Broadband RF Phased Array Design with MEEP: Comparisons to Array Theory in Two and Three Dimensions
AArticle
Broadband RF Phased Array Design with MEEP:Comparisons to Array Theory in Two and ThreeDimensions
Jordan C. Hanson
Whittier College, Whittier, CA, USA; [email protected] February 10, 2021 submitted to Electronics
Abstract:
Phased array radar systems have a wide variety of applications in engineering and physicsresearch. Phased array design usually requires numerical modeling with expensive commercialcomputational packages. Using the open-source MIT Electrogmagnetic Equation Propagation (MEEP)package, a set of phased array designs is presented. Specifically, one and two-dimensional arrays ofYagi-Uda and horn antennas were modeled in the bandwidth [0.1 - 5] GHz, and compared to theoreticalexpectations in the far-field. Precise matches between MEEP simulation and radiation pattern predictionsat different frequencies and beam angles are demonstrated. Given that the computations match the theory,the effect of embedding a phased array within a medium of varying index of refraction is then computed.Understanding the effect of varying index on phased arrays is critical for proposed ultra-high energyneutrino observatories which rely on phased array detectors embedded in natural ice. Future work willdevelop the phased array concepts with parallel MEEP, in order to increase the detail, complexity, andspeed of the computations.
Keywords:
FDTD methods, MEEP, phased array antennas, antenna theory, Askaryan effect, UHEneutrinos
1. Introduction
Radio-frequency phased array antenna systems with design frequencies of order 0.1-10 GHz haveapplications in 5G mobile telecommunications, ground penetrating radar (GPR) systems, and scientificinstrumentation [1–4]. In the one-dimensional case, a series of three-dimensional antenna elements arearranged in a line with fixed spacing [5]. Common antenna designs like loops and dipoles can be used tolimit the elements to two dimensions. In this special case, phased array radiation may be modeled in twospatial dimensions plus time. In the two-dimensional case, a series of three-dimensional antenna elementsare arranged in a two-dimensional pattern, often a grid with fixed element spacing in both dimensions.The elements may be strictly two-dimensional, but there is still an increase in computational complexityand the radiation is calculated in three dimensions plus time.Proprietary RF modeling packages like XFDTD and HFSS are often used to model the responseof elements within phased arrays and the behavior of arrays [6–9]. The XFDTD package, for example,relies on the finite difference time domain (FDTD) method. The FDTD approach is a computationalelectromagnetics (CEM) technique in which spacetime and Maxwell’s equations are broken into discreteform. One variant of the FDTD method is the conformal FDTD method (CFDTD), recently used tostudy phased array concepts on a large scale [9]. The NEC2 and NEC4 family of codes relies on themethod-of-moments (MoM) approach [10]. Aside from the cost, a drawback of proprietary modeling
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Specifications Define elementgeometry MEEP objectsSegmentationDefine array parametersLoad geometry anddesign parametersRun FDTDsimulationNear to far-fieldroutinesComparison toarray theoryE-plane H-planeUpdate frequency Define near-to-far surfaceSave results
Figure 1.
A detailed workflow for phased array design with MEEP. See text for details. software can be a lack of fine control over each individual object in the simulation. Because Maxwell’sequations are scale-invariant, in principle open-source FDTD codes designed for optical regimes could bere-purposd for RF design workflows. One such open-source package is the MIT Electromagnetic EquationPropagation (MEEP) package [11].A recent review [12] covered how open radio design software like openEMS [13], gprMax [14],and the NEC2 family of codes [10] facilitate design workflows. In this work, the radiation patterns ofone-dimensional and two-dimensional phased array designs are simulated with the MEEP package. MEEPtakes advantage of the scale-invariance of Maxwell’s equations. Common MEEP applications are foundin optical wavelength µ m-scale designs, but scale-invariance allows the user to treat designs as cm-scaleRF elements (see Appendix for details). Although MEEP has been used to optimize antenna designs[15], this work appears to be the first to model entire phased arrays in MEEP with a variable index ofrefraction. Two classes of phased array element are considered: Yagi-Uda and horn antennas. The formeris applied to single-frequency designs, while the latter is applied to broadband design. Each element classis treated in both the one-dimensional and two-dimensional cases. The phase-steering properties andradiation patterns of all designs are shown to match theoretical predictions. The appropriate array theoryis shown in Section 2, based on Chapter 1 of Reference [16]. Section 3 contains comparisons between theoryand simulation for one-dimensional cases, and Section 4 contains the corresponding two-dimensionalcomparisons. In Section 5, the varying index of refraction is introduced. Results and future work aresummarized in Section 6.A workflow using MEEP for phased array design is outlined in Figure 1 based on Figure 1 fromReference [12]. Examples of decisions within the specifications category are: single-frequency or broadband,desired directivity and beamwidth, side-lobe tolerance, and number of antenna elements. These decisionslead to the choice of element type which must be implemented in MEEP. Simple shapes like dipolescan be modeled with built-in MEEP objects. Complex shapes like horns and dishes can be assembledfrom groups of objects. Radiation sources and current functions must be defined. For these studies, puresinusoidal currents are passed to radiators which in turn radiate sinusoidal fields. The dielectric constantand boundary conditions of the simulation volume and objects within the volume are defined in thenext step. The information is loaded into a simulation object and run for a number of time-steps. Oncecomplete, near-to-far field routines are called to produce the power at a set of angles. The power versus ersion February 10, 2021 submitted to Electronics
RF Generation Φ = 0 o d y Φ = 10 o Φ = 20 o xy i = 16i = 15i = ... Signal phases:
Element numbers: ● P ΔΦ = Φ - Φ R R i r i Figure 2.
Definitions for the coordinate system, element label i , position vectors, and phase shift perantenna for a one-dimensional phased array of RF radiating elements. An example phase shift per antennaof ∆Φ = Φ − Φ = Φ − Φ = Φ i + − Φ i = ◦ value is shown. Example position vectors for the 12thelement are shown: (cid:126) R = (cid:126) r + (cid:126) R . angle is converted to normalized E and H-plane array radiation patterns and compared to theoreticalmodels. Given a match, the frequency is updated and the process is repeated. If there is not a match,element separation and other array parameters are adjusted.The workflow in Figure 1 represents a non-parallelized approach. Much development has gone intoenhancing the speed, accuracy, and utility of the FDTD method. First, MEEP itself may be run in parallelmode, providing a speed enhancement. In a high-performance computing (HPC) environment, whereeach node has allocated memory (implying local RAM is not the limiting factor) running MEEP in parallelwould speed up results. There has also been CEM research devoted to enhancing the FDTD approachitself. Decreasing memory usage and avoiding repetitive computations in favor of a more subtle approachis presented in [7]. A three-dimensional implementation of FDTD algorithms on GPUs via CUDA hasalso been explored [17]. The results of this work were obtained using the simplest version of MEEP:non-parallel with the python3 interface run in Jupyter notebooks on a laptop. Therefore the results shownin Sections 3 and 4 could benefit from speed and memory enhancements in future studies.
2. Phased Array Antenna Theory
The basic structure of a one-dimensional phased array of RF radiating elements is shown in Figure 2.Two important numerical constants that determine the beam angle ∆ φ of the array are the inter-elementspacing d y and the phase shift per antenna ∆Φ . Letting the subscript i label each of the N elements, theone-dimensional inter-element spacing in Figure 2 is d y ˆ j = (cid:126) r i + − (cid:126) r i , where (cid:126) r i records the position ofelement i . The phase shift per antenna is ∆Φ = Φ i + − Φ i . The relationship between d y , ∆Φ , and ∆ φ isderived in Section 2.1. The radiation pattern for a given ∆ φ is derived in Section 2.2. For all coordinatesystems, the azimuthal angle in the xy-plane is φ , and the polar angle from the z-axis is θ . The beam angle ∆ φ of the array given ∆Φ and d y will now be derived for the coordinate system inFigure 2. First, the relevant far-field approximation will be described. Second, it will be assumed thatthe elements all radiate at the same frequency ω and have the same vector radiation pattern (cid:126) f ( θ , φ ) thataccounts for co-polarized and cross-polarized radiated power. Third, the (cid:126) E -field at point P will be treated ersion February 10, 2021 submitted to Electronics as a sum of the (cid:126) E i radiated from each element. Fourth, the calculations will be restricted to the xy-plane andthe relationship between the beam angle ∆ φ and array parameters will be obtained for a one-dimensionalarray.According to Figure 2, the position of P can be written (cid:126) R = (cid:126) r i + (cid:126) R i (1)Rearranging, the displacement between the i-th element and P is (cid:126) R i = (cid:126) R − (cid:126) r i (2)The magnitude of the displacement is R i = (cid:113) ( (cid:126) R − (cid:126) r i ) · ( (cid:126) R − (cid:126) r i ) = (cid:16) R − (cid:126) R · (cid:126) r i + r i (cid:17) (3)Factoring an R , and neglecting the third term because it is small compared to the others, R i ≈ R (cid:32) − (cid:126) R · (cid:126) r i R (cid:33) (4)Expanding in a Taylor series to first order in 2 (cid:126) R · (cid:126) r i / R , with ˆ r = (cid:126) R / R , yields R i ≈ R (cid:18) − ˆ r · (cid:126) r i R (cid:19) (5)Distributing the R gives the approximation: R i ≈ R − ˆ r · (cid:126) r i (6)The electric field at P due to the i -th element with individual radiation pattern (cid:126) f i ( θ , φ ) is (cid:126) E i ( R , θ , φ ) = (cid:126) f i ( θ , φ ) exp ( − jkR i ) R i (7)Substituting Equation 6 into Equation 7: (cid:126) E i ( R , θ , φ ) = (cid:126) f i ( θ , φ ) exp ( − jkR ) R exp ( jk (cid:126) r i · ˆ r ) (8)The element positions are written in Cartesian coordinates, while P is written in spherical coordinatesusing u = sin θ cos φ and v = sin θ sin φ : (cid:126) r i = ˆ xx i + ˆ yy i + ˆ zz i (9)ˆ r = ˆ xu + ˆ yv + ˆ z cos θ (10)The total field (cid:126) E at P requires summing over elements. The current delivered to the i -th element couldhave a potentially complex amplitude a i . The details of how the currents a i are converted to radiated (cid:126) E -field are taken to be part of (cid:126) f ( θ , φ ) . The summation for (cid:126) E over elements is (cid:126) E ( R , θ , φ ) = exp ( − jkR ) R ∑ i a i (cid:126) f i ( θ , φ ) exp ( jk (cid:126) r i · ˆ r ) (11) ersion February 10, 2021 submitted to Electronics
For identical radiating elements: (cid:126) f i = (cid:126) f : (cid:126) E ( R , θ , φ ) = (cid:126) f ( θ , φ ) exp ( − jkR ) R ∑ i a i exp ( jk (cid:126) r i · ˆ r ) (12)Define the array-factor F ( θ , φ ) = ∑ i a i exp ( jk (cid:126) r i · ˆ r ) : (cid:126) E ( R , θ , φ ) = (cid:126) f ( θ , φ ) exp ( − jkR ) R F ( θ , φ ) (13)Thus, if F =
1, then the (cid:126) E -field is a plane wave, modified only by the elemental radiation pattern.Complex amplitudes a i that cause a plane wave with wavevector pointed to ( θ , φ ) are a i = | a i | exp ( − jk (cid:126) r i · ˆ r ) (14)The notation for beam angle ∆ φ = φ − φ will be introduced shortly. For ˆ r , u and v take thecorresponding θ and φ for the angles: ˆ r = ˆ xu + ˆ yv + ˆ z cos θ . The angles ( θ , φ ) correspond to theplane wave because the phases in the array factor in Equation 13 are cancelled by those in Equation 14,and the summation is over just the magnitudes | a i | . For a linear array in one-dimension, oriented along they-axis as shown in Figure 2, θ = π /2 and (cid:126) r i = id y ˆ y : (cid:126) E ( R , θ , φ ) = (cid:126) f ( θ , φ ) exp ( − jkR ) R ∑ i a i exp (cid:0) jk ( id y v ) (cid:1) (15)The summation is F ( π /2, φ ) , v = sin ( φ ) and v = sin ( φ ) . The weights a i may be arranged toproduce a plane wave at φ : a i = | a i | exp (cid:0) − jkid y v (cid:1) (16)The i-th phase of (cid:126) E in the array factor is Φ i = kid y ( sin φ − sin φ ) (17)The difference ∆Φ = Φ i + − Φ i for angles not far from the x-axis, | φ | < | φ | <
1, is ∆Φ ≈ d y k ( φ − φ ) = π ( d y / λ )( φ − φ ) = π ( d y / λ ) ∆ φ (18)The beam angle is ∆ φ = φ − φ , the angular distance between a reference angle and the angle at whichall contributions to (cid:126) E are in phase. Equation 18 reveals that the relationship between ∆ φ and ∆Φ is linear,with slope λ / ( π d y ) . In Section 3, the relationship between ∆Φ and ∆ φ obtained from FDTD calculationsvia MEEP are shown to match precisely the theoretical prediction. For two-dimensional grid arrays, therelationship “factors,” in that phase shift per element row and phase shift per element column govern ∆ φ and ∆ θ independently. This theoretical prediction is matched precisely by the FDTD calculations shown inSection 4 as well. The radiation pattern, or relative power P emitted versus beam angle, is obtained from the array factor F ( π /2, φ ) summation. Summation over the phased array with identical elements causes the vector elementpattern (cid:126) f ( θ , φ ) and the common phase and amplitude factors exp ( jkR ) / R to cancel upon normalization.The parameters that characterize the radiation patterns of arrays are N , the number of elements, and d y / λ . ersion February 10, 2021 submitted to Electronics
The magnitude of the complex current to each element is assumed to be the same, | a i | = a . Recall the arrayfactor from Equation 13, with θ = π /2 and a i = a : F ( φ , φ ) = a ∑ i exp (cid:0) jkid y ( v − v ) (cid:1) (19)Let χ = kd y ( v − v ) so that z = exp ( j χ ) . The sum is a geometric series from i = i = N , thenumber of elements: F ( z ) = a N ∑ i = z i = a (cid:18) − z N − z (cid:19) (20)Using the Euler formula for sin ( χ ) , the array factor simplifies to F ( χ ) = − a exp ( j ( N − ) χ /2 ) (cid:18) sin ( N χ /2 ) sin ( χ /2 ) (cid:19) (21)The radiation pattern is proportional to power, so it is prudent to take the magnitude of F ( φ ) : | F ( χ ) | = a (cid:18) sin ( N χ /2 ) sin ( χ /2 ) (cid:19) (22)The normalized radiation pattern will be ( F / F max ) , so it is necessary to find F max :lim χ → | F ( χ ) | = a lim χ → (cid:18) sin ( N χ /2 ) sin ( χ /2 ) (cid:19) = aN (23)So | F ( χ ) | / F max is F ( χ ) F max = sin ( N χ /2 ) N sin ( χ /2 ) (24)Finally, with χ = kd y ( v − v ) , v = sin ( φ ) , and v = sin ( φ ) , the radiation pattern P is | F ( χ ) / F max | : P ( φ ) = (cid:18) sin ( π N ( d y / λ )( sin ( φ ) − sin ( φ ))) N sin ( π ( d y / λ )( sin ( φ ) − sin ( φ ))) (cid:19) (25)The -3 dB beamwidth is 0.886 λ / L , where L = ( N − ) d y . In fact, Equation 19 is a function of v − v ,so altering the ∆Φ in the a i only rotates P ( φ ) in φ -space, corresponding to a translation in v-space . Theradiation pattern in Equation 25 is shown to match precisely the main beam of FDTD calculations via MEEPfor one-dimensional arrays in Section 3. For two-dimensional grid arrays, the E and H plane radiationpatterns “factor,” in that P ( θ , φ ) = P ( θ ) P ( φ ) . In Section 4, precision matches for two-dimensional gridarrays are shown. Because one-dimensional and two-dimensional arrays are considered, some notes about radiationpatterns are necessary. First, all one-dimensional array radiation patterns correspond to the E-plane (thexy-plane). The arrays are specified using elements situated in the xy-plane, and the array extends alongthe y-axis. Radiators are linearly polarized such that the E-plane at some radius r is ( r cos ( φ ) , r sin ( φ ) , 0 ) .The H-plane at r would be ( r sin ( θ ) , 0, r cos ( θ )) , but this data is not relevant for a one-dimensional array.Second, the MEEP python routine get_farfield is evaluated at a radius r (cid:29) L , the length of the array, ersion February 10, 2021 submitted to Electronics
Yagi-Uda Horn Scan-loss, N=16 Horn array
Parameter Value Parameter Value
Frequency (GHz) ∆Φ (degrees) d y / λ SL dB N N L a d c r d y dx z n = c / dx d y d y d resolution 6 resolution 6 Table 1. Yagi-Uda : The first and second columns contain the geometric parameters describing the antennaelements for the Yagi-Uda array.
Horn : The third and fourth columns contain those for the horn array.
Scan-loss : The fifth through eighth columns contain scan loss ( SL dB ) data, reported for different frequenciesand different d y / λ values for the N =
16 horn array. to obtain the far-fields (cid:126) E and (cid:126) H . Notice that not all open-source FDTD codes offer near-field to far-fieldtransition modeling [12].All two-dimensional phased array elements presented in Section 4 are arrayed in the yz-plane, andthe E and H-planes have the same definitions as the one-dimensional case. However, the H-plane resultshave been shifted so that the main beam occurs at θ = P ( θ ) with θ =
0, and does not mean the phasedarray is radiating orthogonally to broadside. Equation 25 is matched to E and H-plane two-dimensionalpatterns, and both are normalized to 0 dB at peak power.
3. Phased Array Designs in One Dimension: Two-dimensional Fields
Two antenna designs were considered in modeling one-dimensional phased arrays: Yagi-Uda andhorn, corresponding to narrowband and wideband applications, respectively. The two designs are depictedin Figure 3 with associated parameters described in Section 3.1 below. The Yagi-Uda antennas have 6elements with the same radius, oriented in the xy-plane: one reflector, one radiatior, three directors and aconnecting boom. The current a i is connected only to the radiator. The horn antennas have three structures:the box containing the linearly polarized radiator, the radiator which is connected to a i , and the curvesof the horn. An exponential function y = f ( x ) = k exp ( k ( x − a )) describes the curves (see Figure 3),and the origin is taken to be at the center of the back edge of the box. The constants are k = a /2 and k = ( c ) ln ( d / a ) . The curves are built from n slices where n = c / dx . All objects comprising the antennaelements have the same metallic conductivity, and the surrounding volume has an index of refraction n =
1. At the edge of the space is a layer 1 ∆ x unit thick called the perfectly matched layer (PML) whichcancels reflections. As described in Section 2, the beam angle is controlled by the phase shift per antenna. Simulationresults were run with the parameters in Table 1 for the one-dimensional arrays. The main results areshown in Figure 4. A discussion about scan loss below references data from Table 1.The phase-steering results are shown in Figure 4. The y-axes of Figure 4 (top left) and (top right) arethe beam angles of the Yagi-Uda arrays, divided by the beam widths. The x-axes for these graphs are thephase shifts per element. The top left plot and top right plots corresond to N = N =
16, respectively.For the N =
16 horn case (bottom left and right), the value of d y / λ = f d y / c varies because the elementscan radiate from ≈ − ersion February 10, 2021 submitted to Electronics r d1 d1 d1 d1 yz L (a) The Yagi-Uda antenna, and the N =
16 array. a a c f(x) = k exp(k (x-a)) k = a/2 k = (1/c) ln(2d/a) (b) The horn antenna, and the N =
16 array.
Figure 3.
The two-dimensional antenna designs used in the one-dimensional phased array simulations.(See Section 3.1 for details). The black region has (cid:101) = (cid:101) , the green borders are perfectly matched layers(PML), and the blue surface is a MEEP Near2FarRegion where flux is recorded for near-to-far projection. Thewhite lines represent metal structures, and the red lines represent the radiating elements. ersion February 10, 2021 submitted to
Electronics
Figure 4. (Top left) The beam angle ∆ φ divided by the beam width BW for the N = ∆Φ , the phase shift per element. (Top right) The same results for the N =
16 array. (Bottomleft) ∆ φ versus ∆Φ for the N =
16 version of the one-dimensional horn array, for several frequencies.(Bottom right) The dependence of the beam width on frequency for the one-dimensional N =
16 horn array. ersion February 10, 2021 submitted to
Electronics
10 of 25 linear fits to the Yagi-Uda data. The gray lines represent the function f ( x ) = bx , with b = λ / ( π d y ) . Forthese models, d y = λ = ∆ φ . At such large ∆Φ values, side lobes canshift the location of the main beam by O ( ) degree by merging with the main beam. In the N = N =
16 case the fitted slope is slightly lower. In each model, theobserved beamwidths are within 1% of the value predicted by Equation 18.For the broadband horn case in the bottom left of Figure 4, three frequency cases are shown: 0.3,1.5, and 3.0 GHz. The intercepts are all consistent with zero and the slopes scale correctly: dividing thefrequency by a factor of 2 increases the slope by a factor of 2, and dividing by a factor of 10 increasesit by a factor of 10. Graphs like the top left and top right of Figure 4 would be misleading for hornantennas since the beamwidth depends on frequency (bottom right). The fit parameters for beam widthwere a = ± b = ± f -dependence is a good description of the beamwidth across the [0.3 - 5 GHz] bandwidth. The constantterm b is only necessary since the array has finite length L . The beamwidth ( BW ) scales inversely witharray length L : BW ≈ λ / L , from Equation 25.A discussion of scan loss is merited when analyzing normalized radiation patterns, which are shownbelow in Section 3.2. Scan loss may be quantified as the peak power at the given beam angle divided bythe peak power at a beam angle of zero degrees. In the form of an equation in decibels, scan loss becomesa subtraction: SL dB = P φ − P φ = (26)The scan loss SL dB is shown for the N =
16 one-dimensional horn array in Table 1 (right), as it varieswith frequency and d y / λ . The conservative value ∆Φ =
80 degrees was chosen because it is associatedwith the largest beam angles that do not generate side lobes larger than -15 dB. Given the beam width ofthe N =
16 design (5.04 degrees), this corresponds to a scan range of ± d y / λ values that begin to admit largeside lobes (Section 3.2). Radiation patterns in the E-plane from N =
16 one-dimensional Yagi and horn arrays are shown inFigures 5 and 6, respectively. As described above, the x-direction ( ∆ φ =
0) corresponds to no phase shiftper element ( ∆Φ = N -value and d y / λ -value. Equation25 is symmetric, with identical forward and backward lobes. The front-to-back or FB ratio would be 1.0 or0 dB for a row of ideal point sources. Although there is no backplane in either simulated one-dimensionalarray, the FB ratios of ≤ −
15 dB are observed.The Yagi-Uda results are shown for 2.5 and 5.0 GHz frequencies in Figure 5, with ∆Φ =
0, 20, 40,and 60 degrees. Though the radiating elements are 6 cm long, good agreement between simulationand Equation 25 is observed at both 2.5 GHz and 5.0 GHz, including side-lobes. The beamwidth isproportionally larger at 2.5 GHz relative to 5.0 GHz, and at 5.0 GHz, the theoretical -3 dB beam width of5.0 degrees is achieved. The amplitudes of all side-lobes are limited to ≈ −
15 dB, except at the highestbeam angles where scan losses are experienced. Finally, the effect of frequency on beam steering is evident.The same ∆Φ does not generate as large a ∆ φ at higher frequencies because the slope implied by Equation18 is proportional to λ . ersion February 10, 2021 submitted to Electronics
11 of 25
Figure 5. Yagi-Uda results, two-dimensional elements, one-dimensional array. (Top row) f = ∆Φ =
0, 20 degrees from left to right. (Second row) f = ∆Φ =
40, 60 degrees from left toright. (Third row) f = ∆Φ =
0, 20 degrees from left to right. (Bottom row) f = ∆Φ =
40, 60 degrees from left to right. The radial units are dB, and the angular units are degrees. ersion February 10, 2021 submitted to
Electronics
12 of 25
Figure 6. Horn results, two-dimensional elements, one-dimensional array. (Top row) f = ∆Φ =
0, 10 degrees from left to right. (Second row) f = ∆Φ =
20, 30 degrees from left toright. (Third row) f = ∆Φ =
0, 10 degrees from left to right. (Bottom row) f = ∆Φ =
20, 30 degrees from left to right. ersion February 10, 2021 submitted to
Electronics
13 of 25 - . + . + . Figure 7. (From left to right) The N =
16 horn one-dimensional linearly polarized electric field | (cid:126) E ( x , y , t ) | at t = t = t = t = ×
150 cm , with resolution of 6pixels per distance unit (480 ×
900 pixels). The frequency is 2.5 GHz, and the beam angle is 9 degrees frombroadside. (Far right) The corresponding radiation pattern.
The horn results are shown in Figure 6 for 0.5 GHz and 5.0 GHz frequencies, corresponding to thelower and upper end of the bandwidth. The phase shifts per element are ∆Φ =
0, 10, 20, and 30 degrees.The angular range of ∆Φ is restricted relative to the Yagi-Uda case. Wideband systems experience a naturaltrade-off in angular range versus bandwidth. A d y / λ value that is acceptably smaller than one at lowfrequencies can grow larger with increasing frequency, leading to interference patterns. At 5.0 GHz, thehorns radiate at ±
45 degrees from ∆ φ =
0. The prediction from Equation 25 is that these side-lobes, or grating lobes , are equal in relative power to the main beam. The actual array limits them to −
15 dB, butonly if | ∆Φ | <
35 degrees. For larger phase shifts per element, the opposite side-lobe grows above −
15 dB.If the beam is steered too far in the − ˆ φ -direction, the side-lobe on the ˆ φ side grows, and vice versa.The general features of the radiation pattern compare well to the theoretical prediction. The1/ f -dependence of the main beamwidth is evident in Figure 6. Like the Yagi-Uda array, the minimumtheoretical beamwidth is reached at the highest frequencies (Figure 4 bottom right). The mini-lobes thatare partially merged with the main beam widen the beam, however, the beamwidth is calculated at anglescorresponding to -3 dB relative power. Since the mini-lobes are below -3 dB, the beamwidth calculation isunaffected. The simulation also matches the location and width of side-lobes to the theoretical predictionacross the bandwidth. The six grating lobes at 5 GHz are a result of the pattern multiplication theorem,which states that the normalized radiation pattern is a product of the horn pattern and the pattern ofan array of point sources. At 5 GHz, this multiplication suppresses the horn element pattern in themultiplication.The field magnitude | (cid:126) E ( x , y , t ) | for the N =
16 horn array is shown in Figure 7 for t = ± | (cid:126) E ( x , y , t ) | is ± ∆ φ is 9 degrees, with ∆Φ = −
35 degrees, and the beamwidth is 5.5 ± × ×
150 cm with a resolution of 6 pixels per ∆ x . The dimensions of the box (Tab. 1: a = λ =
12 cm. As the radiation escapes to free space, the wavefront formsseveral λ in front of the horns. Higher-frequency modes with f (cid:29) t = | (cid:126) E ( x , y , t ) | can be interpreted as physical rather than numerical. These features can be eliminated with amplitudesmoothing near t = CustomSource class. Amplitude smoothing,however, makes the location of the wavefront less precise.The radiation pattern in Figure 7 matches the theoretical prediction for the main lobe and first fewside lobes. The origin of the side lobes is apparent from the | (cid:126) E ( x , y , t ) | images, where diffraction patterns ersion February 10, 2021 submitted to Electronics
14 of 25 at the edges of the array are visible. At 0.5 ns, radiation in an element is confined to the horn. By 1.0 ns,that radiation joins the waves from horns on either side. However, horns at the end of the array have nopartner on one side, and some radiation leaks outside the main lobe. The side lobes can change if the totalrun time is not sufficient. The get_farfield routine in MEEP requires
Near2FarRegion surfaces that formthe near-field box that collects flux information at the radiation frequency. The parameters of the near-fieldbox are set by the add_near2far routine. The get_farfield routine performs a near-to-far field projectionto the given radius ( r = ∆ t so that enough radiation can cross the near-field box. The codeis run for 6.67 ns to generate the radiation pattern in Figure 7. Thus, the side lobes are averaged over manyradiation periods.
4. Phased Array Designs in Two Dimensions: Three-dimensional Fields
For two-dimensional grids of radiating elements, the array-factor F ( u , v ) factors : F ( θ , φ ) = F ( u − u ) F ( v − v ) (27)The radiation pattern in Equation 25 applies to the E and H plane separately. The two-dimensionalarrays modeled below are square N × N arrays, so beamwidths implied by Equation 25 are equal for the Eand H planes. The complex phasing of Equation 27 also indicates that ∆ φ E ∝ ∆Φ E , and ∆ φ H ∝ ∆Φ H , asshown in Equation 18 for the one-dimensional case. For the designs presented, the H-plane correspondsto the xz-plane, and to varying the phase in the z-direction (by array row). The E-plane corresponds tothe xy-plane, and to varying the phase in the y-direction (by array column). In Figure 8, the basic shapeof the two-dimensional array is shown in the yz-plane with ∆Φ E =
15 degrees, and ∆Φ H =
15 degrees.Section 4.1 contains results along the lines of Section 3.1 but for two-dimensional Yagi and horn arrays,and Section 4.2 contains results along the lines of Section 3.2 but for two-dimensional Yagi and horn arrays.As before, Table 1 contains the typical run parameters, with a few important exceptions.The first exception is that for the two-dimensional case N becomes N × N . However, squaring thenumber of antennas raises the memory requirements. In order to stay within a 16 GB memory limit,the two-dimensional horn array results had to be restricted to N × N = ×
8. The 2D horn array stillhas over O ( ) metal objects, compared to the O ( ) objects for the N × N = ×
16 2D Yagi-Uda.The typical memory consumption is listed in Table 2, along with modified run paramters. Further, theresolution parameter was restricted to 4.0 for the horns. Restricting to 4 pixels per ∆ x unit limits memoryconsumption, but then the box containing the radiator has too few pixels. Enlarging the box allows theproper sized radiator to be fully contained. A final object was added to reduce the FB ratio: a back-planewith parameters listed in Table 2. One interesting modification is the doubling of the ratio of the box size( a ) and the final horn width ( d ). This had the effect of limiting the maximum frequency to ≈ d / λ ≈ a . Initial runs were performed with hornelements that simultaneously widened according to the exponential function defined in Section 3. Thatdesign allowed reflections internal to the horns to distort the initial wavefront. Holding the horn-widthconstant in the z-direction produces radiation patterns that match Equation 25 because it follows theone-dimensional example of Section 3. To obtain z-polarized wavefronts, all that is necessary is to rotatethe array. Practically, there are already examples of dually polarized RF band horns used in particleastrophysics [18,19], meaning that if this design were created with such elements, no rotation would benecessary. ersion February 10, 2021 submitted to Electronics
15 of 25 y zdy d z λ (Φ E , Φ H ) = (0,0)(Φ E , Φ H ) = (0,225)(Φ E , Φ H ) = (225,0)(Φ E , Φ H ) = (225,225) Figure 8.
The two-dimensional N × N = ×
16 Yagi-Uda/horn y-polarized array layout. The alignmentwith 3D Cartesian coordinates is depicted, along with the array spacing variables d y and d z , the wavelength λ (to scale), and the phases for each row and column if ∆Φ E = ∆Φ H =
15 degrees.
Horn
Parameter Value System InformationN × N × a c d dx n = c / dx d y
16 MEEP installationresolution 4 Python3 interface (conda)backplane location − a backplane thickness 0.5backplane dim. 142 × Table 2.
The parameters for the N × N = × N × N = ×
16 array did not require modification. The number of CPU cores was 4 in hardware, butwas effectively 8 with hyperthreading. The most memory-intensive simulation was the 8 × ersion February 10, 2021 submitted to Electronics
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Figure 9. (Left) The beam angles ∆ φ E and ∆ φ H versus the phase shifts for the N × N = ×
16 Yagi-Udaarray at 5 GHz. The black lines represent the theoretical prediction of a linear dependence with slope λ / ( π d y ) or λ / ( π d y ) , ( d y = d z ). In each graph, the circles and squares correspond to two different ∆Φ constant values for the other array plane. In these examples, location of zero phase on the array is chosen tocause a negative beam angle. (Right) The data points correspond to beam angles in the E and H-planes, withthe associated beamwidths as errorbars. These data represent one-quarter of the possible scan positionswith ∆Φ E / H =
15 degrees.
The ∆ φ vs. ∆Φ results for the two-dimensional N × N = ×
16 Yagi-Uda array are shown in Figure9. Figure 9 (top left) contains ∆ φ E versus ∆Φ E data at 5 GHz. The data match the theoretical linear slope λ / ( π d y ) and λ / ( π d z ) , with d y = d z . The phase shift per antenna is varied over [
0, 75 ] degrees in 15degree increments independently by row and column. The circles and squares correspond to ∆Φ H = ∆Φ H is held at either constant, ∆ φ E still varies with ∆Φ E correctly. Figure 9(bottom left) contains ∆ φ H versus ∆Φ H data at 5 GHz. The circles and squares correspond to ∆Φ E = ∆Φ E =
75 degrees or ∆Φ H =
75 degrees, side lobes appear ( > − × ∆Φ E / H =
15 degrees. The xy-errorbars correspond tothe beamwidths.The ∆ φ vs. ∆Φ results for the two-dimensional N × N = × ∆ φ E versus ∆Φ E data at 1 GHz. The larger horn size relative to those inSection 3 means the upper frequency is ≈ ∆Φ H = ∆ φ H versus ∆Φ H data at 1 GHz. The circles and squares correspond to ∆Φ E = a = ± b = ± c = ± d = ± ersion February 10, 2021 submitted to Electronics
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Figure 10. (Left) The beam angles ∆ φ E and ∆ φ H versus the phase shifts per column (for the E-plane)and per row (for the H-plane) for the N × N = × ∆Φ constantvalues for the other array plane. (Right) The beamwidth in the E and H-planes versus frequency. The radiation patterns in the E and H plane for the two-dimensional Yagi-Uda array are shown inFigure 11. The phase combinations ( ∆Φ E , ∆Φ H ) = (
0, 0 ) , (
30, 60 ) , (
60, 30 ) degrees are shown for E andH planes at 3 and 4 GHz. As in Section 3.2, Equation 25 is shown in red, and the simulation results areshown in blue. The main beam and first several side lobes are modeled correctly in each case, and theFB ratio is ≤ −
15 dB. The side lobes are also at the ≈ −
15 dB level. Following Figure 10 (right), the mainbeam is narrower at 4 GHz than at 3 GHz. Though not generally designed to be broadband elements, theYagi-Uda elements do display some flexibility in frequency. The log-periodic dipole array (LPDA) is abroadband example constructed from dipoles as the Yagi is [20].Producing the radiation patterns in Figure 11 requires only O ( ) seconds to run near-fieldcalculations, and only another ≈ get_farfield routine over the E and H-planes.Modeling arrays constructed from dipole elements is orders of magnitude faster than for the array ofhorns, due to the two-dimensional nature of the dipole elements. Producing the radiation patterns ofFigure 12 for the two-dimensional horn array requires ≈
60 minutes combined for the E and H-planepatterns, per frequency . Unlike the Yagi case, the vast majority of time is not dedicated to the get_farfield routine, but to the near-field calculations. The near-field calculations require “sub-pixel smoothing” forthe many edges of the blocks that comprise the horn structure.The radiation patterns in the E and H plane for the two-dimensional horn array are shown in Figure12. The phase combinations ( ∆Φ E , Φ H ) = (
0, 0 ) , (
30, 60 ) , (
60, 30 ) degrees are shown for E and H planesat 0.5 and 1.0 GHz. As in Section 3.2, Equation 25 is shown in red, and the simulation results are shownin blue. The main beam and first several side lobes are modeled correctly in each case, and the FB ratiois ≤ −
15 dB. The side lobes are also at the ≈ −
15 dB level. Due to the higher bandwidth, a wider rangeof beamwidths is available (see Figure 10). The main beam is narrower at 1 GHz than at 0.5 GHz. Thehorns produce the correct pattern for ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees from 0.1 to 1.2 GHz. However, gratinglobes above −
15 dB are a known problem that occur when attempting to steer phased arrays built frombroadband horns to wide angles (see Chapter 9 of Reference [16]). The addition of the backplane limitsdiffraction of the radiation around the edges of the array and therefore limits the FB ratio, but grating lobesappear at ±
45 degrees from the main beam. There is occasionally a back lobe, which can be attributed to ersion February 10, 2021 submitted to
Electronics
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Figure 11. Yagi-Uda results, two-dimensional array (First row) f = ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees, E-plane ( ∆Φ E , ∆Φ H ) = (
60, 30 ) degrees, E-plane ( ∆Φ E , ∆Φ H ) = (
30, 60 ) degrees. (Second row) f = ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees, H-plane ( ∆Φ E , ∆Φ H ) = (
60, 30 ) degrees, H-plane ( ∆Φ E , ∆Φ H ) = (
30, 60 ) degrees. (Third row) f = ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees, E-plane ( ∆Φ E , ∆Φ H ) = (
60, 30 ) degrees,E-plane ( ∆Φ E , ∆Φ H ) = (
30, 60 ) degrees. (Fourth row) f = ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees, H-plane ( ∆Φ E , ∆Φ H ) = (
60, 30 ) degrees, H-plane ( ∆Φ E , ∆Φ H ) = (
30, 60 ) degrees. ersion February 10, 2021 submitted to Electronics
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Figure 12. Horn results, two-dimensional array (First row) f = ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees, E-plane ( ∆Φ E , ∆Φ H ) = (
60, 30 ) degrees, E-plane ( ∆Φ E , ∆Φ H ) = (
30, 60 ) degrees. (Second row) f = ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees, H-plane ( ∆Φ E , ∆Φ H ) = (
60, 30 ) degrees, H-plane ( ∆Φ E , ∆Φ H ) = (
30, 60 ) degrees. (Third row) f = ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees, E-plane ( ∆Φ E , ∆Φ H ) = (
60, 30 ) degrees,E-plane ( ∆Φ E , ∆Φ H ) = (
30, 60 ) degrees. (Fourth row) f = ( ∆Φ E , ∆Φ H ) = (
0, 0 ) degrees, H-plane ( ∆Φ E , ∆Φ H ) = (
60, 30 ) degrees, H-plane ( ∆Φ E , ∆Φ H ) = (
30, 60 ) degrees. ersion February 10, 2021 submitted to Electronics
20 of 25 the diffraction of fields around the edge of the backplane. This effect is more pronounced when the mainbeam is steered to a wide angle and occurs in the hemisphere opposite to the main beam.
5. Variation of the Index of Refraction
The behavior of a one-dimensional phased array embedded within a dielectric medium withspatially-dependent index of refraction n ( z ) is interesting to the ultra-high energy (UHE) neutrinocommunity [3,5]. Phased arrays represent an opportunity to lower the RF detection threshold for RF pulsesgenerated by UHE neutrinos via the Askaryan effect. Antarctic ice is the most convenient and naturalmedium for Askaryan pulse detection, due to the RF transparency and large pristine volumes located inAntarctic and Greenlandic ice sheets and shelves [21–23]. The index of refraction varies within the icebecause of the transition between surface snow ( ρ ≈ ) and the solid ice below ( ρ = ).Most recent and intricate studies of phased array beam behavior still assume a uniform medium [24–26].Embedded phased arrays with varying n ( z ) emit signals that curve in the direction of increasing n ( z ) .The shadow zone is the volume of ice from which RF signals do not reach a receiver due to the excesscurvature of the ray trace [27]. While there is evidence that RF signals can propagate horizontally throughAntarctic ice [28], data from Greenland suggests the relative strength of the effect is small compared tothe curved radiation [29]. Using the tools developed in this work, it is possible to map out the shadowzone for an embedded phased array radiating sinusoidal signals at fixed frequency. Intriguingly, when thephased array radiates , the grating lobe power reflect downward from the snow-air interface, and radiatesinto the shadow zone. Grating lobe power also refracts into the air above the interface. Grating lobe powerleaves the array at a different angle than the main beam, so their presence in the shadow zone does notrepresent forbidden RF propagation.A two-parameter fit to the n ( z ) data versus depth z below the surface is given by [28] n ( z ) = (cid:40) z > n ice − ∆ n exp ( z / z ) z ≤ ∆ n = ± z = ± n ice = in-situ value of λ so the λ /4 dipoles were spaced by λ /2 according to their free space λ value. Atthe selected frequency of 200 MHz, the dipole length is 0.375 meters, and the spacing is 0.75 meters.Figures 13 and 14 contain the results of a N = ∆ z is 0.1 meters. The units in Figure 13 are meters, and the unit-less frequency in MEEPwas scaled accordingly, to correspond to 200 MHz. The distances between the air-snow interface and thefirst phased array element is 15 meters (top) and 35 meters (bottom).The color scale in Figure 14 is ± ± where the radiation has penetrated the ice after 200 timesteps. In Figure 14 (top), the main beam has curved downwards in the direction of increasing n ( z ) , whilegrating lobes have both diffracted to the air and reflected into the shadow zone. The rate of curvature of ersion February 10, 2021 submitted to Electronics
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Figure 13.
The simplified N = d z = λ /2.0, and the length of the dipole radiators is λ /4.0. The colorscale represents n ( z ) inEquation 28. the main beam is controlled by the fit parameter z in Equation 28. In Figure 14 (bottom), the physics is thesame as Figure 14 (top), but the effect of n ( z ) curvature is weakened. The beam travels farther horizontallybecause the gradient of n ( z ) is smaller at the larger depth. The geometry of the larger depth is such that thereflected grating lobe power is interfering with grating lobe power that was curved downwards withoutreflection. This can be seen just above the C marker in Figure 14 (bottom).
6. Summary and Future Analysis
Four phased array designs have been modeled with the MIT Electromagnetics Equation Propagation(MEEP) package in non-parallel mode. Two types of individual radiating element were explored: thenarrow-band Yagi-Uda and broadband horn antennas. Two phased array geometries were explored: one-dimensional and two-dimensional . The one and two-dimensional Yagi-Uda phased arrays were designedfor ≤ ∆ φ and ∆Φ (Figure 4) (top left and right). Although any row of pointsources would obey the relationship in Equation 18, a row of point sources has two main beam solutionsby symmetry. Thus Figure 4 could not be interpreted correctly were it not for the proper functioning of theYagi elements. The radiation patterns produced with the one-dimensional Yagi array were compared to Figure 14.
The magnitude of the z-component of the z-polarized dipoles as they radiate as a phased arraywith ∆Φ = n ( z ) . ersion February 10, 2021 submitted to Electronics
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Equation 25 in Figure 5. The radiation pattern in the E-plane is shown to agree with Equation 25 in boththe main beam and the first several grating lobes. The calculation takes place in two-dimensions, so anH-plane comparison is not relevant.The one-dimensional array of horn antennas was analyzed in Section 3. The array demonstrated thecorrect linear relationship between ∆ φ and ∆Φ (Figure 4). In that case, the slope of ∆ φ vs. ∆Φ was increased by a factor of 2 and then 10 by decreasing the frequency by a factor of 2 and then 10. The bandwidth of thetwo-dimensional versions of the horns allows the variation of scan range. The scan range is smaller athigh frequencies, as indicated in Figure 4 (bottom left). However, the beamwidth is also smaller at highfrequencies, as indicated in Figure 4 (bottom right). The design trade-off is between small beamwidth andlarge scan range. In Figure 6 the one-dimensional horn array radiation pattern is shown to match Equation25 at both 0.5 and 5.0 GHz. There are 2-4 side lobes at 0.5 GHz to match per pattern, and the simulationresults match them as well as the wide main beam. At 5.0 GHz, the main beam is accompanied by twoprominent grating lobes at ±
45 degrees that should be as powerful as the main beam. The simulationfinds them at the -15 dB level. The grating lobes are being suppressed by the the pattern null from thehorn element pattern [16]. At lower frequencies, however, scan loss takes a toll on radiated power (Tab. 1).The two-dimensional, N × N = ×
16 Yagi-Uda array was analyzed in Section 4. The arraydemonstrated the correct linear relationships between ∆ φ E and ∆Φ E , and ∆ φ H and ∆Φ H (Figure 9) (left).Given the narrow beamwidth, the array design can be scanned ± ± N × N = × ∆ φ E and ∆Φ E , and ∆ φ H and ∆Φ H (Figure 10) (left). The beamwidthis again inversely proportional to frequency (Figure 10) (right). It is not surprising that the fits differ slightlyin the E and H-planes, since the horn width changes in the E-plane but does not in the H-plane. The qualityof the fits to 1/ f + const are excellent. The additive constants in these fits are only necessary because thearray cannot be infinitely long. Technically, Equation 25 implies that the beamwidth would go to zero as N → ∞ . The radiation patterns of the two-dimensional horn array are displayed in Figure 12 at 0.5 and1.0 GHz, for the same sampling of scan angles as in Figure 11. The high-frequency beam is narrower andaccompanied by grating lobes at ±
45 degrees. The patterns agree with theoretical expectations, with theexception of the H-plane lobes at ±
90 degrees. At low frequency, the beam is wider and is accompanied bygrating lobes at ±
45 degrees from the main beam. The results match in the main lobe, but the simulationdoes not match the theoretical grating lobes. This is pronounced when the beam is moved far frombroadside in the H-plane.Finally, a simplified version of the N = n ( z ) . The model for n ( z ) was a simple fit to the profile of the ice atthe South Pole, which is a location of interest for planned phased array detectors designed to recordAskaryan signals from UHE neutrinos passing through ice. Though the studies in this work are restrictedto phased-arrays as transmitters, and not receivers, the shadow zone of the array was mapped at 200 MHzunder realistic conditions. An interesting side effect of the phased array being the radiating system wasthat the grating lobes managed to propagate into the shadow zone.Future work would include several enhancements to the simulations. Calculations of S-parametersfor individual elements should be added, and optimization studies on horn and Yagi geometric parametersare warranted. However, other RF element types should also be studied. Due to the relevance of ersion February 10, 2021 submitted to Electronics
23 of 25 one-dimensional phased array receivers for UHE neutrino physics, one interesting choice is the wide-radiusdipole used by the Radio Neutrino Observatory Greenland (RNO-G) collaboration [30]. Such elementsalready have low VSWR measurements in the relevant bandwidth. Finally, upgrading the simulationcode to utilize parallel MEEP capabilities will increase the potential speed and complexity. Additionalcomplexity will come in the form of more accurate antenna structure modeling, thereby improving thetrustworthiness across a wide bandwidth.
Funding:
This research was funded by the Office of Naval Research (ONR) under the Summer Faculty ResearchProgram (SFRP).
Institutional Review Board Statement:
Not applicable.
Informed Consent Statement:
Not applicable.
Data Availability Statement:
The data presented in this study are available on request from the corresponding author.Due to large file sizes and restricted bandwidth, please contact corresponding author to set up collaborative sharing.
Acknowledgments:
We would like to thank the Office of Naval Research (ONR) for helping to support this research.In particular, we would like to thank the Naval Surface Warfare Center Corona Division and their continued supportof the ONR Summer Faculty Research Program (SFRP). Conversations with Christopher Clark and Gary Yeakley wereespecially helpful. We are also grateful to Van Nguyen, Jeffery Benson, and Golda McWhorter. Karon Myles deservesour special thanks for helping to coordinate the SFRP program.
Conflicts of Interest:
The authors declare no conflict of interest.
Appendix A
MEEP FDTD is more often applied to 1 µ m scale lengths than the 1 cm-scale RF elements, soscale-invariance must be highlighted. Scale invariant units with c = µ m are said to have an “a-value of 1 µ m.” A value of a = f = = a , or simply 4.0 with f = λ = c = λ = f − = T . The periodis 4.0, so a simulation run of 50 T =
200 time units corresponds to 6.67 ns. Assuming 1 pixel/a-value,simulated radiation would therefore propagate 200 units of ∆ x in a straight line before time was up. A resolution parameter sets the number of pixels per distance unit and is usually larger than 1.0. Selecting theright resolution is often a subtle balance between capturing the most relevant effects while limiting thememory usage of the simulation results. References
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