Scattering statistics in nonlinear wave chaotic systems

Concepts from the field of wave chaos have been shown to successfully predict the statistical properties of linear fields in enclosures with dimensions much larger than the wavelength. This includes the properties of closed systems, such as eigenvalues and eigenfunctions, as well as the scattering properties of open systems.¹ The Random Coupling Model (RCM) describes the scattering properties by incorporating both universal features described by the Random Matrix Theory (RMT)^1–7 and the system-specific features of particular system realizations.^8–10 The RCM is formulated in terms of the Wigner reaction matrix, directly analogous to the electromagnetic or acoustic impedance, rather than the scattering matrix.^11,12 This allows the RCM to be expanded and appended in a simple additive or multiplicative manner, creating opportunities to describe increasingly complicated scattering scenarios. Examples of such complications include taking account of “short orbits” between the ports (or scattering channels) that survive the ensemble averaging process^13–15 and modifications of scattering statistics due to losses localized in the ports, rather than in the scattering system.^16–18 

Nonlinearity in wave-chaotic systems has been studied in several aspects. For example, rouge waves can appear in linear wave chaotic scattering systems.^19,20 However, such waves can also appear in a variety of physical contexts and are enhanced by nonlinear mechanisms.^21,22 In acoustics, Time-Reversed Nonlinear Elastic Wave Spectroscopy (TR/NEWS) is based on the nonlinear time reversal properties of a wave chaotic system.²³ TR/NEWS is proposed as a tool to detect micro-scale damage features (e.g., delaminations, micro-cracks, or weak adhesive bonds) via their nonlinear acoustic signatures.^24,25 Applying this idea to electromagnetic waves,²⁶ the nonlinear electromagnetic time-reversal mirror shows promise for novel applications such as exclusive communication and wireless power transfer.^27–30 The theoretical study of stationary scattering from quantum graphs has been generalized to the nonlinear domain, where the nonlinearity creates multi-stability and hysteresis.³¹ A wave-chaotic microwave cavity with a nonlinear circuit feedback loop demonstrated sub-wavelength position sensing for a perturber inside the cavity.³² Nonlinearity is a key ingredient in various machine learning protocols, including neural networks^33,34 and reservoir computing.^35,36 Utilizing wave chaotic layers, along with nonlinearity, offers an attractive way to enable physical realizations of deep learning machines.^37–39 

Nonlinear effects in wave chaotic systems manifest as harmonic and sub-harmonic generation, driving amplitude dependent responses, etc. We have recently studied the statistics of harmonics generated in a wave chaotic system by adding an active frequency multiplier to the 1/4-bowtie microwave billiard,⁴⁰ which is a vertically thin (less than a half-wavelength) microwave cavity whose horizontal shape resembles a quarter of a bowtie (Fig. 1). This is quite relevant to the work that investigates the electromagnetic field statistics created by nonlinear electronics inside a wave chaotic reverberation chamber, and it has a number of applications in the EMC (Electromagnetic Compatibility) community, such as the electromagnetic immunity testing of digital electronics.^41,42 Another approach to observe the nonlinear effects is to create a scattering system with amplitude dependent response. To achieve this, we have introduced different sources of nonlinearity into the billiards, and in this paper, we focus on a high frequency diode. Reaching the nonlinear regime usually requires high amplitude inputs, hence we have implemented a high power vector network analyzer (VNA) which is able to measure the scattering (S) parameters for signals up to $∼+43dBm$ (⁠$≃20W$⁠).

In this paper, we first briefly review the salient features of the Random Coupling Model and several of its extensions. We then discuss the microwave billiard experiment with a nonlinear diode element attached to the single port. The raw data are presented and discussed, and then a nonlinear generalization of the RCM is presented and applied to the data. The results are discussed and conclusions are drawn in Sec. VI.

The basic idea of the RCM is as follows. For an N-port ray chaotic system, the statistical properties of the $N×N$ cavity impedance matrix $Z¯¯cav$ are described by a universally fluctuating complex normalized impedance $ξ¯¯$ and the system specific properties $Z¯¯avg$ through the following equation:^9,10,13–15

where $Z¯¯avg$ is the average impedance over an ensemble of cavity realizations. $Z¯¯avg$ contains the non-universal features of the system, including the radiation impedance of the port (which fully characterizes the port-specific properties that determine the “prompt response” to an input excitation^{8,13,14,43,68}), and short orbits that survive the ensemble average.^13–15 The “radiation impedance” $Z¯¯rad$ describes the impedance of the port when only outgoing waves are present. Practically speaking, it is measured in the case that the waves get into the cavity through the port but do not return. This can be realized, for example, by covering the boundary of the billiard with perfect microwave absorbers. A “short orbit” describes a ray trajectory that leaves the port and immediately returns to it, or another port, without ergodically visiting the chaotic system. It is the result of the port-to-port interaction that introduces deterministic field components which can remain fixed in the ensemble.¹⁴ Under the assumption that losses are uniform, the statistics of the universally fluctuating complex impedance $ξ¯¯$ is determined by a single parameter named the loss parameter $α$⁠.^8,9,43,44 For a two-dimensional electromagnetic system (i.e., a vertically thin cavity), it is given by $α=k2A/(4πQ)$ and can be interpreted as the ratio of the typical 3-dB bandwidth of the resonant modes to the mean mode spacing. Here, $k=2πf/c$ is the wave number at frequency $f$⁠, $A$ represents the area of the billiard, and $Q$ is the typical loaded quality factor of the enclosure. The loss parameter $α$ can vary from 0 (isolated resonances) to infinity (many overlapping resonances). The universal statistical properties of a chaotic system with loss parameter $α$ is given by RMT.^9,10,45–47 Note that we can invert Eq. (1) and create an experimental approximate to the statistics of $ξ¯¯$⁠, called $ξ¯¯exp$ by gathering an ensemble of $Z¯¯cav$⁠, and constructing $Z¯¯avg$⁠. By fitting the statistics of $ξ¯¯exp$ to theoretical predictions, the loss parameter of the system can be estimated.

The success of the principles underlying the RCM has been experimentally verified in linear wave chaotic systems including microwave systems^48–50 and acoustic systems,^51–55 from 1D quantum graphs,^56–58 2D electromagnetic billiards,^14,15,43,59 and 3D cavities.^60–63 Based on its success and flexibility, it is of great interest to extend the RCM to other systems. One area of extension is to nonlinear systems.

In this work, we show the results for measurements of the nonlinear scattering parameters in a diode-loaded 1/4-bowtie quasi two-dimensional microwave cavity. The 1/4-bowtie cavity is a ray-chaotic billiard that displays universal statistical properties predicted by RMT and RCM.^{9,10,45,49,64–70} In this case, the diode acts as a nearly point-like nonlinearity in a wave chaotic system. Attaching a diode to the excitation port, we observed that the raw cavity statistics of the impedance change substantially with the excitation power. We extend the RCM to this situation and use it to analyze our experimental results. We find that when the radiation impedance becomes nonlinear, short orbits between the port and a nearby wall, and the raw impedance statistics are strongly modified. We also find that many of these changes are due to the fact that the admittance of the diode changes with the excitation power. The nonlinear diode competes with the cavity admittance, substantially altering the response of the system. By implementing the lossy port model extension of the RCM,^16–18 the results are well explained by the changing radiation efficiency of the diode-loaded port. As a result, the diode effectively acts like a protection element in this configuration.

In the small signal limit, our system can be approximated as linear. To observe a nonlinear response, the system must have some sort of nonlinear property, and a large excitation signal is required. In our earlier studies of wave chaotic systems with one port or multiple ports, we measured the scattering parameters and used these measurements to study the statistical properties of the system. Here, we measure the high power S-parameters including a nonlinear element in the wave system, at power levels up to $+43dBm$ (see the supplementary material).

To induce strong nonlinearity, a diode (Infineon BAS7004 with two diodes in the package but only one is electrically connected) is soldered between the center pin and cavity ground, as shown in the inset to Fig. 1. From the datasheet,⁷¹ this diode has low transition capacitance, $C∼1.5pF$ at 1 MHz, which decreases nonlinearly to $∼0.5pF$ as the reverse voltage increases. Its differential resistance also changes nonlinearly as a function of the forward current. For typical forward currents $IF=1∼15mA$⁠, the resistance $R$ changes from 80 to 20 $Ω$⁠. A rough estimate for the time constant $τRC=RC∼100ps$⁠, which is close to the charge carrier life-time as given in the data sheet. Thus, this diode can respond in the GHz frequency range and produce clear nonlinear responses, making it suitable for our microwave wave chaos experiments.^72,73 In addition, the diode package is significantly smaller than the wavelengths used in this study (30-75 mm), rendering it approximately “point like.” The connection shown in Fig. 1 has advantages in terms of stability and reproducibility, due to the fact that when the bowtie billiard is opened, the antenna and the top plate are attached together as one piece, and the bottom plate is a separate piece. This in turn allows for excellent reproducibility of $Z¯¯rad$ (radiation impedance, see the supplementary material), $Z¯¯cav$⁠, and $Z¯¯avg$ measurements.

With this diode loaded nonlinear port, we observed that both the system specific properties and universal statistics change dramatically with the input microwave amplitude. The detailed analysis of the system specific properties, including the radiation impedance and short orbits, is presented in the supplementary material. Here, we focus mainly on the scattering statistics, which are significantly affected in the $4∼10GHz$ range.

To analyze the statistics of the cavity, the metallic perturbers shown in Fig. 1 are moved around to create 120 distinct static realizations. Figure 2(a) shows reflection vs. frequency results for a typical realization for low (blue) and high (green) input power. They have similar shapes as in the radiation case, but are “decorated” with many resonance fluctuations. The linear RCM approach applies well in the low power case, and we follow the RCM normalization process to determine an experimental approximate, $ξexp$⁠.^8,10,14 Figure 2(b) shows PDFs of $Re(ξexp)$ in the 6.5 to 7.5 GHz range for several different input powers. Clearly, the PDF of the normalized impedance $Re(ξexp)$ changes substantially with power, being more widely distributed in the low power case, indicating stronger fluctuations, a property which is associated with lower loss parameter $α$⁠. Note that the $Re(ξexp)$ distribution is more concentrated near unity as power increases, consistent with a high loss (high $α$⁠) situation. If we naïvely fit this distribution function to the RCM prediction using $α$ as the sole fitting parameter, the fitted loss parameter $α$ systematically increases with power, as shown in the inset. The raw statistics of the system change substantially with power because of the presence of the nonlinear port. However, it should be noted that these $ξexp$ PDFs show substantial deviations from RMT predictions (note the low fit $R2$ values at high power), making it clear that the naïve application of the RCM breaks down in the nonlinear regime.

In the RCM treatment presented above, we expect the loss parameter of the system to be independent of the excitation power as long as the properties of the cavity remain unchanged. The nonlinear property in this case is only associated with the port. The RCM described in Eq. (1) is derived assuming a lossless linear port. But this is no longer the case in this experiment. As we can see from Fig. 2(a) particularly in the vicinity of 6 GHz, the fluctuations are suppressed in the high power case, indicating that excitations of the cavity modes are suppressed. In this case, the port must be considered as a lossy port. References 16 and 17 have derived a generalization of the RCM to account for the loss of the port. A radiation efficiency $η$ is introduced to quantify the ratio of the power radiated by the port to the input power to the antenna, $η=Prad/Pin$ (⁠$η$ is real and $0⩽η⩽1$⁠). In a high loss system (i.e., $α≫1$⁠), it can be shown that the impedance of a lossy antenna inside a complex enclosure can be approximated as

where $η$ is the radiation efficiency of the antenna, $δξ¯¯=ξ¯¯−I¯¯$⁠, $I¯¯$ is the identity matrix with diagonal elements $1+i0$⁠, and $Z¯¯ant$ is the input impedance of the lossy antenna radiating in free space. Reference 18 has successfully applied this model to a scaled cavity, where the radiation efficiency accounts for the loss in free-space propagation suffered through a remote injection path. In our case, $Z¯¯in$ can be considered as $Z¯¯cav$ and $Z¯¯ant$ can be considered as $Z¯¯avg$⁠, therefore, Eq. (2) can be modified as

which is valid in the limit $α≫1$⁠. To determine $η$ for the nonlinear port, we first measure $ξ¯¯cav$ of the billiard when there is no diode attached to the antenna. In that case, $ξ¯¯cav$ describes the properties of the billiard alone (because all system-specific properties have been removed), and as such it is a linear system. We use the linear RCM approach, creating 120 realizations with the two perturbers, then applying Eq. (1) to extract $ξ¯¯cav$ and fit to RCM to find the corresponding loss parameter $α$⁠.^8,10,14 Additionally, to make the bowtie billiard a high loss system, the perimeter of the billiard is partly (but uniformly) covered with microwave absorbers, tuning the loss parameter to be $α=3$ (4.5 GHz) to $α=7$ (9.5 GHz). Then, using the value of $α$ which characterizes loss in the cavity and $ξ¯¯cav$⁠, we go back to the diode-loaded nonlinear port case, utilizing Eq. (3) and varying $η$ so that the statistics of $ξ¯¯(η)$ agree with $ξ¯¯cav$⁠. Figure 3(a) shows the fitted radiation efficiency from $Im(ξ)$ statistics [the $Re(ξ)$ statistics give similar results].

Figure 3(a) shows that the radiation efficiency is strongly power-dependent in the frequency range $4∼10GHz$⁠. Between 6 and 9 GHz, the radiation efficiency decreases with increasing power, meaning the port is getting more lossy as the excitation power increases. There is a cross-over regime at a low frequency of $4∼6GHz$⁠, where the radiation efficiency increases at high powers. And although it is not shown in the plot, at 10 GHz and above, the radiation efficiency is almost independent of power.

To explain the results, we are interested in characterizing the diode admittance under different excitation powers. The diode is connected between the center pin and the ground of the port. The billiard radiation admittance can also be considered as being connected between the center pin and ground. Therefore, a simple model is constructed by considering the diode and the billiard to be connected in parallel. By measuring the port radiation admittance with (⁠$Ywithdioderad$⁠) and without the diode (⁠$Ynodioderad$⁠), the diode admittance can be approximated as $Ydiode≃Ywithdioderad−Ynodioderad$⁠.

Figure 3(b) shows the experimental results of $|Ydiode|$ compared with $|Ynodioderad|$⁠. Between $4and6GHz$⁠, $|Ydiode|$ has similar values to $|Ynodioderad|$ and changes little with power. Thus, the diode admittance competes roughly equally with the billiard admittance. A small change in the admittance value may result in a big change in the fitted radiation efficiency. However, between 6 and 9 GHz, the diode admittance dominates the billiard admittance and generally increases with power. In this frequency range, because the diode admittance is much larger than the billiard admittance and increases with power, the radiation efficiency of the port decreases, consistent with the results in Fig. 3(a).

To better understand the nonlinear port, we built a circuit model for simulation in the finite difference time-domain code called CST (Computer Simulation Technology). The SPICE models of the diode as well as the package are provided by the manufacturer. In addition to the SPICE models, the dielectric properties of the package near the port also affect the radiation impedance, and this was added to the model. The internal capacitances of the package SPICE model were altered because the diode is being used beyond its design frequency range. The simulated amplitude dependent radiation S-parameters show relatively good agreement with the experimental results (see the supplementary material). In addition, the resultant radiation efficiency that is directly calculated as $Prad/Pin$ is in general agreement with the experimental results. Based on the circuit model, the nonlinearity arises from the diode, which is approximated with an exponential I-V diode function.⁷⁴ The diode nonlinearity is shorted by parasitic capacitance in the package SPICE model at high frequencies, thus the port model does not have power dependence at 10 GHz and above, consistent with the data.

We have shown that by attaching a diode to the center pin and ground of an antenna, the port shows dramatic nonlinear behavior. For the statistical results, if we blindly apply the RCM to the billiard with a diode-loaded port, one finds that the loss parameter increases with the input power. The RCM posits that the loss parameter determines the universal properties of the chaotic system. In our case, the billiard properties should not change with power because the nonlinearity is only associated with the port. We applied the newly developed radiation efficiency model to the port, using the radiation efficiency $η$ to quantify the proportion of power from the source radiated into the billiard. At high power in the nonlinear region, the diode consumes most of the power, causing the radiation efficiency to decrease. The diode thus prevents high power signals from getting into the billiard.⁷⁵ 

There are several interesting questions for further study of this system. First, the billiard had to be intentionally modified into a high loss system in order to use the radiation efficiency model. The statistics of low-loss nonlinear systems cannot be addressed at this time. In addition, another behavior we have observed is the loss of reciprocity in a two-port version of this system, where one port is a nonlinear port and the other is linear. We observed that $S12≠S21$ when large amplitude signals are applied. We note that there is no general reciprocity theorem that holds for nonlinear systems.⁷⁶ This behavior can be understood by considering that with equal powers injected in both ports, the diode will be driven to nonlinearity when power is injected in the port hosting the diode, and to a lesser extent when the linear port is excited.

Besides the approach we used in this paper to analyze the nonlinear system, we mention for completeness that high power S-parameters are sometimes called hot S-parameters,⁷⁷ and the nonlinear effects can be fully characterized by the so-called X-parameters measured by the nonlinear VNA.^78,79 However, we believe that the present treatment is best suited for understanding the statistical properties of nonlinear wave chaotic systems in the semi-classical regime.

To conclude, a diode based nonlinear port alters the radiation impedance, short orbits, and raw impedance statistics of a wave chaotic system from those observed in a linear system. By using the Random Coupling Model with incorporation of the diode nonlinear properties and a lossy port model, these nonlinear phenomena are well explained and verified by the simulation. The nonlinear property of the port can be applied to protect delicate circuits from high power electromagnetic microwave interference. This represents an important step forward in understanding the statistical scattering properties of a semi-classical wave chaotic system with a nonlinear coupling channel. The raw data are available in Ref. 80.

The supplementary material contains more information on the implementation of high-power vector network analyzer (VNA), detailed analysis of the nonlinear effects on the system specific properties, and simulation results of the diode-loaded port in computer simulation technology (CST).

This work is supported by the AFOSR under COE Grant No. FA9550-15-1-0171, the ONR under Grant No. N000141512134, the COST Action IC1407 “ACCREDIT” supported by COST (the European Cooperation in Science and Technology), and the Maryland Center for Nanophysics and Advanced Materials (CNAM).