**The Random Coupling Model: A User's Guide (updated 12 July, 2023)**

The University of Maryland Wave Chaos Group, in collaboration with the University of Illinois

__Research funded by ONR, AFOSR, DARPA, and DURIP programs__

This page provides an overview of the Random Coupling Model (RCM) and its use in predicting Electromagnetic Interference (EMI) effects. The RCM is a method for making *statistical* predictions of induced voltages and currents for objects and components contained in complicated (ray-chaotic) over-moded enclosures and subjected to RF fields. It is based on simple universal predictions of wave chaos theory and is quantitatively supported by random matrix theory. The system-specific (non-universal) aspects of the problem are quantified by means of the radiation impedance of the "ports" involved in the problem, as well as prominent short orbits. We continue to extend and improve the RCM, and welcome your input on which problems need to be addressed.

Please take a look at our papers, presentations, Frequently Asked Questions (FAQs), and caveats below. We hope to make this model useful and accessible to all interested parties, so please give us your feedback.

A comprehensive paper describing the Random Coupling Model for the engineering community has been published as IEEE Trans. Electromag. Compat. **54**, 758-771 (2012). A recent review article on the Random Coupling Model: Gabriele Gradoni, *et al*., Wave Motion **51**, 606-621 (2014). pdf

**FAQ's about the Random Coupling Model**:

In this web site we will attempt to answer questions frequently asked about the Random Coupling Model (RCM). The RCM is a model for making predictions about the values of the elements of the impedance (and scattering) matrices for voltages and currents (and incident and reflected waves) at ports in electromagnetic enclosures. Mainly, people are interested in knowing when the RCM applies, and conversely when it fails. As the RCM is a statistical model; it does not predict precise values for specific, well-defined geometries. Rather, it predicts probability distribution functions for measuring values of the elements of the impedance and scattering matrices. Thus, for it to be determined to have failed, a number of measurements must be taken to see that their statistics are not in agreement with those predicted by the RCM. This statistical aspect contributes to the uncertainty in the validity of the RCM, as people are accustomed to deterministic predictions for particular quantities. If the weather forecast is a 50% chance of rain, and it doesn’t rain, is the forecast wrong?

As stated, the RCM predicts the distribution of an ensemble of measured values. Generally, there are two ways to create an ensemble of measured values that can be compared with RCM predictions. One way is to vary some physical dimensions of the enclosure under consideration. For example, an ensemble can be created by rotating a stirrer or moving a perturbing object in the enclosure. For this to be effective the stirrer or object has to be big enough such that its displacement generates a sufficiently large change in the system’s response. A second way an ensemble can be created is by varying the frequency of excitation. This latter case requires that frequencies be sufficiently high so that the system response varies over the range of frequencies being tested, but the range of frequencies should be small enough so that average properties don’t vary within that range. The requirements on using a range of frequencies to make an ensemble needs to be addressed on a case-by-case basis. Often, the best approach is to vary both the position of a stirrer and the frequency of excitation.

Q1: What assumptions are made in deriving the RCM? Q2: What is missing in the RCM making it only approximate? The RCM is derived by starting with a formally exact expression for the elements of the frequency dependent impedance matrix. In this exact expression each element of the impedance matrix is written as an infinite sum over the modes of the enclosure. Each term in the infinite sum has a denominator containing the difference between the excitation frequency and the complex frequency of the mode in question. The mode frequencies are complex in the presence of losses, and different modes have different loss rates. Each term in the sum has a numerator that contains a product of two integrals. The integrals give the overlap of the mode in question with the current density or electric field distribution of a transmitting or receiving port. As stated, the starting expression is formally exact. However, to exactly evaluate the expression is both computationally intensive and requires complete knowledge of the environment in the enclosure. This is what one hopes to avoid. Instead, a series of simplifying assumptions, approximations, and replacements is made. First, the exact spectrum of mode frequencies is replaced by a generic spectrum of frequencies. The generic spectrum is based on the eigenvalues of a certain type of random matrix (one drawn from the Gaussian Orthogonal Ensemble, GOE). The generic spectrum is adjusted in two ways to replicate properties of the exact spectrum. The mean spacing between mode frequencies is made to match that in the enclosure under consideration. This is done by using the so called Weyl formula that gives an approximation for the number of mode frequencies less than a specified frequency value in an enclosure of a given volume. Second, the mode frequencies are given imaginary parts to model losses. As an approximation, each mode in a general frequency range is given the same value of imaginary part. This is based on the idea that in an enclosure where the field energy can be thought of as distributing itself throughout the enclosure as a random superposition of plane waves (the random plane wave hypothesis), each mode will have the same damping rate.

The concept that the mode profiles have the properties of a random superposition of plane waves is used to evaluate the overlap integrals between the cavity modes and the port current or field distributions that appear in the numerators of the terms in the infinite sum. The result is that each numerator in the sum over modes contains a product of two random numbers. Each of these is a zero mean, Gaussian distributed random number according to the central limit theorem, as each results from the overlap of a port profile with a superposition of plane waves. What remains is to calculate the variance of the gaussian random variables. Miraculously, it turns out the variance is given in terms of the radiation impedance of the port. This is the impedance that would be seen at the port if the enclosure volume were infinite, and no signal leaving the region of a port would ever return.

These are the assumptions made in deriving the RCM. Based on the assumptions one can see what is missing. Any property of the response based on the specific shape or contents of the enclosure is lost when the exact modes are replaced by the generic modes that are in the form of superpositions of plane waves. This means that if the exact values of the resonant frequencies are important, the RCM will not reveal them. Similarly, for the exact damping rates. Generally, generic features of the spectrum apply only when the frequency is high and high order modes are excited. Further, in the RCM, communication between two ports is carried out through the eigenmodes. Thus, the effect of direct propagation of energy from one port to another, or propagation with only a few reflections from port to port, is absent. This information can be included in a revised version of the RCM. However, it obviously requires more knowledge about the internal layout of the enclosure. This is known as the short orbit effect.

References for the above discussion are as follows:

The basics of so called “wave chaos” and Random Matrix Theory are discussed in the text book: Chaos in Dynamical Systems, by E. Ott, Cambridge University Press, 2002.

Xing Zheng, Thomas M. Antonsen Jr., Edward Ott, "**Statistics of Impedance and Scattering Matrices in Chaotic Microwave Cavities: Single Channel Case**," Electromagnetics **26**, 3 (2006). pdf.

Xing Zheng, Thomas M. Antonsen Jr., Edward Ott, "**Statistics of Impedance and Scattering Matrices of Chaotic Microwave Cavities with Multiple Ports**," Electromagnetics **26**, 37 (2006). pdf.

Sameer Hemmady, Xing Zheng, Edward Ott, Thomas M. Antonsen, and Steven M. Anlage, "**Universal Impedance Fluctuations in Wave Chaotic Systems**," Phys. Rev. Lett. **94**, 014102 (2005). pdf

S. Hemmady, X. Zheng, T.M. Antonsen, E. Ott, S.M. Anlage, "**Universal Properties of 2-Port Scattering, Impedance and Admittance Matrices of Wave Chaotic Systems**," Phys. Rev. E **74 **, 036213 (2006). pdf.

The effect of “short orbits” was first seen in computations by Zheng above. It was later given a mathematical basis and verified experimentally in the papers:

James A. Hart, T. M. Antonsen, E. Ott, "**The effect of short ray trajectories on the scattering statistics of wave chaotic systems**," Phys. Rev. E **80**, 041109 (2009). pdf This paper presents the Extended Random Coupling Model.

Jen-Hao Yeh, James Hart, Elliott Bradshaw, Thomas Antonsen, Edward Ott, Steven M. Anlage, “**Experimental Examination of the Effect of Short Ray Trajectories in Two-port Wave-Chaotic Scattering Systems**,” Phys. Rev. E 82, 041114 (2010). pdf

Two reviews of the RCM have appeared in the archival literature:

S. Hemmady, T.M. Antonsen, E. Ott, S.M. Anlage, "**Statistical Prediction and Measurement of Induced Voltages on Components within Complicated Enclosures: A Wave-Chaotic Approach,” **IEEE Trans. Electromag. Compat. **54**, 758-771 (2012). pdf This paper offers an accessible introduction to the RCM for the engineering community.

Gabriele Gradoni, Jen-Hao Yeh, Bo Xiao, Thomas M. Antonsen, Steven M. Anlage, Edward Ott , “**Predicting the statistics of wave transport through chaotic cavities by the Random Coupling Model: a review and recent progress**,” Wave Motion** 51**, 606-621 (2014). pdf

Q3: How and when does the RCM break/fail, in general? Q4: How does the RCM degrade as the wavelength increases and you are no longer in the “short wavelength” regime? As discussed above, in the derivation of the RCM key approximations are made, first to the spectrum of resonant modes and second to the spatial properties of the resonant eigenmodes.

Concerning the spectrum of resonant modes, the average spacing between eigenfrequencies is determined by the volume of the cavity and the frequency range under study. The RCM further assumes that the mode spectrum has the characteristics of the eigenvalues of a certain type of random matrix: one drawn from the GOE. The important property of the eigenfrequencies is the distribution of eigenfrequency spacing values. Eigenvalue spacings for these matrices, as well as for wave systems in which the corresponding wave ray trajectories are chaotic, show what is called level repulsion. That is, it is extremely unlikely to find two eigenvalues much closer to each other than the average spacing between eigenvalues. Systems for which the corresponding wave trajectories are not chaotic show a distribution of spacings that is peaked at zero spacing (a Poisson distribution). As it turns out, the spacing distribution is only important when the Q-width of the modes is less than the average spacing. In the RCM this occurs when the so-called loss parameter is less than unity, which is relatively rare.

The more significant breakdown of the RCM occurs as a result of the eigenmodes of the cavity not having the spatial character of a random superposition of a large number of plane waves. This can occur in a number of ways. First, at low frequency the eigenmodes have the character of a superposition of plane waves, but the effective number of plane waves is too small for the RCM to apply, even in cases where the cavity shape is such that all ray trajectories are chaotic. The effective number of plane waves can be estimated as follows. The field can be represented as a sum over a number of plane waves. The relative amplitudes of the plane waves are determined by satisfying boundary conditions on the internal surface of the enclosure. Each plane wave varies over a distance corresponding to half a wavelength. The number of independent points on the boundary is then determined by the surface area of the boundary as measured in square half wavelengths. Thus, **the effective number of plane waves is four times the surface area measured in square wavelengths**.

In a study (B. Addissie, 2017) the mean and variance of the measured impedance values for a cavity of volume 1.9 m^{3} and surface area 8.6 m^{2} were compared with RCM predictions. The RCM results were deemed accurate for frequencies above 0.7 GHz corresponding to a wavelength of 0.42 m. The estimated number of plane waves in this case was 100. The finite number deviations would then be estimated to be at 10%. Applying the same reasoning to the DARPA cylinder yields a frequency estimate of 3 GHz above which the RCM statistics are expected to apply.

There are other reasons the random plane wave hypothesis might not be valid. As mentioned, the random plane wave hypothesis is argued on the basis that the waves, when thought of as rays, propagate through the enclosure chaotically and ergodically. This means that nearby rays diverge from each other exponentially, and eventually each ray visits every point on the interior surface at all angles. This requires that the internal surfaces of the enclosure not be shaped so as to create periodic trajectories which are stable in the sense that nearby trajectories do not diverge exponentially. An extreme example of this would be if a Fabry-Perot resonator were placed inside the enclosure. Rays trapped in the resonator would never visit anywhere else in the enclosure, and vice versa. In this case the statistics of the fields in the enclosure depend on where the measurements are being made. Prediction of these statistics requires substantially more information about the enclosure than its volume and loss parameter. (M-J Lee, 2013)

Finally, even if the classical rays in the cavity are ergodic and chaotic, the RCM statistics may not be realized. This can happen if the damping rate of the waves is too high, the port launching the rays is too directive, or some combination of these conditions. In a recent two-dimensional numerical study (F. Adnan, 2021) it was found that for waves launched from a port in a narrow sector of angles such that the waves damped before hitting every point on the perimeter significant deviations from RCM statistics were obtained if the loss rate were too high. RCM statistics could be restored by decreasing the damping rate, or by launching waves from a less directive port. Again, predicting coupling in cases where the RCM fails leads to a requirement that more specific information about the enclosure is needed than the volume, loss factor, and port impedances.

The validity of the RCM is studied in the following papers and theses.

Ming-Jer Lee, Thomas M. Antonsen, Edward Ott , "**Statistical model of short wavelength transport through cavities with coexisting chaotic and regular ray trajectories**," Phys. Rev. E 87, 062906 (2013). pdf

Ming-Jer Lee Ph.D. thesis, **Statistical Modeling of Wave Chaotic Transport and Tunneling**. Appendix A has an algorithm for efficeintly diagonalizing large random matrices. Available on UMD/DRUM

** Bisrat Addissie, Methods for Characterizing Electromagnetic Coupling Statistics in Complex Enclosures ** DOI: https://doi.org/10.13016/M2XD0R03Q

** Adnan, Farasatul (Ph.D. with Prof. Antonsen, UMD), Wave Chaos Studies for Two-Dimensional Cavities Using the Random Coupling Model (RCM) and Other High Frequency Methods ** DOI: https://doi.org/10.13016/zram-mtz9

**Here are a set of shorter questions and answers about the RCM**:

How do I know my enclosure is chaotic for ray orbits? Most enclosures __are__ ray-chaotic (its hard to make them otherwise!) The
“soda can effect” is a good operational test - in other words the impedance of the cavity is a strong function of frequency and details of the internal configuration of the components. See also Fig. 8.2 and related discussion in Sameer Hemmady's Ph.D. thesis.

**Suppose I don't know the Q, volume, Z _{Rad} precisely, is this a problem? **
Most likely no. The predicted PDFs are not highly sensitive to the loss parameter. Z

**I have a wide-band incident signal. Is the RCM still valid?** Yes. First of all, the loss parameter is usually not a very strong function of frequency because it depends on the fixed geometry of the system as well as the fact that the quality factor is a smoothly varying function of frequency with small fluctuations (see Fig. 7 in the paper of Barthelemy, Legrand and Mortessagne, Phys. Rev. E **71**, 016205 (2005). This shows a nice example of how the Ohmic losses vary with frequency in a ray-chaotic microwave enclosure.) Secondly, the frequency dependence of the radiation impedance accounts for the variation of the coupling of the signal in/out of the enclsoure. Rather than assuming the loss parameter is constant, one could extend the RCM to include the variation of the loss parameter k^{2}/(Dk^{2} Q) with frequency to make a more accurate estimate of the PDF of induced voltages from the broadband excitation.

**What is the minimum number of modes for the RCM model to be applicable?** The higher the mode number the better (for best results use overmoded cavities). Generally you would need more than about 50 propagating modes below your lowest frequency of interest. We have empirical evidence that the RCM "degrades gracefully" in the limit of low frequencies. A rough rule of thumb is that the enclsoure should be at least 3 wavelengths long in each dimension.

**How do I know if my enclosure is overmoded?** A general rule-of-thumb would be to look at the ratio of the maximum to minimum transmitted power at a given frequency for your measured cavity ensemble. The enclosure (cavity) is overmoded if this ratio is more than about 20 dB in magnitude. See also Fig. 8.2 and related discussion in Sameer Hemmady's Ph.D. thesis.

**What about cross-talk between ports? **This is included in the RCM.
See section IV of the paper Phys. Rev. E **74 **, 036213 (2006).

**Does a port have to be on the surface of the enclosure? **No, it can be inside and away from the walls of the enclosure. In fact, this is often the most interesting case because you would like to know the statistics of induced voltages on an electronic component inside the enclosure.

**Does the RCM work for pulsed excitations? **Yes, we have developed the theory for the time-domain. The long-term behavior of a cavity excited with a pulse has been studied in detail (Phys. Rev. E. 79 016208 (2009)). We have also examined the contributions of short-orbits to the impedance (Phys. Rev. E **80**, 041109 (2009), see below), and this will influence the short-time behavior of excited systems. Experimental verification is largely complete (Phys. Rev. E 81, 025201(R) (2010)). We have investigated the sensitivity of time-domain signals to the scattering properties of ray-chaotic enclosures (see the YouTube movie). We have also developed a new sensor paradigm based on the sensitivity of wave scattering in such systems to small perturbations (Appl. Phys. Lett. **95 **, 114103 (2009)), and Biniyam Taddese's Ph.D. thesis.

**What happens if there is a port in the enclosure that is overlooked? **
It is incorporated in the model through the scattering it produces, as well as modifications to the Q and cavity volume.

**How do I identify the presence of a new object inside my enclosure? **The addition of a new object to the enclosure can be detected by a change in the loss parameter (since the object takes up a certain electromagnetic volume and will have some degree of loss), by the creation or destruction of short-orbit trajectories, or by modification of the radiation impedance of a nearby port.

**How many ports do I need to include in the RCM to describe my enclosure?** We believe that the most important ports to explicitly include are those that are actively adding energy to the system, and those that represent the sensitive/susceptible object(s) in the system. On a related note, one might also consider adding single ports that represent classes of more-or-less identical objects in the enclosure. The present code (Terrapin RCM Solver v1.0) considers only 2-port systems. An extension of this code to a higher number of points can be arranged through Dr. Sameer Hemmady (see below).

**Suppose I don't have an ensemble of enclosures, can I still use the RCM? **
Yes, it is often a good approximation to use a frequency average to substitute for an ensemble average. In addition, our work on removing the influence of short-orbits (Phys. Rev. E **80**, 041109 (2009), Phys. Rev. E 81, 025201(R) (2010), see below) allows one to use single-realizations and smaller ranges of frequency averaging to uncover the universal statistical fluctuations. This means that predictions of induced voltage PDFs will be improved if basic knowledge of short orbits in the system is also available.

I am worried about using a single Q-value. Suppose Q varies from mode to mode?
Ray-chaotic systems tend to show small fluctuations of Q with mode number.
Also, remember that what counts is the loss parameter k^{2}/(Dk^{2} Q), not Q by itself. See also the "wide-band" FAQ above.

**What about antenna polarization for 3-D enclosures?** This effect is included in the radiation impedance of the ports of interest.

**Does the RCM take into account field variations associated with the presence of a wall?** Yes. The presence of a wall is included in the radiation impedance of the port located near the wall, edge, or corner of a structure.

**Does the short-orbit extension of the RCM take into account multiple short-orbits produced by waves that bounce off of the port?** Yes, these multiple bounce short orbits are naturally included through consideration of the impedance.

**Can the RCM work for multiple enclosures conected by apertures?** Yes, the RCM has been extended to study the statistics of fields in a cascade of connected reverberant systems. See the paper of Gradoni, Phys. Rev. E **86**, 046204 (2012).

**Can the RCM work for enclosures that have both regular and chaotic ray trajectories?** Yes, the RCM has been extended to study the statistics of systems with "mixed phase space." Research continues on studies of more generic mixed systems. See the paper of Ming-Jer Lee, Phys. Rev. E **87**, 062906 (2013).

**Random Coupling Model: CAVEATS
and additional details **

**What could possibly go wrong? **

If you need to predict the outcome of a specific measurement in a specific situation, then the RCM cannot help. The RCM provides only statistical predictions. The 'extended RCM' now includes short-orbit system specific information, in addition to the system specific port information.

If there are strong periodic contributions to the ray dynamics (e.g. short periodic orbits from parallel planes), these will lead to deviations from RCM predictions. Such orbits are now included in the extended RCM. However, scars, cusps, caustics, singularities, and perhaps “Freak Waves” can produce large local enhancements of electromagnetic fields, and they fall outside of the Random Coupling Model.

What is the low-frequency limit of the model? Rough rule of thumb: Apply the RCM to mode #50 and greater. One should have at least 3 wavelengths along each dimension of the enclosure. The low-frequency limit of the RCM is still being explored. The RCM should merge with deterministic FDTD solvers in this limit.

One __failure mode__ of the RCM arises when there is poor coupling to the enclosure (i.e. |S_{Rad,11}|~1 or |S_{Rad,22}|~1). The poor coupling makes it very difficult to de-embed the antenna properties from the impedance data. When S_{Rad} approaches 1 in magnitude, the Z_{Rad} will become very large in magnitude and sensitive to small errors in S_{Rad}. This propagates into large values and large errors for Z_{Cav}. This is likely to cause deviations in the induced voltage PDF. However, poorly-coupled enclosures will not have much of an EMI/EMC problem!

Another __failure mode__ of the RCM arises if the losses in the system are NOT uniformly distributed. If losses are highly localized (e.g. an electrically large aperture that allows many ray trajectories to exit the system), the assumption of uniform loss may be violated, and the statistical predictions of the RCM may not be strictly valid. Another way to look at it, the properties of the system will be dominated by short orbits (which are non-universal and non-statistical) because the longer "chaotic" orbits will exit the system through the localized loss mechanism (e.g. an electrically large aperture). We have now addressed this problem experiemtnally in collaboration with NRL, and the results are in the paper by Jesus Gil Gil listed below.

**When do you NOT want to use this model?
**

Enclosure Q ~ 1 or less. No reverberation, no chaos, very lossy. The system impedance does not fluctuate. It simply reduces to the radiation impedance.

Enclosure size **NOT** much larger than wavelength l. A direct numerical solution of such a problem is not sensitive to details and should be employed rather than the RCM. Rule of thumb for validity of the RCM: cube root of enclosure volume > about 3 wavelengths.
Remember: dielectrics inside the enclosure increase it's effective volume.

**Random Coupling Model: Does it work in 3 Dimensions?
**

YES! Our results are not in any way predicated on the peculiar 2D “bow-tie” cavity employed in some of our early verification work, nor on it's shape, thickness, or the antenna configuration, coupling or position. The theory is well established and works in 3D situations. In addition to our own 3D verification results, several other groups have performed demonstrations of RCM-like results in 3D: **Sandia ** (Warne, *et al *.): Demonstrated the utility of Z_{Rad} in removing the effects of coupling in 3D enclosures. They independently discovered that the 3D statistics are governed by a single loss parameter. **ONERA ** (Parmentier, *et al *.): Demonstrated the equivalence of the S_{Rad} and <S> in 3D. They also independently discovered an S-variance ratio relation in 3D data analogous to the Hauser-Feshbach relation of nuclear physics (see the paper: Phys. Rev. E **73**, 046208 (2006)).

**Loss Parameter: Does it depend on dimensionality?**

The general expression for the loss parameter is k^{2}/(Dk^{2} Q), where k is the wavenumber (w = k c), Dk^{2} is the mean spacing between squared wavenumbers, and Q is the typical Q-factor of the modes (see above). In 2D cavities the loss parameter can be written as k^{2}*A*/(4pQ) (where *A* is the area of the cavity), whereas in 3D it can be written as k^{3}*V*/(2p^{2}Q) (where *V* is the volume of the cavity), using the Weyl formula for mean spacings of the corresponding closed systems. A general discussion of different ways to determine the loss parameter for a given system is presented in this appendix of Sameer Hemmady's Ph.D. thesis.

Random Coupling Model Publications:

Theory:

Xing Zheng, Thomas M. Antonsen Jr., Edward Ott, "**Statistics of Impedance and Scattering Matrices in Chaotic Microwave Cavities: Single Channel Case**," Electromagnetics **26**, 3 (2006). pdf. This and the following paper are the seminal papers on the Random Coupling Model.

Xing Zheng, Thomas M. Antonsen Jr., Edward Ott, "**Statistics of Impedance and Scattering Matrices of Chaotic Microwave Cavities with Multiple Ports**," Electromagnetics **26**, 37 (2006). pdf.

Xing (Henry) Zheng, Ph.D. thesis, "**Statistics of Impedance and Scattering Matrices in Chaotic Microwave Cavities: The Random Coupling Model**," University of Maryland, 2005.

Xing Zheng, Sameer Hemmady, Thomas M. Antonsen Jr., Steven M. Anlage, and Edward Ott, "**Characterization of Fluctuations of Impedance and Scattering Matrices in Wave Chaotic Scattering**," Phys. Rev. E **73**, 046208 (2006). pdf This paper also includes experimental verification of the raw-S and raw-Z variance ratios.

James Hart, Thomas M. Antonsen, Jr., Edward Ott, "**Scattering a pulse from a chaotic cavity: Transitioning from algebraic to exponential decay**," Phys. Rev. E **79**, 016208 (2009). pdf The first paper on the time-domain version of the random coupling model.

James A. Hart, T. M. Antonsen, E. Ott, "**The effect of short ray trajectories on the scattering statistics of wave chaotic systems**," Phys. Rev. E **80**, 041109 (2009). pdf This paper presents the Extended Random Coupling Model.

Gabriele Gradoni, Thomas M. Antonsen, Jr., and Edward Ott, “**Impedance and power fluctuations in linear chains of coupled wave chaotic cavities**,” Phys. Rev. E **86**, 046204 (2012). pdf.

T. M. Antonsen, G. Gradoni, S. M. Anlage E. Ott, “**Statistical Characterization of Complex Enclosures with Distributed Ports**,” proceedings of the 2011 IEEE International Symposium on Electromagnetic Compatibility, pp. 220-225. pdf

G. Gradoni, Jen-Hao Yeh, T. M. Antonsen, S. M. Anlage, E. Ott, “**Wave Chaotic Analysis of Weakly Coupled Reverberation Chambers**,” proceedings of the 2011 IEEE International Symposium on Electromagnetic Compatibility, pp. 202-207. pdf

Jen-Hao Yeh, Thomas M. Antonsen, Edward Ott, Steven M. Anlage, “**First-principles model of time-dependent variations in transmission through a fluctuating scattering environment**,” Phys. Rev. E (Rapid Communications) **85**, 015202 (2012). pdf An application of the RCM to model fading in communications.

Gabriele Gradoni, Jen-Hao Yeh, Bo Xiao, Thomas M. Antonsen, Steven M. Anlage, Edward Ott , “**Predicting the statistics of wave transport through chaotic cavities by the Random Coupling Model: a review and recent progress**,” Wave Motion** 51**, 606-621 (2014). pdf

G. Gradoni, Xiaoming Chen, T. M. Antonsen, Steven M. Anlage, Edward Ott, “**Random coupling model for wireless communication channels**,” 2014 International Symposium on Electromagnetic Compatibility (EMC Europe), pp. 878-882 (2014). pdf

C. Kasmi, O. Maurice, G. Gradoni, T. Antonsen Jr., E. Ott and Steven Anlage, “**Stochastic Kron's model inspired from the Random Coupling Model**,” 2015 IEEE International Symposium on Electromagnetic Compatibility (EMC), pages 935-940, (2015). pdf

Gabriele Gradoni, Thomas M. Antonsen, Steven M. Anlage, and Edward Ott, “**A Statistical Model for the Excitation of Cavities Through Apertures**,” IEEE Trans. Electromag. Compat., **57** (5) 1049-1061 (2015). pdf

Shen Lin, Zhen Peng, and Thomas Antonsen, “**A Hybrid Method for Quantitative Statistical Analysis of In-situ IC and Electronics in Complex and Wave-chaotic Enclosures**,” 2016 Progress In Electromagnetic Research Symposium (PIERS), Shanghai, China, 8–11 August. pdf

Experimental Tests and Verification of the RCM:

Sameer Hemmady, Xing Zheng, Thomas M. Antonsen, Edward Ott, and Steven M. Anlage, "**Universal Statistics of the Scattering Coefficient of Chaotic Microwave Cavities**," Phys. Rev. E **71**, 056215 (2005). pdf

Sameer Hemmady, Xing Zheng, Edward Ott, Thomas M. Antonsen, and Steven M. Anlage, "**Universal Impedance Fluctuations in Wave Chaotic Systems**," Phys. Rev. Lett. **94**, 014102 (2005). pdf

S. Hemmady, X. Zheng, T.M. Antonsen, E. Ott, S.M. Anlage, "**Universal Properties of 2-Port Scattering, Impedance and Admittance Matrices of Wave Chaotic Systems**," Phys. Rev. E **74 **, 036213 (2006). pdf.

Sameer Hemmady, Xing Zheng, Thomas M. Antonsen Jr., Edward Ott and Steven M. Anlage, "**Aspects of the Scattering and Impedance Properties of Chaotic Microwave Cavities**," Acta Physica Polonica A **109**, 65 (2006). pdf.

S. Hemmady, J. Hart, X. Zheng, T.M. Antonsen, E. Ott, S.M. Anlage, "**Experimental Test of Universal Conductance Fluctuations by means of Wave-Chaotic Microwave Cavities,” **Phys. Rev. B **74**, 195326 (2006). pdf. The RCM applied to a classical analog of quantum transport.

Jen-Hao Yeh, James Hart, Elliott Bradshaw, Thomas Antonsen, Edward Ott, Steven M. Anlage, “**Universal and non-universal properties of wave chaotic scattering systems**,” Phys. Rev. E **81**, 025201(R) (2010). pdf

Jen-Hao Yeh, James Hart, Elliott Bradshaw, Thomas Antonsen, Edward Ott, Steven M. Anlage, “**Experimental Examination of the Effect of Short Ray Trajectories in Two-port Wave-Chaotic Scattering Systems**,” Phys. Rev. E 82, 041114 (2010). pdf

S. Hemmady, Ph.D. thesis, "**A Wave-Chaotic Approach to Predicting and Measuring Electromagnetic Field Quantities in Complicated Enclosures**," University of Maryland, 2006.

T. Firestone, M.S. Thesis, "**RF Induced Nonlinear Effects in High-Speed Electronics**," University of Maryland, 2004.

S. Hemmady, T.M. Antonsen, E. Ott, S.M. Anlage, "**Statistical Prediction and Measurement of Induced Voltages on Components within Complicated Enclosures: A Wave-Chaotic Approach,” **IEEE Trans. Electromag. Compat. **54**, 758-771 (2012). pdf This paper offers an accessible introduction to the RCM for the engineering community.

Sun K. Hong, Biniyam T. Taddese, Zachary B. Drikas, Steven M. Anlage, Tim D. Andreadis, “**Focusing an arbitrary RF pulse at a distance using time-reversal techniques**,” J. Electromag. Waves and Apps. **27**, 1262-1275 (2013). pdf

Jen-Hao Yeh, Thomas M. Antonsen, Edward Ott, Steven M. Anlage, “**First-principles model of time-dependent variations in transmission through a fluctuating scattering environment**,” Phys. Rev. E (Rapid Communications) **85**, 015202 (2012). pdf

Jen-Hao Yeh, Edward Ott, Thomas M. Antonsen, Steven M. Anlage, “**Fading Statistics in Communications - a Random Matrix Approach**,” Acta Physica Polonica A, 120, A-85 (2012). pdf

Zachary B. Drikas, Jesus Gil Gil, Hai V. Tran, Sun K. Hong, Tim D. Andreadis, Jen-Hao Yeh, Biniyam T. Taddese and Steven M. Anlage, “**Application of the Random Coupling Model to Electromagnetic Statistics in Complex Enclosures**,” IEEE Trans. Electromag. Compat. **56**, 1480-1487 (2014). pdf

Bisrat D. Addissie, John C. Rodgers, Thomas M. Antonsen, Jr., “**Application of the Random Coupling Model to Lossy Ports in Complex Enclosures**,” IEEE Metrology for Aerospace (MetroAeroSpace) Conference, 214-219 (2015). pdf

Bo Xiao, Thomas M. Antonsen, Edward Ott, and Steven M. Anlage, “**Focusing Waves at an Arbitary Location in a Ray-Chaotic Enclosure Using Time-Reversed Synthetic Sonas**.” Phys. Rev. E **93**, 052205 (2016). pdf See our related work on time-reversed electromagnetic wave propagation.

Jesus Gil Gil, Zachary Drikas, Tim Andreadis, Steven M. Anlage, “**Prediction of Induced Voltages on Ports in Complex, 3-Dimensional Enclosures with Apertures, using the Random Coupling Model**,” IEEE Trans. Electromag. Compat. 58, 1535-1540 (2016). pdf

Min Zhou, Edward Ott, Thomas M. Antonsen, and Steven M. Anlage, “**Nonlinear wave chaos: statistics of second harmonic fields**,” Chaos 27, 103114 (2017). pdf

Additional Information about the RCM:

The Anlage Statistical Methods Meeting presentation can be downloaded here.

The Hemmady MURI Final Review Meeting presentation can be downloaded here.

The Anlage EUROEM 2016 presentation can be downloaded here.

**Computer Code for implemetation of the RCM**

This matlab code generates an ensemble of normalized 2x2 impedance (z) matrices for a given value of the loss parameter k^{2}/(Dk^{2} Q), called "Ktwiddle" in the code. Generating this large number of matrices and finding their eigenvalues is much much faster in matlab than mathematica. The code assumes the system is time-reversal symmetric (GOE). It runs in Matlab 6.5, and generates a file for input into the program NormtoSZ.nb below.

The first mathematica analysis code (StoNorm.nb) takes the measured 2x2 radiation S data (S_{Rad}) and 2x2 ray-chaotic cavity S data (S_{Cav}) and finds the eigenvalues of the normalized impedance (z, EigZnorm.txt) and scattering (s, EigSnorm.txt) matrices. This code is written in Mathematica 5.0. Example input files are Srad.txt and Scav.txt. This cavity data set is just one rendition of the cavity. Ordinarily one wants to analyze a large number of renditions (~100) of the ray-chaotic cavity to compile statistics for the resulting normalized s and z matrices. This code essentially allows you to find the "hidden" universal statistical properties of your enclosure, by removing the non-universal coupling.

The second mathematica analysis code (NormtoSZ.nb) takes the ensemble of normalized 2x2 z matrices generated form the MatLab code above, along with the measured or calculated 2x2 radiation S matrix (S_{Rad}), and generates the eigenvalues of the 2x2 S_{Cav} (EigScav.txt) and Z_{Cav} (EigZcav.txt) matrices. Example input files are the ensemble of 2x2 z matrices with loss parameter k^{2}/(Dk^{2} Q) = 1.5 (RMTZ_1_5loss2.txt) and Srad.txt. This code essentially allows you to predict the statistical properties of the raw S and Z of your cavity, given the loss parameter of the enclosure and the radiation properties of the 2 ports.

**The revised (26 July, 2006) "Terapin RCM Solver v1.0" User's Guide is available here. **

Please direct questions and comments to 'anlage "at" umd.edu'

Link to the UNM-Maryland / AFOSR Center of Excellence on the Science of Electronics in Extreme Electromagnetic Environments web site

This work is supported by ONR under Grant No. N000141512134, ONR/DURIP Grant No. N000141410772, AFOSR COE Grant FA9550-15-1-0171, the DoD MURI for the study of microwave effects under AFOSR Grant F496200110374, as well as AFOSR DURIP Grants FA95500410295 and FA95500510240, and the ONR / Maryland Center for Applied Electromagnetics, Task A2, Grant No. N000140911190.