Data-driven Identification of Parametric Governing Equations of Dynamical Systems Using the Signed Cumulative Distribution Transform.

This paper presents a novel data-driven approach to identify partial differential equation (PDE) parameters of a dynamical system. Specifically, we adopt a mathematical "transport" model for the solution of the dynamical system at specific spatial locations that allows us to accurately estimate the model parameters, including those associated with structural damage. This is accomplished by means of a newly-developed mathematical transform, the signed cumulative distribution transform (SCDT), which is shown to convert the general nonlinear parameter estimation problem into a simple linear regression. This approach has the additional practical advantage of requiring no a priori knowledge of the source of the excitation (or, alternatively, the initial conditions). By using training data, we devise a coarse regression procedure to recover different PDE parameters from the PDE solution measured at a single location. Numerical experiments show that the proposed regression procedure is capable of detecting and estimating PDE parameters with superior accuracy compared to a number of recently developed machine learning methods. Furthermore, a damage identification experiment conducted on a publicly available dataset provides strong evidence of the proposed method's effectiveness in structural health monitoring (SHM) applications. The Python implementation of the proposed system identification technique is integrated as a part of the software package PyTransKit [1].

Parametric Signal Estimation Using the Cumulative Distribution Transform.

We present a new method for estimating signal model parameters using the Cumulative Distribution Transform (CDT). Our approach minimizes the Wasserstein distance between measured and model signals. We derive some useful properties of the CDT and show that the resulting estimation problem, while nonlinear in the original signal domain, becomes a linear least squares problem in the transform domain. Furthermore, we discuss the properties of the estimator in the presence of noise and present a novel approach for mitigating the impact of the noise on the estimates. The proposed estimation approach is evaluated by applying it to a source localization problem and comparing its performance against traditional approaches.

Calculation of the impulse response and phase noise of a high-current photodetector using the drift-diffusion equations.

We describe a procedure to calculate the impulse response and phase noise of high-current photodetectors using the drift-diffusion equations while avoiding computationally expensive Monte Carlo simulations. We apply this procedure to a modified uni-traveling-carrier (MUTC) photodetector. In our approach, we first use the full drift-diffusion equations to calculate the steady-state photodetector parameters. We then perturb the generation rate as a function of time to calculate the impulse response. We next calculate the fundamental shot noise limit and cut-off frequency of the device. We find the contributions of the electron, hole, and displacement currents. We calculate the phase noise of an MUTC photodetector. We find good agreement with experimental and Monte Carlo simulation results. We show that phase noise is minimized by having an impulse response with a tail that is as small as possible. Since, our approach is much faster computationally than Monte Carlo simulations, we are able to carry out a broad parameter study to optimize the device performance. We propose a new optimized structure with less phase noise and reduced nonlinearity.

Frequency dependent harmonic powers in a modified uni-traveling carrier photodetector.

We use a drift-diffusion model to study frequency dependent harmonic powers in a modified uni-traveling carrier (MUTC) phododetector. The model includes external loading, incomplete ionization, the Franz-Keldysh effect, and history-dependent impact ionization. In three-tone measurements, the bias voltage at which a null occurs (bias null) in the second-order intermodulation distortion (IMD2) is different for the sum frequency and difference frequency. We obtained agreement with the experimental results. The bias null that appears in the IMD2 is due to the Franz-Keldysh effect. The bias voltage at which the bias null is located depends on the electric field in the intrinsic region, and the difference in the location of the bias null for the sum frequency and difference frequency is due to the displacement current in the intrinsic region. When the frequency is large, the displacement current is large and has a large effect on the harmonic powers. We also found that the bias null depends on the recombination rate in the p-absorption region because the electric field decreases in the intrinsic region when the recombination rate in the p-region decreases.

Impact of the Coulomb interaction on the Franz-Keldysh effect in high-current photodetectors.

The Franz-Keldysh effect has been recognized as the largest contributor to oscillations in the responsivity of high-current photodetectors as a function of the applied bias or the incident light wavelength and to device nonlinearity. Prior work only considered the effect of the electric field without considering the Coulomb interaction. We show that it is not possible to obtain agreement with experiments at all optical wavelengths without including this effect in the effective mass equation. We find the maxima and minima in the absorption of the applied electric field shift when the Coulomb interaction is included. We then use the calculated absorption with the drift-diffusion equations to calculate the responsivity in a partially depleted absorber (PDA) photodetector, and we obtain excellent agreement with experiments at all biases and optical wavelengths.

Power photodiodes for high dynamic range photonic links.

High-power photodiode applications for multioctave high dynamic range links are presented. A review of modulator and photodiode distortion analysis is given as well as an introduction to polarization-dependent loss distortion as it pertains to such systems. A new analysis of the photodiode distortion contributed degradation of spurious free dynamic range (SFDR) is developed. Experimental data covers high-power photodiodes for zero-bias high dynamic range links, showing significant improvement in SFDR. A link is presented showing the degradation of link performance when polarization-dependent loss is added into the system. A summary of state-of-the-art device performance is covered as well as the outlook on future applications for power photodiodes in analog photonic links requiring high SFDR.

Simulation of a partially depleted absorber (PDA) photodetector.

We use a 2D drift-diffusion model to study the nonlinear response of a partially depleted absorber (PDA) phododetector. The model includes external loading, incomplete ionization, the Franz-Keldysh effect, and history-dependent impact ionization. It also takes into account heat flow in the device. With all these effects included, we obtain excellent agreement with experiments for the responsivity and for the harmonic power at different modulation frequencies. The role of these different physical effects is elucidated, and we find that both the Franz-Keldysh effect and the load resistance play a key role in generating higher harmonic power at larger reverse biases. Increasing the size of the p-region absorption layers reduces the impact of the Franz-Keldysh effect. Decreasing the effective load resistance also decreases the higher harmonic powers. We also show that the model can suggest design changes that will improve device performance.

Procedure for aligning polarization modulator link for amplitude modulation applications.

A procedure is detailed for aligning the transmitted output states of a polarization modulated signal to the analyzer states of a polarizing discriminator in an analog photonic link. The steps in the procedure insure optimal amplitude modulation in the link. Experimental results are presented for biasing in two ways: either the DC bias on the modulator or a rotatable half-wave plate can be used. The corresponding theory is included.

Analysis of magnitude and relative phase of photodiode IMD2 using amplitude matched MZM-distortion cancellation technique.

We present a detailed look at using Mach-Zehnder modulator generated distortion for identifying the magnitude and relative sign of photodiode generated second order intermodulation distortion (IMD2). Previous discussions introduced the concept for characterizing a test device. Analysis is expanded to IMD2 as a function of voltage, photocurrent and frequency.

Suppression of even-order photodiode distortions via predistortion linearization with a bias-shifted Mach-Zehnder modulator.

A new technique to cancel photodiode-induced even-order distortion in microwave photonic links is demonstrated. A single Mach-Zehnder modulator, biased slightly away from the quadrature point, is shown to suppress photodiode second-order intermodulation distortion in excess of 40 dB without affecting the fundamental power. The technique is theoretically described with supporting experimental results.