least-squares). This is achieved by pre-estimating the range of frequency occupation or region of ‘spectral activity’ with an initial estimation employing the broadest analysis range. Then, successively, number of unknown frequencies are reduced with progressively smaller frequency range till we reach a single column of Rfrep at Rm. The final estimation is carried with non-negative analysis showing both frequency localization and accurate power estimation.
Fig. 7 Iterative procedure combining Tikhonov regularizer and non-negative estimator to increase the probability of a successful estimation. This begins with the broadest spectral analysis range to search for the region of ‘spectral activity’ and progressively narrowing it down to a single frequency search which is then estimated with a non-negative estimator. The figure shows successful estimation of a single tone excitation with only 20 nW of incidence power at 650 GHz (~ 15 dB below spectrum noise floor in least-squares)
This probabilistic nature of the estimation process is illustrated in Fig. 8. The figure demonstrates the histogram of the estimated spectrum in the first step of estimation process that searches for the region of ‘spectral activity’ when the chip is excited by different power levels at 650 GHz. Below 5 nW, there is no separation of the histogram of the power spectrum at 650 GHz and the rest of the frequencies. At 20 nW of incidence power, the histogram starts separating which can be iteratively processed to more accurate predictions of frequency and power. At ~1 µW of incidence power, the probability distribution converges into the incident spectrum with an impulse at 650 GHz with the accurate incident power level in the first estimation step.
Fig. 8 Histogram of estimated spectrum by the chip when excited at 650 GHz with various incidence power levels. The figure shows that above 20 nW, the probability distribution of spectral activity at 650 GHz starts separating from the rest and can be iteratively processed to more accurate predictions of frequency and power.
The probability of successful estimation using the above three regression methods are summarized in Fig. 9 for various incident power levels, when the chip is excited by single-tone across 0.04–0.99 THz. As can be seen, particularly for lower incidence power levels, a significant enhancement in success rate can be observed for Tikhonov regression and non-negative least squares regression leading to a significant improvement of spectral sensitivity of the chip through regression analysis techniques.
Fig. 9 Probability of successful estimation for different incident power levels across 40–990 GHz using least squares, Tikhonov and non-negative estimators. The regression methods allow us to increase the success rate by reducing the spectral noise floor.
4.4. Spectral noise floor and estimation quality for broadband THz signals
The presented analysis, so far, has considered single frequency excitations with regards to the achievable spectral sensitivity. However, for wideband excitations, evidently the iterative process for increasing accuracy cannot be directly applied. Figure 10 shows the estimation quality (EstQ) for the three different regression methods across 40–990 GHz, when the chip is excited by a wideband signal with Gaussian spectrum and bandwidth varying from 2 to 80 GHz. The total incident power is fixed at 10 µW, 50 µW and 100 µW for 40–300 GHz, 300–600 GHz and 600–900 GHz, respectively.
Fig. 10 Estimation Quality (EstQ) of spectral estimation using least squares, LASSO and non-negative estimators across 40–990 GHz, when the chip is excited by a wideband signal with Gaussian spectrum.
It can be seen from the figure that the estimation quality decreases with an increasing bandwidth as expected, since the total power spreads over a large range of frequency. The figure also demonstrates that regression methods with Tikhonov regularization and non-negative least squares regression methods can provide higher accuracies in spectral estimation results compared to classical least squares regression. Interestingly, non-negative least squares performs better than Tikhonov regression for lower bandwidth signals and vice-versa. Figure 11 presents examples for estimation of Gaussian spectra with bandwidth of 40 GHz and total power of 5 µW centered at 260 GHz and 100 µW centered at 950 GHz. As the figure shows, a strong improvement in estimation can be achieved using Tikhonov and non-negative least squares regression methods. The presented examples demonstrates that through a co-design approach of THz electromagnetics and electronics by combining deep sub-wavelength near-field sensing and regression analyses can enable a new class of THz chip-scale sensory systems.
Fig. 11 Examples for estimation of Gaussian Spectra with bandwidth of 50 GHz and center frequencies at 260 GHz and 950 GHz.
5. Conclusion
The paper demonstrates a single-chip THz spectroscope that extracts incident THz spectral signatures through spatial sensing of on-chip antenna near fields. It combines methods to integrate massively multi-port radiating surfaces on-chip and employs regression analyzes on the measured sensor data for spectral estimation across 0.04–0.99 THz. This allows the architecture to eliminate complex THz and optical sources, nonlinear mixing and amplifiers and reduces the spectroscopic receiver to single chip system. The chip is powered by a low power battery, dissipates 212 mW of DC power, operates at room temperature across 0.04–0.99 THz with 10 MHz accuracy in spectrum estimation of THz tones across the entire spectrum. Combining THz sub-wavelength electromagnetics and electronics in a co-design methodology can lead to a new class of THz chip-scale systems for a wide range of applications.
Funding
National Science Foundation (NSF) (ECCS-1408490); National Science Foundation (NSF) (ECCS-1509560), Office of Naval Research Young Investigator Program (N00014-17-1-2694), DURIP: (N00014-16-1-2655).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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