# The Systematic Design of Non-commensurate Impedance Matching Tapers for
Ultra Wideband Gradient-Index (GRIN) Lens Antennas

Wei Wang, , Nicolas Garcia, , Jonathan Chisum Manuscript received August 28,
2020; revised February 13, 2021; revised April 28, 2021; accepted June 5,
2021. Date of publication ??; date of current version June 9, 2021. This work
supported in part by the Department of the Navy, Office of Naval Research
under contract N00014-20-C-1067. (Corresponding author: Jonathan Chisum)The
authors are with the Department of Electrical Engineering, University of Notre
Dame, Notre Dame, IN, 46556 USA. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>versions of one or more of the figures in this paper are available online at
http://ieeexplore.ieee.org.Digital Object Identifier ??.????/???.????.???????

###### Abstract

We propose a general method for designing wideband matching tapers in
inhomogeneous media where phase velocity is coupled to the taper impedance
profile. Such tapers are used to match wideband gradient index (GRIN) lens
antennas. To simplify fabrication tapers are often constrained to physically
uniform layers wherein commensurate line theory cannot predict the frequency
response. Therefore, we present a new design algorithm which derives an
effective permittivity $\epsilon_{eff}$ to equalize the electrical length of
commensurate and non-commensurate line tapers. The algorithm provides a
systematic design method with predictable frequency response for non-
commensurate line tapers. Nevertheless, there are several unavoidable
nonidealities present in such discretized tapers which we discuss and provide
recommendations for mitigation. The algorithm is used to design a Klopfenstein
taper with return loss better than 15 dB from 8 to 78 GHz. The design is
fabricated and measurements agree with simulation across the WR90, WR28, and
WR12 bands. An approximate efficiency formula is proposed which predicts
aperture efficiency of taper-matched lenses without the need for time-
consuming full-wave simulations. Various lenses are designed and compared to
highlight the advantages of Klopfenstein tapers in GRIN lens design. The
results demonstrate the usefulness of the proposed design method.

###### Index Terms:

matching taper, inhomogeneous media, metamaterial, GRIN lens, millimeter-wave

##  I Introduction

We present a method for the design of broadband impedance matching tapers
constrained to uniform layer thickness for application in quasi-optical
systems such as gradient index (GRIN) lens antennas. GRIN lenses provide a
low-profile means of realizing high aperture efficiency over wideband
operation but only if they are properly impedance matched over the band [1].
Because of the wide variety of applications which use GRIN lenses, their
matching sections should be readily designed across many frequency bands (from
Ku-band to W-band) [1, 2, 3, 4] and with a variety of performance metrics. For
example, cost and size-constrained Ku/Ka-band commercial satellite
communications demand large bandwidth and high aperture efficiency to enable
compact and high-gain apertures [5]. Such applications might prefer low cutoff
frequency over passband match level. In contrast, high power, wideband radar
and communications systems[6] prefer extremely low passband reflections to
reduce power reflected back toward the source. In polarization sensitive
applications very low passband reflections might be required in order to
maintain a nearly equal passband transmission efficiency for TE- and TM-
modes[7]. These demands require a flexible and predictable matching solution
for GRIN lenses.

Figure 1: (a) Symmetrical GRIN lens cross section and permittivity profile
with matching tapers (vert./horiz. axes not to scale). (b) Ideal, continuous
taper $Z(z)$ and discretized PUT $Z[n]$. (c) PUT and PNT in adjacent unit-
cells with core permittivity $\epsilon_{rc}$. (d) Normalized aperture
efficiency $\eta_{ap}^{\prime}$ (see (7)) for unmatched and Klopfenstein taper
GRIN lenses.

For practical considerations, the 3D permittivity profile required to
implement a GRIN lens is often realized on a uniform grid as in [8, 9, 2, 10,
6, 11, 3, 4, 12, 1]. As we will show below, this constraint makes it
impossible to reliably predict the performance of an impedance matching taper
using classical commensurate-line taper theory [13]. The general symmetrical
GRIN lens cross section and permittivity distribution considered in this work
is shown in Fig. 1(a). In such a configuration the GRIN lens comprises a stack
of layers, each with a unique in-plane ($xy$-plane) permittivity profile.
While, in general, the thickness of each layer may be independent from other
layers, in order to practically realize a continuous 3D index gradient the
GRIN profile is often discretized on a uniform grid. As such, the tapers in
this work are constrained to be of uniform layer thickness $l_{N}$ along the
$z$-axis.

Figure 1(b,top) shows a simplified taper in which a continuously varying
characteristic impedance, $Z(z)$, is realized by varying the width of the
transmission line. In order for such a taper to serve as the surface impedance
matching section of a GRIN lens (dashed rectangles of Fig. 1(a)),
$Z(z),z\in[0,L]$ is usually fabricated as $N$ discrete segments as shown in
Fig. 1(b,bottom). For transmission lines in which impedance can be set by
cross section geometry (e.g., stripline) the phase velocity may remain
constant for each segment and thus commensurate line (electrically uniform)
tapers are also physically uniform tapers (PUT). In contrast, tapers realized
with transmission lines in which the phase velocity and characteristic
impedance are simultaneously modified through the refractive index of the
medium (e.g., microstrip or quasi-optical systems) should be physically non-
uniform tapers (PNT) to maintain the desired performance characteristics of
the taper. For planar transmission lines such as microstrip, it is simple to
realize the required PNT by adjusting layer thickness [14], but for quasi-
optical matching tapers which prefer a PUT realization, modifying the
permittivity to realize the impedance taper results in non-commensurate lines
(electrically non-uniform) and the overall response deviates from the intended
design [15]. Therefore this work presents a method for maintaining the key
performance parameters of an impedance matching taper realized with non-
commensurate lines.

Figure 1(c) shows detail of two hypothetical adjacent GRIN lens unit-cells
from Fig. 1(a), which provide a match from free-space ($\epsilon_{r}=1$) to a
radial varying core permittivity, here $\epsilon_{rc}=4$ and
$\epsilon_{rc}=3$. Each unit-cell is shown in a PUT realization and a PNT
realization. The comparison shows PNT layer thickness varies with the layer
permittivity and its overall thickness depends on the core permittivity, but
the PUT layer thickness***layer “thickness” and layer “length” will be used
interchangeably throughout. is uniform and the overall thickness is constant
making PUTs convenient for fabrication of GRIN lenses. Figure 1(d) shows the
approximate aperture efficiency (see (8)) of such a lens with and without the
proposed matching sections—the PUT-matched lens, described in Section V-C
below, is simple to fabricate and provides a significant increase in passband
efficiency.

Because of the benefit of PUTs in quasi-optical structures, we seek a design
method for realizing PUTs. In section II we present the theory and propose an
algorithm for their design. In section III we discuss the unavoidable
nonidealities which result from such discretized and non-commensurate tapers
and suggest methods for mitigating the impairments when possible. In section
IV we present the design and measurement of a prototype taper section
comprising nine physically uniform layers for use in GRIN lens antennas
operating across the 8–78 GHz band. Section V presents a complete GRIN lens
design procedure, proposes an approximate efficiency formula which is useful
for guiding design, and provides a detailed comparison of Klopfenstein taper-
matched GRIN lenses with other common matching approaches (e.g., exponential
tapers, quarter wave sections).

##  II Theory

Tapers provide an impedance match over a prescribed operating band which
results in increased transmission efficiency and aperture efficiency. However,
taper designs used in the recent literature [1][16][17] only consider the
impedance profile $Z(z)$ and neglect the fact that impedance and phase
constant are coupled through the permittivity of the medium. As stated above,
if PNTs are used to match GRIN lenses the taper response can be predicted from
well-established commensurate line theory [13]. However, since nearly all
realized GRIN lenses use discretized PUTs (owing to their reduced complexity
of fabrication and design), they exhibit nonuniform electrical length of each
taper layer which causes taper performance $\Gamma(\theta)$ to deviate from
the corresponding ideal taper response. Indeed there is no predictive formula
for the frequency response $\Gamma(\theta)$ of PUTs. The lack of a predictable
response means tapers are designed approximately and therefore not optimally.
This is especially problematic for Klopfenstein tapers which provide a design
tradeoff between low-frequency cutoff and passband match—the tradeoff cannot
be accurately studied without a predictable response.

To develop a method which considers the coupling of impedance and phase
velocity, we need to not only acquire the impedance profile but preserve the
performance of the ideal taper in a PUT. Consider the ideal taper structure
shown in Fig. 1(b,top). TEM electromagnetic waves travel through the ideal
taper with uniform phase velocity $v_{p}$ while the local characteristic
impedance $Z(z)$ varies according to the taper matching profile. Then the
impedance profile $Z(z)$ of the taper (e.g., exponential, Klopfenstein) is
discretized on a uniform grid of $N$ layers resulting in $Z[n]$. Now if each
layer is realized with a uniform electrical length by varying the physical
layer thickness we arrive at a PNT. If each layer is realized with a uniform
physical length (and correspondingly, varying electrical length) we arrive at
a PUT. In order to derive a PUT with a predictable frequency response we
present the following procedure, summarized in Algorithm 1. Note that while
the algorithm is demonstrated with a Klopfenstein taper (because it provides a
prescribed match level in the shortest possible length), other tapers could be
used.

First consider the ideal Klopfenstein taper profile and response: for a given
taper length $L$ and matching range $Z_{01}$ to $Z_{02}$ ($Z_{01}$ $>$
$Z_{02}$), the Klopfenstein profile achieves a tradeoff between maximum
passband ripple, $\Gamma_{max}$, and cutoff frequency, $f_{c}$†††As is
standard for a Klopfenstein taper, the cutoff frequency is defined as the
lowest frequency for which the input reflection coefficient is
$\leq\Gamma_{max}$. It is not the frequency of the first null or the first
sidelobe peak., according to,

$\Gamma_{max}=\frac{\Gamma_{0}}{\cosh A},A=\frac{2\pi
f_{c}}{c_{0}}\epsilon_{eff}L,$ (1)

where $\Gamma_{0}=\left(Z_{01}-Z_{02}\right)/\left(Z_{01}+Z_{02}\right)\approx
0.5\ln\ \left(Z_{01}/Z_{02}\right)$, $c_{0}$ is the speed of light in vacuum,
and $\epsilon_{eff}$ is the homogeneous, uniform permittivity of an ideal
continuous TEM Klopfenstein taper. With $Z_{01}$, $Z_{02}$, $\epsilon_{eff}$,
$\Gamma_{max}$ and $f_{c}$, one can derive the unique impedance profile and
corresponding response of the ideal Klopfenstein taper [13].

Next, to realize a PNT the ideal taper profile is discretized with $N$ layers
of equal electrical length. For large $N$, assigning the impedance of each
layer $Z[n]$ as the corresponding central value from the original continuous
impedance profile $Z(z)$, results in a trivial modification of the ideal
continuous taper response. For quasi-optical realizations with non-magnetic
media the permittivity value of the $n^{\textrm{th}}$ layer is
$\epsilon_{r}[n]={(377/Z[n])}^{2}$ with phase velocity
$v_{p}[n]={c_{0}}/{\sqrt{\epsilon_{r}[n]}}$. To maintain commensurate layers,
the physical thickness $l[n]$ can be found for each layer using
$\epsilon_{r}[n]$, $n\in\left[1,N\right]$. These layers can be thought of as
quarter wavelength transformers at a common central frequency $f_{o}$,

$f_{o}=\frac{c_{0}}{4\sqrt{\epsilon_{r}[n]}l[n]}.$ (2)

We note that the discretized, N-layer Klopfenstein PNT is a Chebyshev
transformer for large $N$.

Finally, to realize a PUT $N$ layers of equal physical thickness $L/N$ are
used instead of $l[n]$. Suppose $Z_{01},Z_{02},L,N,f_{c}$ are known: the
general approach we pursue for realizing a PUT from the ideal taper impedance
profile is to find the permittivity $\epsilon_{eff}$ of the ideal taper which,
when discretized as a PUT results in equal total phase,
$\theta_{PUT}=\theta_{\textrm{ideal}}$, and equal total thickness,
$L_{PUT}=L_{\textrm{ideal}}$. These conditions taken together maintain the
cutoff frequency $f_{c}$ which is considered the key parameter for quasi-
optical components which are ideally thin relative to the lens diameter (and
thus achieving a low $f_{c}$ is challenging). Equality of total phase
requires,

$\sum_{n=1}^{N}{\frac{2\pi
f_{c}}{c_{0}}\sqrt{\epsilon_{r}[n]}l_{N}}=\frac{2\pi
f_{c}}{c_{0}}\sqrt{\epsilon_{eff}}L.$ (3)

Notice that the uniform $\epsilon_{eff}$ of the ideal taper means the phase
velocity is constant along the taper while the non-uniform $\epsilon_{r}[n]$
of the PUT results in a non-uniform phase velocity, $v_{p}[n]$. (3) can also
be expressed in terms of layer thickness $l_{N}$ as:

$l_{N}=\frac{\sqrt{\epsilon_{eff}}L}{\sum_{n=1}^{N}\sqrt{\epsilon_{r}[n]}},$
(4)

which is a constant $l_{N}=L/N$ for the effective permittivity which satisfies
$\sqrt{\epsilon_{eff}}=\frac{1}{N}\sum_{n=1}^{N}\sqrt{\epsilon_{r}[n]}$. This
condition results in the electrical length and physical length of the PUT
being equal to that of the ideal taper. Therefore the transform starts from
phase equality and, with the proper $\epsilon_{eff}$, ends up with uniform
layer thickness equality.

Algorithm 1 Non-commensurate Taper Design

0: $Z_{01}$, $Z_{02}$, $L$, $N$, $f_{c}$44footnotemark: 4, $TOL$

0: $Z[n],\epsilon_{r}[n]$, $\epsilon_{eff}$

1: calculate $\epsilon_{r01}$ and $\epsilon_{r02}$

2: set $\epsilon_{eff,L}=\epsilon_{r01}$, $\epsilon_{eff,R}=\epsilon_{r02}$,
$\epsilon_{eff,M}=\frac{\epsilon_{eff,L}+\epsilon_{eff,R}}{2}$, $e>TOL$

3: while $e>TOL$ do

4: $A_{L/R/M}$ using (1)44footnotemark: 4

5: $Z(z)_{L/R/M}$ using [13]

6: $Z[n]_{L/R/M}$ for PUTs

7: $\epsilon_{r}[n]_{L/R/M}$ for PUTs

8: $l_{N,L/R/M}$ using (4)

9: errors, $e_{L/R/M}=\frac{L}{N}-l_{N,L/R/M}$

10: if $e_{L}$ and $e_{R}$ have opposite signs then

11: $\epsilon_{eff,R}=\epsilon_{eff,M}$, exact solution is in this interval

12: else

13: $\epsilon_{eff,L}=\epsilon_{eff,M}$, exact solution is in this interval

14: end if

15: $\epsilon_{eff,M}=\frac{\epsilon_{eff,L}+\epsilon_{eff,R}}{2}$,

16: $e$ = min{$e_{L/R/M}$},

17: choose $\epsilon_{eff}$, $\epsilon_{r}[n]$ and $Z[n]$ with the smallest
$e$

18: end while

19: return $Z[n],\epsilon_{r}[n]$, $\epsilon_{eff}$ ††footnotetext: $\S$These
are only for Klopfenstein design.

However, (4) is a transcendental function (because $\epsilon_{r}[n]$ must be
derived for the Klopfenstein profile) which cannot be directly solved to find
$\epsilon_{eff}$. Therefore Algorithm 1 presents an iterative bisection method
for the design of PUTs, which also facilitates the design of other types of
tapers. The inputs are the desired port impedances, $Z_{01}$ and $Z_{02}$ ,
taper length $L$, number of layers $N$, and taper cutoff frequency $f_{c}$.
Since the bisection method is iterative, an error tolerance, $TOL$, is set for
convergence and is specific to different taper parameters. The proper
${\epsilon_{eff}}$ and corresponding $\epsilon_{r}[n]$ are found through
iteration—to realize bisection three ideal tapers are calculated in each
iteration: ${\epsilon_{eff,L}}={\epsilon_{r01}}$ is the left (or lower)
boundary, ${\epsilon_{eff,R}={\epsilon_{r02}}}$ is the right (or upper)
boundary, and ${\epsilon_{eff,M}=\frac{\epsilon_{eff,L}+\epsilon_{eff,R}}{2}}$
is the mean. Then three PUTs are designed with three layer thicknesses,
$l_{N,L/R/M}$. A suitable solution is reached when the error between
$l_{N,L/R/M}$ and $\frac{L}{N}$ is less than $TOL$ so in each iteration of the
algorithm a length error is calculated for the three cases as
$e_{L/R/M}=\frac{L}{N}-l_{N,L/R/M}$. Then new boundary permittivity values are
selected for the next loop and the PUT permittivity associated with the
current loop is assigned that profile with the smallest error. For the
bisection method, errors $e_{L/R/M}$ with different signs enclose the exact
solution so the new range is selected to include the exact solution. Precise
calculations are presented in Algorithm 1. Note that except $f_{c}$ and Step 4
(unique to Klopfenstein), the discretization and transformation methods are
generally applicable for any taper if the corresponding impedance profile is
chosen in Step 5. Once $\epsilon_{eff}$ is found to equalize electrical and
physical length, the frequency response of the ideal Klopfenstein taper for a
non-commensurate line can be found as:

$\left|\Gamma(f)\right|=\left|\Gamma_{0}\exp(-j\beta_{eff}L)\frac{\cos\sqrt{(\beta_{eff}L)^{2}-A^{2}}}{\cosh{A}}\right|\\\
=\left|\Gamma_{0}\exp(-j\frac{2\pi
f\sqrt{\epsilon_{eff}}}{c_{0}}L)\frac{\cos\sqrt{(\frac{2\pi
f\sqrt{\epsilon_{eff}}}{c_{0}}L)^{2}-A^{2}}}{\cosh{A}}\right|.$ (5)

$\beta_{eff}$ is the effective propagation coefficient and depends upon
$\epsilon_{eff}$. This expression is similar to the classical expression for
the frequency response of a Klopfenstein taper except that it has been written
in terms of the proper $\epsilon_{eff}$ and can now predict the correct cutoff
frequency and passband performance.

In the following sections three different Klopfenstein tapers will be
discussed so we wish to clarify the purpose behind the three different tapers:
i) The first taper, discussed in Section III, is a nine layer taper matching
$Z_{01}=308\Omega$ to $Z_{02}=184\Omega$—this taper was chosen because it
clearly shows the impairments we wish to discuss. ii) The second taper is the
fabricated taper, discussed in Section IV, and also comprises nine layers but
matches a more extreme range from $Z_{01}=308\Omega$ to
$Z_{02}=140\Omega$—this is representative of a maximum core permittivity in
our typical fabricated lenses. iii) The third taper is a six layer taper used
in the simulated Klopfenstein-matched demonstration lenses in Section V—this
taper only uses six layers in order to reduce the complexity (and also cost)
of the matched GRIN lenses to make them as realistic as possible. In practice
we try to minimize the number of layers in a matching section because cost is
dominated by the layers in the taper (the core can be fabricated from a small
number of thick substrates).

##  III Nonidealities of PUTs

As mentioned, the case study PUT uses nine layers to provide an impedance
match from the realizable permittivity values of $\epsilon_{r}$ = $1.5$
($Z_{01}$ = $308\Omega$) to a core permittivity of $\epsilon_{rc}$ = $4.2$
($Z_{02}$ = $184\Omega$), which is instructive to show the taper properties as
stated above. The schematic is shown in the inset of Fig. 2(a) where each of
the nine substrate layers is $l_{N}=0.030"$ (total thickness is $0.27"$) and
$f_{c}$ = $11$ GHz. As mentioned above, the idealized impedance profile was
discretized by selecting the central impedance of $N$ = $9$ sections from the
ideal taper. Algorithm 1 produced the following outputs: $\epsilon_{eff}$ =
$2.56$, $Z[n]$ = 290, 279, 266, 252, 238, 224, 212, 202, 195 $(\Omega)$,
$\epsilon_{r}[n]$ = 1.68, 1.82, 2.00, 2.23, 2.51, 2.82, 3.15, 3.47, 3.75, and
$\Gamma_{max}$ = $0.04$. These results satisfy the electrical and physical
length equality condition in (4). At $f_{0}$ the Klopfenstein taper achieves a
perfect impedance match where $\widetilde{Z_{in1}}$ = $Z_{01}$ (see inset of
Fig. 2(a)) and $\widetilde{Z_{in2}}$ = $Z_{02}$. Figure 2(a) shows the
response of the ideal taper (calculated with (5) and using $\epsilon_{eff}$),
the PNT (using (2)) and the PUT (calculated with wave-transfer matrices [18]).
There are a number of nonidealities present in both the PNT and PUT which are
indicated with letters “A” to “D” in Fig. 2(a) and (c). The remainder of this
section discusses each nonideality in detail and provides suggestions for
mitigating each, if possible.

Figure 2: Frequency response of the proposed taper. (a) Ideal, PNT, and PUT
with $Z_{01}$=308$\Omega$ and $Z_{02}$=184$\Omega$. (b) First zero shift of
PUT caused by different relative permittivity matching ranges. (c) Comparison
among the original PUT with $Z_{01}$=308$\Omega$,
$\widetilde{Z_{in1}}$=308$\Omega$ and $Z[1]$=289$\Omega$; method (i) with
$Z_{01}$=377$\Omega$, $\widetilde{Z_{in1}}$=308$\Omega$ at $f_{0}$ and
$Z[1]$=289$\Omega$; and method (ii) with $Z_{01}$=377$\Omega$,
$\widetilde{Z_{in1}}$=377$\Omega$ at $f_{0}$, $Z[1]$=289$\Omega$.

“A”: The continuous Klopfenstein taper is a high-pass structure while the
discretized PNT is approximately a Chebyshev transformer with a finite
passband resulting from discretization with finite-thickness layers. From (2),
each layer of the PNT is a quarter-wavelength transformer at
$f_{0}=61.6\,$GHz. Therefore, at $2f_{0}$ each layer will be a half-wavelength
long and the taper comprising $N$ layers will be $N\lambda/2$ in length. The
taper then mirrors the impedance of the second port at the first resulting in
$\Gamma$ at “A” being equal to $\Gamma_{0}$.

“B” and “C”: These impairments are similar to “A” but for a non-commensurate
taper (the PUT) in which multiple reflections add incoherently over a range of
frequencies around $2f_{0}$. For a PNT the center frequency $f_{0}$ at which
each layer is a quarter-wavelength long (from (2)) is 61.6 GHz while for a PUT
$f_{0}$ and $\theta$ (at 62.1 GHz) of each layer varies as shown in Table I.
For the fixed layer thickness of the PUT and defining the center-most layer
(with $\epsilon_{r}=2.51$) to be a quarter wavelength at $62.1\,$GHz, the
corresponding $\theta$ of the outermost layers ranges from $110^{\circ}$ to
$74^{\circ}$ (in other words, they are non-commensurate). Layers in the center
of the PUT are approximately 90∘, so that the central part of the passband
response is consistent with the PNT. Toward the edges layers deviate more
significantly from 90∘ resulting in an increase in $\Gamma$ and distortion at
both the low frequency band edge (“B”) and the high frequency band edge (“C”).
The distortion at “C” was also observed in [19, 17]. At “C” there is a
smearing of the band edge compared to “A” because there is a range of
frequencies at which various layers are a half-wavelength. This smeared high-
frequency cutoff approximately spans from $2\times 50.8$ GHz ($101.6$ GHz) to
$2\times 75.9$ GHz ($151.8$ GHz). The result of these incoherent reflections
is a lower reflection (compared to $\Gamma_{0}$ at “A”) but a significant
reduction in the high frequency band edge which must be considered in a
design. The impedance transformation ratio influences the central frequencies
of each layer, thus influencing “B” and “C”. A smaller impedance
transformation ratio minimize this effect (see zoom of Fig. 2(b)).

TABLE I: Layer profile of a PUT†

Layer | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---
$f_{0}$ (GHz) | 50.8 | 52.9 | 53.7 | 56.7 | 62.1 | 65.9 | 69.6 | 73.0 | 75.9
$\theta^{\ddagger}(\deg)$ | 110.0 | 105.8 | 104.2 | 98.6 | 90.0 | 84.9 | 80.3 | 76.6 | 73.6

††footnotetext: $\dagger$For a PNT, each layer has an identical $f_{0}=61.6$
GHz and $\theta=90^{\circ}$.††footnotetext: $\ddagger$Phase at central
frequency $f_{0}=62.1$ GHz.

“D”: As mentioned above, a perfect impedance match implies
$\widetilde{Z_{in1}}=Z_{01}$ at $f_{0}$. However, since most artificial
dielectrics have a lower permittivity limit (suppose $\epsilon_{r,min}=1.7$),
there is an upper limit on the realizable impedance within the taper
(correspondingly, $Z_{max}=289\Omega$). And yet lenses typically interface
with waves in free-space, $Z_{01}=377\Omega$. There are two approaches to
mitigating this issue, shown in Fig. 2(c): Method (i) is to design the match
to the realizable $Z_{max}$ and then accrue an additional mismatch at the
outer surface relative to free-space causing $Z[1]=Z_{max}$ and
$Z[1]<{\widetilde{Z_{in1}}}<Z_{01}$, or specifically for a Klopfenstein taper
method (ii) is to use (1) to design a taper with increased $\Gamma_{max}$ such
that the edge discontinuity expects a mismatch to free-space where
$Z[1]=Z_{max}$ and $Z[1]<{\widetilde{Z_{in1}}}=Z_{01}$. While method (ii)
seems compelling because it advantageously uses the impedance discontinuity at
the edges of a Klopfenstein taper, Fig. 2(c) shows that it is much worse than
method (i). The increase of $\Gamma$ by mismatch is labeled as “D” and
estimated as
$\Delta\Gamma=\left(Z_{01}-\widetilde{Z_{in1}}\right)/\left(Z_{01}+\widetilde{Z_{in1}}\right)$
in Fig. 2(c).

##  IV Taper Prototype

Figure 3: (a) Fully assembled 4-inch taper. (b) Zoom of each layer of
perforated dielectric. (c) Measured and simulated effective permittivity from
free-space characterization.

To demonstrate the proposed method we designed and fabricated a prototype
Klopfenstein PUT with $N=9$ layers, each of $l_{N}=0.030"$ (total thickness
0.27”). This taper design differs from the case study considered in the
previous section—the case study was designed to clearly demonstrate the
various impairments (and thus it matched to a moderate core permittivity of
4.2) while this prototype matches to a high core relative permittivity of 7.2
in order to validate the performance of one of the most extreme unit-cells in
a hypothetical GRIN lens. Referring to Fig. 1(a), the unit-cell corresponds to
the black dashed boxes and the prototype taper is the impedance matching
layers along the $z-$axis from free-space at the bottom of the GRIN lens to
$\epsilon_{rc}=7.2$ ($Z_{02}=140\Omega$) in the core of the GRIN lens. Since
the core permittivity of a GRIN lens varies radially the required taper across
the bottom (and top) of the lens must also vary radially. Here we only
fabricate a single instance of a 0.27” taper with permittivity variation along
the $z-$axis and uniform permittivity in the cross section ($xy-$plane). The
cross section diameter is 4” to ensure the majority of the energy from our
free-space measurement setup (described below) intercepts a uniform taper
cross section.

Figure 4: Taper response of original and fabricated tapers with -15 dB match
bandwidth indicated from $8$–$78$ GHz.

The original taper was designed using Algorithm 1 to match with
$Z_{01}=308\Omega$ and $Z_{02}=140\Omega$ (method (i) in Fig. 2(c)) with an
$f_{c}=11$ GHz. The resulting permittivity profile is
$\epsilon_{r}[n]=1.70,1.92,2.24,2.69,3.29,4.01,4.82,5.63,6.34$ and
$\epsilon_{ri}=3.46$ (these results also satisfy (4)). The realized taper is
shown in Fig. 3(a) where all nine layers have been stacked and bonded.
Rectangular samples of each layer are shown in Fig. 3(b). The effective
permittivity of each layer was realized with a perforated dielectric as
described in [1, 20] and results in a slight modification to the original
permittivity values. To span the modified permittivity range 6.43–1.74, holes
were drilled on a hexagonal lattice in four substrates (Rogers AD1000, AD600,
AD350, and AD250) with nominal permittivity values of 10.35, 6.15, 3.5 and
2.5, respectively. Drill diameters were 10.5, 11.8 mil in AD1000 (18 mil
lattice constant), 10.5, 13.8 mil in AD600 (23 mil lattice constant), 8.3, 16
mil in AD350 (32 mil lattice constant), and 13.8, 21.7, and 25.6 mil in AD250
(37 mil lattice constant). The frequency response of the originally designed
permittivity profile and the modified profile considering fabrication
(“Original” and “Fabricated”) are shown in Fig. 4 with very little deviation.
The passband of a 15 dB return loss, is 8-78 GHz.

The 4” diameter disk of each separate layer was measured in a free-space
material characterization setup, shown in Fig. 5. Since the tapers are
intended for use in quasi-optical structures and lens antennas it is important
that the measurement setup closely mimic the intended use case with nearly TEM
incident radiation. This prevented us from measuring the tapers in a
rectangular waveguide as was done in [1, 20]. Instead, the free-space setup
employs spot-focusing antennas in the WR90, WR28 and WR12 waveguide bands to
produce approximately collimated TEM waves within a confined area.
Specifically, 83% of the energy from the spot antennas is contained within a
2.51”, 1.2” and 0.28” diameter, respectively. To validate the TEM assumption
in the measurement, transmission efficiency versus incident angle of the
fabricated taper was simulated for both TE- and TM-modes. At normal incidence
transmission efficiency is 98.7% for both modes, and 99.8%(95.3%) for TM(TE)
mode at 45∘ (note that the TM-mode is approaching Brewster’s angle and thus
transmission increases off-broadside). Since spot-focusing antennas are
specifically designed to achieve a nearly collimated wavefront at their focal
distance $F_{spot}$, the wave will have close to normal incidence at the
samples. However, to be conservative we calculate the largest possible
incident angle from a spot-focusing antenna of diameter $D_{spot}$ as
$\arctan(D_{spot}/2F_{spot})$. For the X, Ka and W-band antennas used in the
measurements the angle is 30∘, 21∘ and 30∘, respectively with transmission
efficiency greater than 99.1%(97.7%) for TM(TE) mode. Therefore the TEM
assumption is valid in the free space measurement setup.

Each layer of the taper was individually mounted to one side of the low
dielectric mounting foam shown in Fig. 5. A 3” slot was removed from the
middle of the foam to further decrease the influence of the foam. A two tier
calibration was used: first an SOLT calibration was used to deembed systematic
errors in the VNA and the coaxial cable, then the Gate-Reflect-Line (GRL)
calibration method [21] was used to extend the phase reference plane from the
coaxial cable to the surface of the taper and decrease multipath reflections.
The measured permittivity values of each layer, extracted at 10.3 GHz in the
WR90 band and 33.3 GHz in the WR28 band, are shown in Fig. 3(c) along with the
simulated permittivity profile and they show very close agreement in both
waveguide bands. This measurement validates that each layer was fabricated
correctly and that the measurement setup and calibration method were
effective.

Figure 5: Measurement Setup in the WR90 band: (a) Schematic of free-space
measurement setup. (b) Side-view without DUT. (c) Top-view with DUT. (d)
Front-view with DUT.

After confirming accurate fabrication of each layer the complete taper was
bonded with a total thickness of 284 mil (12 mil thicker than nominal). The
increase in thickness causes a decrease in the cutoff frequency and each null
of the passband response. Fig. 6(a) shows the simulated response of the taper
using the predicted permittivity values (“Fabricated”, same as blue curve in
Fig. 4) and a fitted taper (“Fitted”) which uses the average measured
permittivity values reported in Fig. 3(c) and slightly thicker PCB layers to
account for manufacturing tolerances (discussed in detail below).

Figure 6: (a) Comparison between the fabricated taper and fitted taper.(b)
Measured taper response and renormalized fitted response from free-space spot-
antenna measurements in the WR90, WR28 and WR12 waveguide bands.

The prototype taper was measured with the same free-space measurement setup as
described above. The simulated taper match bandwidth spans at least seven
waveguide bands (from WR137 to WR12). Due to setup limitations the match
response was measured in three bands: near taper cutoff, $f_{c}$, in the WR90
band (8.2–12.4 GHz), in the taper pass-band in the WR28 band (26.5–40 GHz) and
at the taper high-frequency cutoff in the WR12 band (76–86 GHz, limited by the
spot antenna). The measured results are shown in Fig. 6(b). Since the second
port impedance was designed to be $Z_{02}=140\Omega$, but was measured in
free-space ($Z_{02}=377\Omega$), we renormalized the simulated taper response
[22] to confirm the measurement. In order to achieve good agreement between
the measured prototype and simulation (“Fabricated”), Monte-Carlo simulations
with layer thicknesses ranging over the expected tolerances were conducted and
the fitted response in Fig. 6(b) corresponds to a taper with the following
substrate thicknesses: 31.6 mil for AD1000, 30.8 mil for AD600, 31.6 mil for
AD350, and 31.4 mil for AD250. These thicknesses increase the overall
thickness of the taper by 10.2 mil, close to the measured thickness increase
of 12 mil. Fig. 6(a) shows the nominal design (“Fabricated”) and the fitted
design with good agreement, confirming both the design and the fabrication.
The results of this measurement confirm the validity of the design method
presented in Algorithm 1.

##  V Demonstration Lens Design & Simulation

In this section we demonstrate the effectiveness of the proposed design method
with simulations of several complete GRIN lens designs. First the GRIN lens
modeling and simulation setup are introduced, then we present an approximate
expression for the efficiency of a GRIN lens which serves as a design guide.
Finally, several Klopfenstein taper GRIN lenses are designed and their
performance is compared with an unmatched lens, as well as GRIN lenses matched
with quarter-wave sections and exponential tapers.

###  V-A Lens Design and Simulation Setup

All of the lenses designed in this section follow the procedure outlined in
[1] and summarized as follows: a GRIN lens comprises 250 concentric rings
arranged radially from the center of the lens across a diameter of 8”
($2.7\lambda_{0}$ at 4 GHz). Each ring is a unit-cell comprising a matching
layer on the bottom and top of a uniform core permittivity and has an overall
thickness of 1.2”. The unit-cells are arranged such that their respective
absolute phase delays equalize the optical path length from a central feed
element (at a focal distance of 4” from the bottom surface of the lens) to the
top surface of the lens to produce a collimated wave and far-field beam at
broadside. The unit-cells are selected from a library of unit-cells which
include a six layer matching taper on the top and bottom of a uniform core
with permittivity values ranging from $\epsilon_{rmin}$ to $\epsilon_{rmax}$
(corresponding to e.g., the perforated dielectrics as described in [1]). The
focal length and diameter of the GRIN lens ($F/D=0.5$) determine the amount of
phase which must be collimated. The thickness of a unit-cell is determined by
the range of available permittivities and the amount of phase to be
collimated. In order to demonstrate the influence of the taper design on GRIN
lens performance, several lenses with different matching sections are
presented in the next two subsections. We chose to use six 30-mil layers in
the tapers (both for Klopfenstein and Exponential taper matched lenses)
because it clearly reveals the taper properties and reduces overall complexity
of the lens. In order to maintain clear and predictable trends in the
simulated lens responses $\epsilon_{rmin}=1$. In practice a higher minimum
permittivity would result in slightly higher reflections.

The designed lenses were simulated with high fidelity over a very wide
frequency range using an in-house two-dimensional (2D) finite-difference time
domain (FDTD) code with an idealized feed. Two-dimensional simulations were
used instead of three-dimensional in order to manage the computational
complexity of the GRIN lenses over such large frequency spans. It is supposed
that in the simulation the $z$-plane (normal to the 2D simulation plane) is
infinite and there is no $z$ variation in the fields—this 2D assumption does
not sacrifice the generality of the conclusions drawn in this section since
most lenses are designed with azimuthal symmetry. The simulation frequency is
from 4 GHz to 60 GHz and the grid was chosen to be $\lambda_{0}/50$ at 60 GHz
($\Delta X=0.1$ mm) in order to provide for very accurate aperture fields.
This was necessary to be able to differentiate between several lenses which
all have extremely high efficiency. The TE-mode was simulated to obtain the
worst-case transmission result (the TM-mode has a transmission peak at
Brewster’s angle while the TE-mode does not). An ideal 2D isotropic point
source excitation was used so that the illumination efficiency remained
constant over frequency and all variation in efficiency over frequency could
be attributed to the lens and matching sections. The far-field radiation
pattern was computed from the aperture fields along a line of extent $D=8$”
and compared to the gain from an ideal uniform line source, $2D/\lambda$. For
a fair comparison of different matching approaches, the efficiency of each
lens should be dominated by matching performance and not, e.g., poor phase
collimation. Therefore, each lens was designed with an iterative method
described in more detail in Section V-B to achieve nearly perfect phase
collimation (relative to a wavelength at 60 GHz).

###  V-B Performance Estimation and Full-wave Simulation

Figure 7: Iterative design of a Klopfenstein taper GRIN lens. Taper are six
layers per side with $f_{c}=11$ GHz. Blue and red traces are
$\eta_{ap}^{\prime}$ (see (7)) and $\widetilde{\eta_{ap}}$ (see (8)).

The performance of the designed lenses can be evaluated with a full-wave
simulation code (e.g., the 2D-FDTD code described above or a 3D code). But,
even an efficient 2D code can take hours to complete due to the complexity of
GRIN lenses (our 2D-FDTD code with the fine simulation grid mentioned requires
approximately 4 hours to solve from 4 GHz to 60 GHz) and, due to
approximations made in the design algorithms, the result will be a GRIN lens
with imperfect phase collimation. The challenge is that since GRIN lenses
provide so many degrees of freedom, analytical design expressions are
typically derived from simplified straight-line ray tracing [11, 23] or curved
ray tracing without matching layers [24]. Due to the approximations in these
methods phase collimation is imperfect and therefore gain suffers over very
wide operating bandwidths. In these cases iterative or optimization methods
are used to modify the initial lens design. Suppose the GRIN lens materials
are non-dispersive: then if the phase distribution is derived and modified
(through iteration) at the highest frequency of interest, the lower frequency
band will also exhibit good phase collimation. Thus a general way to modify
the wideband design is to record the phase distribution error at the design
frequency in the current iteration and then modify the phase correction by the
current error until the efficiency converges.

Figure 7 shows lens efficiency (defined in (7) and (8), below) for the first,
second, and final (6th) iteration of the design method in [1]. The blue traces
show $\eta_{ap}^{\prime}$ (see (7), below) improving over a wide bandwidth
because of increasingly better phase collimation throughout the iteration.
However, it is time consuming to evaluate different taper designs in order to
identify one that meets the design specification. For this reason we seek an
approximate expression for aperture efficiency which does not require a full-
wave simulation and can serve as a target for iterative designs. Specifically,
based on the initial lens profile (first iteration), a formula for approximate
aperture efficiency, $\widetilde{\eta_{ap}}$ (red traces in Fig. 7) can be
derived which then serves as a target for iteration—convergence is reached
when the simulated lens is close enough to the target. The approximate
aperture efficiency also guides the initial specification of the unit-cell
(e.g., total thickness, number of discrete layers in the tapers, required
minimum and maximum permittivity) to achieve a desired passband efficiency and
low-frequency cutoff.

Figure 8: (a) Transmission efficiency $\eta_{T}$ and normalized aperture
efficiency $\eta_{ap}^{\prime}$ versus frequency of a Klopfenstein taper
matched lens and an unmatched lens. The inset table shows the marked
$\eta_{ap}^{\prime}$ values. (b) Gain patterns of the Klopfenstein matched
lens (solid) versus the unmatched lens (dotted), which are normalized to
$\eta_{ap}^{\prime}=1$ at each frequency. The inset figure showsshows a zoom
view of the pattern peaks where the vertical axis of the inset is Normalized
gain in dB. (c) Gain maxima $G_{o}$ and sidelobe level of the Klopfenstein-
matched lens versus the unmatched lens.

In order to derive the approximate formula, we begin with a general expression
for aperture efficiency as in [1]:

$\eta_{ap}=e_{r}\eta_{a}\eta_{s}\eta_{t}\eta_{T},$ (6)

where $e_{r}$ is radiation efficiency and $\eta_{a}$ is achievement
efficiency, both of which are approximately 1. $\eta_{s}$ is spillover
efficiency, which is $\eta_{s}$=0.25 for all lens designs and frequencies
considered in this work because the feed is an ideal 2D-isotropic line source
with an $F/D=0.5$. $\eta_{t}$ is taper efficiency [25]. $\eta_{T}$ is
transmission efficiency determined by the lens impedance matching sections. We
note that this is an effective lens transmission efficiency since the
permittivity profile varies radially and thus each unit-cell has a unique
transmission efficiency which combines to yield an overall lens transmission
efficiency. With these assumptions we normalized the aperture efficiency, (6),
as:

$\eta_{ap}^{\prime}=\frac{\eta_{ap}}{\eta_{s}}=\eta_{T}\eta_{t}.$ (7)

Compared to $\eta_{ap}$, $\eta_{ap}^{\prime}$ has divided out spillover
efficiency ($\eta_{s}$), thus $\eta_{ap}^{\prime}$ considers only the lens
performance and not the feed. The use of $\eta_{ap}^{\prime}$ also provides a
more complete description of the frequency dependence of the matching
efficiency compared to transmission efficiency $\eta_{T}$. While it may seem
natural to characterize the matching performance of a lens antenna with
$\eta_{T}$, in fact the matching has a small but noticeable affect on the
amplitude distribution (e.g., a slightly higher reflection in the middle of
the lens relative to the edge of the lens would result in a flattening of the
amplitude distribution) and phase collimation of the lens. These effects are
captured in the taper efficiency and included in $\eta_{ap}^{\prime}$.
Therefore we prefer to use $\eta_{ap}^{\prime}$ to characterize the matching
performance of a lens.

There is another advantage to considering $\eta_{ap}^{\prime}$ to characterize
a GRIN lens—an approximate design expression $\widetilde{\eta_{ap}}$, can be
derived. Suppose the phase collimation has been optimized through iteration
such that taper efficiency is dominated by the amplitude distribution. Then
the lens performance $\eta_{ap}^{\prime}$ can be estimated by an effective
$\widetilde{\eta_{ap}}$ which is a spatially averaged efficiency across the
entire lens:

$\widetilde{\eta_{ap}}\approx\frac{1}{N}\sum_{n=1}^{N}(1-|\Gamma_{n}(f,\theta_{inc,n})|^{2})^{2}\cdot\frac{1}{N}\sum_{n=1}^{N}\cos{\theta_{inc,n}},$
(8)

where $N$ is the number of rings in the GRIN lens (i.e., the number of unit-
cells in the 2D cross section, here $N=250$). $\theta_{inc,n}$ is the incident
angle from the point feed to the $n^{\textrm{th}}$ unit-cell. The first
summation is the square of the effective transmission coefficient averaged
over all $N$ unit cells, $\bar{\eta_{T}}$. This quantity is squared to account
for transmission through two tapers (bottom and top of lens). The second
summation is an average taper efficiency, $\bar{\eta_{t}}$, where phase is
assumed to be perfectly collimated (and thus dominated by amplitude taper).
$\Gamma_{n}(f)$ from (5) can be written in terms of design parameters and
$\theta_{inc}$:

$\begin{split}\left|\Gamma(f,\theta_{inc})\right|=&\Biggl{|}\Gamma_{0}\exp\left(-j\frac{2\pi
f\sqrt{\epsilon_{eff(\theta_{inc})}}}{c_{0}}L\right)\cdot\Biggr{.}\\\
&\Biggl{.}\frac{\cos\sqrt{\left(\frac{2\pi
f\sqrt{\epsilon_{eff(\theta_{inc})}}}{c_{0}}L\right)^{2}-A^{2}}}{\cosh{A}}\Biggr{|},\end{split}$
(9)

where $\epsilon_{eff}$ is a function of $\theta_{inc}$ and
$\theta_{inc}=\arctan(r/F)$ is the incident angle of the unit-cell at radius
$r$. Once $\theta_{inc}$ is determined, $\epsilon_{core}$ can be determined by
the lens design algorithm in [1] and $\epsilon_{eff}$ of that unit-cell can be
derived by Algorithm 1‡‡‡Formula (9) uses the quasi-TEM assumption, which is
valid within 45∘. Therefore, (8) can be rewritten as:

$\widetilde{\eta_{ap}}\approx\bar{\eta_{T}}\bar{\eta_{t}}\approx\eta_{T}\eta_{t}=\eta_{ap}^{\prime}.$
(10)

Referring back to Fig. 7, $\widetilde{\eta_{ap}}$ (red traces) is largely
invariant between iterative refining of the core permittivity profile but it
is sensitive to the matching section specification. This indicates that
$\widetilde{\eta_{ap}}$ is not dominated by the GRIN lens core profile so long
as the lens design algorithm achieves approximate collimation in each
iteration. Using $\widetilde{\eta_{ap}}$ as a target, iteration is performed
to improve phase collimation and increase $\eta_{ap}^{\prime}$ until its
minima match $\widetilde{\eta_{ap}}$ in the final result. Therefore
$\widetilde{\eta_{ap}}$ can be used as a design reference and provides a quick
performance estimation.

###  V-C Performance Comparison

Implementing the design and simulation methods above, a Klopfenstein taper
matched lens (designed in Fig. 7) and an unmatched lens are simulated to
investigate the influence of taper operation. Figure 8 shows the performance
comparison between these two lenses. Based on the discussion of
$\eta_{ap}^{\prime}$ above, Fig. 8(a) shows the transmission efficiency
$\eta_{T}$ and normalized aperture efficiency $\eta_{ap}^{\prime}$ to be
nearly the same which indicates the taper efficiency is almost unity
(specifically in the passband $\eta_{t}$ is between $0.96$ and $0.98$). We
also note that the shape of the curves is almost identical further confirming
that $\eta_{ap}^{\prime}$ is a good metric for comparing matching efficiency.

Referring again to Fig. 8(a), below the designed low-frequency cutoff of 11
GHz, $\eta_{ap}^{\prime}$ of both the Klopfenstein matched and the unmatched
lenses oscillates rapidly due to multiple reflections. Above the taper cutoff
frequency, however, $\eta_{ap}^{\prime}$ of the matched lens improves
significantly with much smaller oscillation due to the high transmission
efficiency of the tapers. In order to further compare gain patterns of the two
lenses five frequencies, 4.8 and 9.0 GHz (below cutoff) and 16.0, 34.8, and
50.2 GHz (within the passband) are indicated on the $\eta_{ap}^{\prime}$ trace
with colored markers and the associated $\eta_{ap}^{\prime}$ values are
included in the inset table. The corresponding gain patterns are shown in Fig.
8(b) and the inset shows a zoom view of the pattern peaks. The solid traces
are the Klopfenstein-matched lens and dotted traces are the unmatched lens.
Gain patterns are each normalized to the peak gain of an aperture with
$\eta_{ap}^{\prime}=1$ at each frequency to make it simple to observe the
reduction in aperture efficiency below cutoff. Below the cutoff frequency, we
select frequencies where $\eta_{ap}^{\prime}$ of both lenses are nearly the
same and as a result the main lobes of two lenses are nearly identical.
However, in the passband (above the cutoff frequency), gain maxima $G_{o}$ of
the matched lens is around 1 dB higher and the sidelobe levels (SLL) of the
matched lens are better than those of the unmatched lenses.

Further comparison of $G_{o}$ and SLLs, defined as the ratio of the main beam
to the first sidelobe, are shown in Fig. 8(c). Below the cutoff frequency the
Klopfenstein-matched lens performs as an unmatched lens with nearly the same
$G_{o}$. However, above the cutoff frequency the matched lens maintains a
higher $G_{o}$ with around 1 dB difference compared to that of the unmatched
lens. As for SLL, below cutoff both lenses exhibit oscillatory SLL behavior
versus frequency while in the passband the Klopfenstein-matched lens has
improved SLL. The SLL of the unmatched lens oscillates to as low as 13.5 dB
across the entire band while the matched lens maintains a SLL above 15.5 dB
across the passband. We attribute the degradation in SLL for the unmatched
lens to a flattening of the aperture amplitude distribution due to more
pronounced reflections in the center of the lens. In general flat GRIN lenses
of this type do not significantly alter the feed element amplitude taper as
the field propagates through the lens (in part because as rays become more
collimated the power density is maintained (rays do not diverge)). However,
since the permittivity is highest in the center of the lens, the unmatched
lens will have the largest reflection magnitude there and thus the higher
fields in the center (due to the feed distribution) will be reduced due to
reflections while the fields at the edges (where the permittivity is lowest)
will have very little reflection. The result is a flattening of the amplitude
distribution which results in a rise of the sidelobes (and corresponding drop
in SLL as we have defined it here). Note that due to multiple reflections
there are spot frequencies where $\eta_{ap}^{\prime}$ is high and the SLL
improves (large reflections in the middle of the lens are canceled due to
advantageous multiple reflections) but in general, in order to maintain a
consistent SLL versus frequency lens matching is necessary.

We note that the 2 GHz periodic ripple in the Fig. 8(a) and (c) is caused by
the resonance between lens lower and upper surfaces. Finally, note that
$G_{o}$ is below 0 dB at 4 GHz and yet the inset pattern shows directionality.
This is because the actual aperture efficiency is
$\eta_{ap}\approx\eta_{s}\eta_{ap}^{\prime}=0.25\times 0.65=-7.9\,$dB
(accounting for the 25% spillover efficiency of the 2D isotropic source used
in the simulation).

Figure 9: (a) The comparison of $\eta_{ap}^{\prime}$ for lenses with different
types of tapers: Klopfenstein tapers (6 layers per side) with 4 different
cutoff frequencies, Exponential tapers (6 layers), Quarter-wave matching
sections (2 layers) and unmatched lens (0 layer). (b)The comparison of
$\eta_{ap}^{\prime}$ (dotted traces) and $\widetilde{\eta_{ap}}$ (solid
traces) of all lenses plotted in (a).

To further investigate the influence of taper types on overall lens
performance Fig. 9(a) shows $\eta_{ap}^{\prime}$ of GRIN lenses with
Klopfenstein taper matching sections (labelled “Klop.11G” through “Klop.17G”),
exponential taper matching sections (as in [1]) (labelled “Exp.”), quarter-
wave matching sections (as in [11, 26]) (labelled “Quart.”), and unmatched
lenses (as in [4, 27]) (labelled “Unmatched”). Klopfenstein tapers with cutoff
frequencies of 11, 13, 15, and 17 GHz are included to show the tradeoff
between cutoff frequency and passband efficiency. For the Klopfenstein tapers,
based on formula (1), the larger the cutoff frequency, the smaller the
passband ripples (and vice-versa). The traces show that the efficiency of an
entire lens which uses many different Klopfenstein tapers (a unique one for
each ring of the design) exhibits the expected tradeoff and has the same
cutoff frequency as a single taper in isolation.

Because of the flexibility of Klopfenstein tapers, one can design lenses with
similar passband efficiency as the Exponential matched lens but with lower
cutoff frequency (see “Klop.17G”). Here the 17 GHz cutoff Klopfenstein taper
achieves nearly identical passband efficiency but the low frequency cutoff is
about 1–2 GHz lower. A GRIN lens with quarter-wave matching sections was also
designed for comparison. The matching sections were realized as a stack of two
identical 0.030” layers so that the match frequency was within the passband of
the taper-matched lenses (a single 0.030” quarter wave match has a match
frequency at approximately 65 GHz and is outside of our simulated band). At
the exact match frequency the efficiency is similar to the Exponential and
Klopfenstein matched lenses but degrades rapidly above and below this
frequency. Finally, the unmatched lens is presented as a baseline with
efficiency oscillating around approximately 72%—the average efficiency is
determined by the range of core permittivities used in the lens such that the
unmatched efficiency would degrade if higher permittivities were used. The
dramatic increase in efficiency of the taper-matched GRIN lenses compared to
the unmatched case highlights the necessity of impedance matching for GRIN
lenses. It is important to note that, in contrast to a one-dimensional taper
which will exhibit a well-behaved and predictable frequency response, the GRIN
lenses shown here have complicated responses which are the aggregate response
of many unit-cells and have the added influence of a spatially displaced feed
element (at the focal distance). Therefore we would not expect the GRIN lens
efficiency to follow exactly the simpler trends of a single taper.
Nevertheless, the overall trends hold which means that the taper design
methods presented in this work can be used in the design of complete lenses.

Figure 9(b) shows dotted semi-transparent $\eta_{ap}^{\prime}$ versus solid
$\widetilde{\eta_{ap}}$ from the final iteration for all lenses in Fig. 9(a).
As frequency increases the non-idealities of the discretized tapers (as
discussed in Section III) begin to distort the ideal taper response and this
effect is exacerbated by the shorter taper lengths such that
$\widetilde{\eta_{ap}}$ (which is based on idealized theory) deviates from
$\eta_{ap}^{\prime}$ (which is based on full-wave simulations). Nevertheless,
$\eta_{ap}^{\prime}$ and $\widetilde{\eta_{ap}}$ agree well within 5% across
the band. Notabely, $\widetilde{\eta_{ap}}$ shows the correct trends regarding
cutoff frequency and passband efficiency indicating that the approximate
$\widetilde{\eta_{ap}}$ are predictive and therefore useful for design. For
the quarter-wave matched lens and the unmatched lens, $\Gamma(f,\theta_{inc})$
is derived by the wave transfer matrix method due to the lack of an analytical
formula. As a result, the approximate $\widetilde{\eta_{ap}}$ includes the 2
GHz ripple due to multiple reflections within the lens.

##  VI Conclusion

In this work we have proposed a systematic design method for PUTs which
minimally distort the ideal taper performance due to the use of non-
commensurate lines. The design algorithm (Algorithm 1) and equivalent
frequency response formula (8) are derived. The taper design method maintains
key performance parameters of the chosen taper (e.g., $f_{c}$). We have
explained the source of the nonidealities and proposed various means of
mitigating them. The taper design method was validated by fabricating a nine
layer Klopfenstein taper and characterizing it in a free-space (quasi-TEM)
measurement setup across three waveguide bands. Finally, we presented an
approximate expression for GRIN lens efficiency and designed GRIN lenses with
several impedance matching sections, confirming the advantages of Klopfenstein
taper matched GRIN lenses. The methods presented in this work enable designers
to accurately predict the efficiency versus frequency of a single non-
commensurate line taper as well as a complete GRIN lens comprising a large
number of tapers to achieve optimal efficiency in high performance
applications. The method allows for an accurate analysis of performance
tradeoffs which was not previously possible.

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|  Wei Wang (Student Member, IEEE) is currently a Ph.D student in Electrical
Engineering at the University of Notre Dame in Indiana, U.S.A. In 2019, he
graduated with a B.S., Electrical Engineering from Nankai University in
Tianjin, China. His research interests are gradient index(GRIN) lens antennas
design and computational electromagnetics for GRIN lenses.
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|  Nicolas C. Garcia received the B.S. degree in electrical engineering from
the University of Notre Dame in 2017. He is currently pursuing a Ph.D. at the
same institution. His research interests focus on low-profile gradient index
(GRIN) materials and their applications for emerging 5G and millimeter wave
antennas and technologies.
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|  Jonathan D. Chisum (S’02–M’06–SM’17) received the Ph.D. in Electrical
Engineering from the University of Colorado at Boulder in Boulder, Colorado
USA, in 2011. From 2012 to 2015 he was a Member of Technical Staff at the
Massachusetts Institute of Technology Lincoln Laboratory in the Wideband
Communications and Spectrum Operations groups. His work at Lincoln Laboratory
focused on millimeter-wave phased arrays, antennas, and transceiver design for
electronic warfare applications. In 2015 he joined the faculty of the
University of Notre Dame where he is currently an Assistant Professor of
Electrical Engineering. His research interests include millimeter-wave
communications and spectrum sensing with an emphasis on low-power and low-cost
technologies. His group focuses on gradient index (GRIN) lenses for low-power
millimeter-wave beam-steering antennas, nonlinear (1-bit) radio architectures
for highly efficient communications and sensing up through millimeter-waves,
as well as reconfigurable RF circuits for wideband distributed circuits and
antennas. Dr. Chisum is a senior member of the IEEE, a member of the American
Physical Society, and an elected Member of the U.S. National Committee (USNC)
of the International Union or Radio Science’s (URSI) Commission D (electronics
and photonics). He is the past Secretary and current Vice-chair for USNC URSI
Commission D: Electronics and Photonics.
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