Introduction

During my postdoc appointment at Penn State I was tasked with developing the Phase III detector for Project 8. A major change from previous phases of the experiment was that we moved into the far-field of the CRES source to cover a larger volume. Having no structures near the source means we have to rely purely on the radiative power transfer and can not exploit any reactive “modal” coupling (i.e. TE01 in a waveguide) as in the previous phases. A wide variety of antenna topics have been covered while studying the large volume demonstrator. This folder contains most of my presentations while at Penn State. A few helpful references for antennas and array design are in an archived folder here. Below is a list the general research developments in this effort followed by an synopsis of each:

1. CRES Synthesis

2. Power Combining

3. Transmit vs. Receive mode

4. Far/Near-field properties

5. Maximum Power Transfer Theorem

6. From Frequency to Time-domain with Impulse Response (HFSS to Locust)

CRES Synthesis

A Locust CRES point source based on a numerical routine for the Lienard-Wiechert potentials has been rigorously proven to be accurate in the frequency and time-domains (Nick B. and M.Jones Reference). The general properties of the CRES point source are that it has an instantaneous gain (from the relativistic lobing) of 6dBi and a time-avg gain of 1.4dBi. The CRES source is very close to an isotropic radiator and thus naturally introduces the “hairy-ball” theorem. The polarization is similar to a magnetic dipole, but has some dynamic polarization effects equivalent to the classic turnstile antenna in order to achieve this isotropy. Therefore, point source antennas have been developed in frequency domain simulation software with the capability of synthesizing radiation from a 90 degree CRES event in both polarization and power. It is essentially an electrically small, K-band, turnstile antenna, or equivalently a crossed Hertzian dipole.

The CRES synth antenna includes the spatial quadrature we see as a spiral in the analytical cyclotron field which includes the LHCP/RHCP in the +/- z-dir. The polarization in the φ direction is proving difficult to create in practice and we are forced to use advanced fabrication techniques to realize the synth CRES antenna in the lab. This is expected since θ pol from the electric dipole moment is naturally much stronger and as Griffiths points out, making a magnetic dipole radiator basically comes down to just trying to minimize the electric dipole moment. A. Ziegler has continued the work into a new level of precision, A. Ziegler Email: adz6@psu.edu for any questions.

Intellectual Property Evolution: This idea had started with Hamish Robertson and Brent VanDevender as a concept with no real design before I arrived at P8, then Nick (at 2018 PNNL collab meeting) and I talked about the idea as an actual testing device for P8 to more detail (reproducing different CRES features). I prototyped a few ideas with PNNL’s help: mag-dipole, octo-micro-helix array (which made the full field), etc...then Erik van Bronkhorst suggested Turnstile, Helical variations, and a turnstile array. Kirk MacDonald's Hertzian fields matched well to time-avg CRES so we sim prototyped Ideas, did a “overly-lobed” proof of concept in the lab, then Andrew Ziegler took from there and added a slot-guide approach, among other phased structures needed to condition the signal to be in spatial quadrature.

Power Combining

After the incident field on an antenna has been converted to a travelling-wave voltage signal on a transmission line, the voltage wave propagates to the amplifier. Since the PIII design involves passive arrays of antennas, it is often the case that multiple antennas are coupled to one amplifier through a feed network. The purpose of power combining in Locust is to sum and propagate these signals accurately through the network to the amplifier. The signal propagation is done semi-analytically in Locust: the amplitude of the signal is derived either from HFSS or an equivalent circuit model, and the phase is calculated analytically assuming mod(2pi) electrical spacing between elements. Some of the details are described in presentations here including “power combining” (or dividing) in the title such as this patch array study and this slot array study. Many of the Locust results are shown alongside analogous computations in HFSS by Arina T. and Penny S. presentations on Basecamp. This Locust/HFSS interface has served as a general cross-check for the coding of the power combiner, although simple analytic impedance calculations have been used as well.

One fundamental discovery made by the antenna team was that the ideal junction S matrix in a passively combined linear array would need to include a high directivity for the LNA-side output, a single port of a 3 port junction. The following S-matrix represents the ideal junction for a phased array with maximal axial coverage and ideal power combining losses through evading reradiation: S=[1,1,1:0,0,0:0,0,0]. In this matrix one can see that port 1 output is highly favored over the others. Unfortunately, EM laws prevent us from creating a simultaneously matched, reciprocal, and lossless network (Pozar p.318). Circulators and Isolators are devices that come close as they use ferrous materials to enforce non-reciprocal behavior. The Wilkinson divider also has a similar property to that which we are looking for, isolation between two ports.

Moreover, the junction efficiencies highly depend on what mode they are in, even-mode (100%) or odd (0%) or something in between (asymmetric amplitude and/or phase between the inputs). For example, in the Wilkinson divider odd-mode produces total destructive interference while even-mode produces the complete opposite. Perhaps the most important feature we had to address in power combining was the possibility of the incident signals on each junction not only being unequal in phase but also amplitude. This was mostly a near-field effect. We do indeed anticipate this near-field phenomena where the signal’s incident on each of the junctions are not only asymmetric in phase but also in amplitude in the FSCD.

A last but most important part is that these are specifically called Power Dividers and NOT Voltage Dividers, therefore, the outputs are NOT simply a superposition of voltages of the inputs! One way to reconcile this with the irrefutable idea of EM being a linear theory is by considering the transformation of impedances of the junction in question.

Transmit Mode vs Receive Mode

Before discussing the details of the antenna elements for a given array it is important to establish the differences between transmit mode and receive mode analysis. The Friis formula and antenna patterns are all generated under the far-field condition. The detector designed here is designed to be sensitive to near-field phenomena and therefore many of the traditional antenna engineering methods break down.

One particular instance of that breakdown begins in the differences between receive and transmit modes for electromagnetically large arrays. The first place receive mode and transmit mode diverge is in the current distributions on the elements. The current distribution on a transmitting antenna follows that of a pure standing wave. The current distribution on that same antenna in receive mode depends both on the excitation condition and the loading conditions and differs with that on the transmitting antenna. In transmit mode, the field at infinity is the interference of all the differential elements on the antenna with certain amplitude distribution, uniform electric phase (in the case of a half-wavelength dipole), and progressive space phase difference caused by the path difference in space. In receive mode the excitation is a plane wave with uniform amplitude and progressive phase difference caused by the path difference between the wave-front in space and the antenna (note the taper is evident in an array in transmit mode and not in receive mode). Therefore, the electric phase is different at the different points of the receive antenna. The induced current through the load is the superposition of the differential current produced by the differential elements with certain amplitude and electric phase distributions. Although the meaning of superposition is different for transmitting and receiving modes, the final integral expression is the same, resulting in the same pattern.

For the most part, simple array code using electrically small dipole-like elements (even superposition of scalar isotropic sources![“Phase is King” M. Jones]) can be used for analysis of the detector’s coverage. However, a nuance to the detector is that reciprocity breaks down, especially in the near-field with a non-uniform-copolar source. The CRES source has an elliptical polarization up/down stream which affects the up/down stream fields with a -3dB hit of the phi polarization, this changes the reciprocity enough to see significant differences in pure transmit mode and Friis maps using a CRES source in some cases.

Near/Far field Analysis

We have discovered that a major detection sensitivity challenge comes from the CRES source being in the near-field and in some cases the extreme near-field of the receiving array. In the FSCD the general idea is that the fields incident across the array aperture, and therefore the currents induced on the respective elements, are not uniform if the driving source is within the near field of the array. This near-field non-uniformity is exaggerated for a CRES source as the Huygens surface is a changing wave-front of linear to circularly polarized waves on top of that. On top of this the source is also moving relativistically within the near-field of the arrays, in and out of coverage regions producing both amplitude and frequency modulation in the received wave front. It is for these reasons that we employ sanity checks in the Receive mode and do not study the detector focal coverage only in transmit mode as commonly done in HFSS.

In the extreme near-field of a microwave/radio source the fields are reactive. This reactivity can be seen in most easily in the frequency-domain (E, H, V, and I in steady state) in three ways:

• 1) Non-orthogonality of E and H vector phasors

• 2) E_u/H_v not equal to 377

• 3) Phase between the E and H vector phasors

• 4) Non-zero Imaginary part of V/I (in a circuit t-line/port/termination/load)

Bullet points 2 and 3 are connected when you consider instantaneous vs time-avg E/H ratios. Bullet point 1 is related to axions which are coupled to E dot H. 1 is also the reactance one sees in the extreme near-field of a CRES source. The above 3 ways for a field to be reactive are mostly only seen in the extreme near-field, or, reactive near-field. Once you reach the Fresnel distance the fields are mostly all radiative and the only thing that keeps it from being a “far-field” zone is the fact that the pattern is still a function of distance. Bullet 4 is only a concern after the signal has been coupled to a transmission line when matching circuits such as those between the radiating elements in the passive combiner network, or between the antenna output port and the LNA, but it’s still a key reactivity in the full Rx chain nonetheless. At this point one can see the classic ELI the ICE man rule can be used for the free-field reactivity, where inductance “L” is likened to the energy stored in the magnetic field and capacitance “C” to that in the electric field.

Maximum Power Transfer Theorem and the Losses in FSCD

Being as how the Project 8 experiment is challenged by the extremely low power levels of the source, losses by any mechanism should be well understood by the engineer/physicist. Some general electromagnetic losses are shown in the Fig [1] below. One fundamental EE principle that is of constant concern in matching the antenna to the free-field is the Maximum Power Transfer Theorem. It is more typically found in a circuit context, however, for the free-field power propagating to the antenna t-line output it is stated as:

MPTT: Under the condition for which a maximum amount of power is delivered to the antenna terminal impedance, an equal amount of power is reradiated.

This is also true for any impedance transformation along transmission lines in the receiving chain such as LNA input, mixer RF input etc, but the “equal amount of power” being not in the form of reradiation, but rather power dissipated by the “load”. It is not quite intuitively understood as to whether or not this implies a -3dB loss for our already tiny 1fW in the free-field we are working with, as the theorem is in a circuit context. For our application the physical circuit can be drawn to include a source voltage drop across free-space (377 Ohms) as well as the load radiation resistance of the antenna for a full loop.

One thing to consider is the subsequent collection of this equally reradiated power, which indeed is theoretically possible. The conclusion is that there is no law of physics that directly claims we cannot collect more than half of the incident power asymptotically through any arbitrary number scatters. It can be seen with Friis far-field theory that an antenna with a physical aperture of 4pi solid angle (surrounding the transmitter) and matching polarization can capture ~100% of the transmitted power from a point source.

Moreover, for a critically populated sheet of dipoles one can more intuitively see the the limit of Rx power will always be 50% of the incident flux. This is due to the equal amount being reradiated by the dipole on the other side, as stated in the MPTT. However, back that same critically populated sheet with a ground plane (a quarter wavelength away) and you can see that the result gets close to 100%. Therefore, the multiple scatter viewpoint may address the question as to whether or not one can collect 100% of the incident power. This can be seen in this FEM simulation example.

From Frequency to Time-domain with Impulse Response

Traditional antenna engineering and design is dominated by frequency-domain analysis (Balanis, Krauss). Unfortunately for us, the FSCD of Project 8 has a relativistically moving source in the near-field. This includes time-domain effects that cannot be simply imported into a frequency domain software like HFSS. When modeling the antennas in Locust we needed a seamless way to accurately convert the incident E fields into antenna voltages. Since the antennas are Linear Time-Invariant systems we exploited what is called the system's Impulse Response, a concept commonly used in Linear Systems Theory. A Finite Impulse Response filter can be derived from HFSS data with an IFFT of the Transfer Function of the system being modeled. This filter is used to model the antenna's response to any arbitrary time-varying incident radiation field. This is one of the more important contributions I made to Project 8 and the FSCD modeling in Locust and was rigorously validated by J. Tedeschi at PNNL. It is crucial that we accurately model the antenna’s phase response above all since that is most likely how we will localize events through model based processing. Especially for sensitive or resonant structure where the phase response is highly frequency dependent. We have developed a robust connection between HFSS and Locust so we can exploit the best of both worlds, commercial FEM packages and homemade software. A set of tutorials focused on HFSS specific to Project 8 needs has been created for the future Project 8 contributor.