# Near-Field Millimeter Wave Vector Measurements - Experimental Design &
Measurement Interpretation

Laurent Chusseau1, Thibaut Auriac1, Jérémy Raoult1  1IES, Université de
Montpellier, CNRS, Montpellier, France

###### Abstract

Near-field imaging experiments exist both in optics and microwaves with often
different methods and theoretical supports. For millimeter waves or THz waves,
techniques from both fields can be merged to identify materials at the micron
scale on the surface or in near-surface volumes. The principle of such near-
field vector imaging at the frequency of 60 GHz is discussed in detail here.
We develop techniques for extracting vector voltages and methods for
extracting the normalized near-field vector reflection on the sample. In
particular, the subharmonic IQ mixer imbalance, which produced corrupted
outputs either due to amplitude or phase differences, must be taken into
account and compensated for to avoid any systematic errors. We provide a
method to fully characterize these imperfections and to isolate the only
contribution of the near-field interaction between the probe and the sample.
The effects of the mechanical modulation waveform and harmonic rank used for
signal acquisition are also discussed.

###### Index Terms:

Near-field measurement, mm-waves, microwave probes, vector local
characterization, subharmonic IQ mixer.

##  I Introduction

Near-field imaging is now an essential technique for the characterization of
devices and materials. Depending on the targeted application, reflection or
transmission modes are used in the infrared [1], microwave [2] or THz [3], and
for the last two take advantage of the transparency of many materials for non-
destructive testing through the surface [4, 5, 6, 7].

If high spatial resolution is desired in respect to the measurement
wavelength, a probe is required to concentrate the field and overcome the Abbe
diffraction limit by means of evanescent waves. At GHz frequencies, the
scanning microwave microscope (SMM) frequently uses coaxial resonant probes
whose quality factor and resonance can be related to the interaction with the
sample [8, 9, 10], or AFM tips [11, 12, 13] that can be modeled
electromagnetically in 3D [14, 15, 16]. The latter technique is also used in
the millimeter wave (mmW) region, which is the realm where it is possible to
combine optical and microwave techniques, for example the joint use of
propagating waves [17] and guided waves coupled to specific probes including
small antennae capable of concentrating the electric or magnetic field with a
very high resolution [18, 19, 20].

Although most of near-field experiments in the mmW have focused on intensity
measurements [21, 22, 6, 23, 24, 17], local vector characterization is
necessary to distinguish material changes from loss changes. Many such
experiments have recently been proposed in scattering-type scanning near-field
optical microscopy (s-SNOM) applied to the THz domain [25, 26, 27, 28], but
only the latter seems to be easy to translate to mmW. We do the same
transformation on our 60 GHz test bench [24] and chose to include a
subharmonic IQ mixer to simultaneously detect the two quadratures of the
reflected field, but we kept a complete waveguide configuration. This paper
details the new experiment and elaborate the required processing of
experimental signals in order to determine without systematic error and with
reasonable accuracy the vector voltages and reflection coefficients related to
the near-field probe-sample interaction. We show how the rapid phase rotation
due to the path length in mmW can be used to achieve these values, and how
neutralize the IQ imbalance of the mixer, either in amplitude or phase, by its
theoretical account in data processing. In the end, this gives us an overall
figure of merit for the subharmonic IQ mixer and the entire detection system.

The paper is organized as follows. In §II we describe the experimental setup
and its basic operation. §III details the processing to be applied to
experimental data, namely the vector voltage determination and how mixer non-
ideal behavior transforms the observed vector voltage. We then develop the
method for tracing the true reflection coefficient, provided that a quality
reference measurement is available. In §IV, we deal with the effect of the
harmonic rank of the detection in relation to the modulation waveform applied
to the mechanical modulation of the distance between the probe and the sample,
which is a peculiarity of our experiment as compared to s-SNOM. Finally, in §V
we qualify the frequency response of our experiment and the subharmonic mixer
imbalance.

##  II Measurement Setup

The scheme of the new experiment is given in Fig. 1 with its central element
for the vector output: the subharmonic IQ mixer which operates at zero
intermediate frequency in the band 55-65 GHz. Compared to our previous
intensity reflectometer [24], the whole mmW transmitting/detecting system has
changed and now uses an HF synthesizer followed by multiplication chains for
the detection and LO signal. In the measuring path, an active $\times 4$
multiplier delivering 10 dBm is used in the front of a 3 dB coupler which
returns the reflected signal at the input of the mixer. In between, we take
care of the mmW signal level entering the subharmonic IQ mixer by
incorporating a variable attenuator to validate the -10 dBm requirement for
linear operation. The LO path of the mixer is fed by an active doubler
delivering 10 dBm. The $\times 4$ and $\times 2$ active multipliers are not
really perfectly flat in frequency, unlike the reference synthesizer. As a
result, the subharmonic IQ mixer’s conversion loss variations, which are
typically 15 to 18 dB, may experience additional ripple due to these power
variations of the mmW sources. The whole system was provided by LTEQ Microwave
according to our specifications.

As in [24], probes are homemade bow-ties attached to a WR15 open end. Metal
triangles used for the bow-tie were produced using fs-laser cut on a 10 µm
thick tungsten sheet. The Fig. 2 illustrates this realization and the probe
positioning 10 µm above a surface. The lateral and vertical positioning over
the sample is ensured by a motorized step-by-step system (Newport XPS-C8) and
is optically monitored with two cameras at right angle as shown in the
picture. When set at their highest magnification, cameras equipped with
Navitar UltraZoom lenses can observe a displacement of 1 µm, which corresponds
to one motor step.

Figure 1: Scheme of the mmW near-field experimental setup. All mmW links are
made using WR15 waveguides up to the subharmonic IQ mixer.

The modulation of the probe-to-sample distance is carried out by the
piezoelectric actuator (PI P-611.Z Precision Z-stage) driven by a signal
generator whose frequency is set to 40 Hz due to the limited weight capacity
of this actuator. Subharmonic IQ mixer outputs are thus modulated at this low-
frequency that is also the reference of the two lock-in (AMETEK 7265 Dual
Phase DSP). This allows the filtering of the near-field spatial component
ideally split in its real and imaginary parts. In practice both lock-in
outputs are complex and refer to the phase of the low-frequency modulation.
Correct usage thus requires to set the same phase reference on the two lock-
in.

Figure 2: Photos of the experiment. Left: emission/detection board in place
with the waveguide probe and the two cameras. Right up: tungsten bow-tie probe
view from above with the microscope. Right down: Probe as observed by the
camera 10 µm above the gold mirror, the reflection is used to set the minimum
distance to $h=10$ µm.

At first glance, the real part in channel I (respectively, the imaginary part
in channel Q) of the complex near-field reflected voltage $\widetilde{v}$ is
expected to have the module $m_{I}=\sqrt{X_{I}^{2}+Y_{I}^{2}}$ (respectively,
$m_{Q}=\sqrt{X_{Q}^{2}+Y_{Q}^{2}}$). If these values are obtained
straightforwardly, their respective sign must be determined to fully locate
$\widetilde{v}$ in the entire complex plane.

Our first test sample was chosen to obtain the highest possible near-field
reflectivity. It is a gold optical mirror from Edmund Optics with an ultra-
flat substrate ($\lambda/20$) and a thin (a few nm) dielectric protective
layer. A far-field reflectivity of $\approx 97\%$ is certified in the mid-
infrared, a value also expected in the far infrared and in the mmW range since
gold is homogeneous and well known [29]. This sample is intended to be a
reference for reflectivity, to which any other sample can be compared.

When scanning the input frequency in a tiny range of 200 MHz, the four outputs
of the lock-ins oscillate rapidly, as shown in Fig. 3. This is due to the
total electrical length of the system here estimated at about $1.25$ m using
the $240\pm 2$ MHz period deduced for the $X_{i}$ and the $Y_{i}$. Since the
probe senses the reflection on a flat and homogeneous material, it is not
anticipated that it will contribute to a rapidly changing $\widetilde{v}$ with
frequency, but rather that it can be regarded as constant. Consequently, we
assume in what follows that the almost sinusoidal evolution of the $X_{i}$ and
$Y_{i}$ is solely attributed to the electrical length.

Figure 3: Lock-in outputs as measured $h=10$ µm above a gold mirror with a
sinusoidal vibration amplitude $\delta h=20$ µm while scanning the input
frequency.

In addition, a close inspection of Fig. 3 shows a deviation from the expected
ideal subharmonic IQ mixer: at 60.765 GHz channel I cancels ($X_{I}=Y_{I}=0$)
but channel Q is not maximum, which occurs at 60.785 GHz. Conversely, the same
occurs at 60.720 GHz for channel Q and 60.705 GHz for channel I. This is quite
unexpected for a perfect IQ mixer and shows the mixture of channel signals
because of its imperfection. As a result, the measurement of near-field
reflection at a single frequency would inevitably be flawed by a systematic
error regarding the contribution of the material placed just under the probe.
We therefore prefer to carry out frequency sweeps like the one shown in Fig. 3
and process the data to eliminate these errors by assuming that the near-field
contribution is constant over the low measurement frequency band.

##  III Measurement Data Processing

###  III-A Vector Voltage Determination

Proceeding from a measurement like the one shown in Fig. 3, the modulus of the
reflected near-field voltage is directly obtained by

$\left\lvert{\widetilde{v}}\right\rvert=\sqrt{m_{I}^{2}+m_{Q}^{2}}.$ (1)

Accessing the phase is more tedious since the rough estimate
$\angle\widetilde{v}=\arctan\frac{m_{Q}}{m_{I}}$ is inherently limited to
$[0,\frac{\pi}{2}]$ because of the positive $\arctan$ argument. Going further
is possible using the original lock-in outputs, which include the sign changes
for channels I and Q when $m_{I}$ and $m_{Q}$ go through zero. Noting that
these outputs always present opposite $X$ and $Y$ values, the sign are
straightforwardly obtained from
$\operatorname{sgn}(I)=\operatorname{sgn}(X_{I}-Y_{I})$ and
$\operatorname{sgn}(Q)=\operatorname{sgn}(X_{Q}-Y_{Q})$. Keeping track of all
these sign changes, we define the phase of the reflected voltage using the
heuristic formula

$\angle\widetilde{v}=\arctan\frac{\operatorname{sgn}(Q)m_{Q}}{\operatorname{sgn}(I)m_{I}}.$
(2)

We have applied these formulas to the experimental case of the gold mirror in
Fig. 3 and also for an unintentionally doped GaAs wafer (ITME, Warsaw). The
same geometrical conditions were applied for the measurement, _i.e._ $h=10$
µm, $\delta h=20$ µm and a sinusoidal modulation. The results are shown in
Fig. 4 with the modulus and phase of $\widetilde{v}$ and their locations in
the complex plane. As expected, $\angle\widetilde{v}$ covers the whole range
$[-\pi,\pi]$ which leads to elliptic representations in the complex plane.

Figure 4: Evolutions of the complex reflected near-field voltages
$\widetilde{v}$ for a gold mirror and a GaAs undoped wafer. Left up: modulus,
left down: phase, right: complex representation. Dots are for gold and open
triangles for GaAs.

Although the two materials can be easily distinguished on these curves,
especially by their average values $\left\lvert{\widetilde{v}}\right\rvert$
which are clearly different and by their shifted $\angle\widetilde{v}$
evolutions, it is impossible at this stage to characterize a material with a
single complex value whereas this was hoped by assuming a constant influence
of the material on the limited frequency range considered. In this framework,
centered ellipses are distorted views of circles usually obtained in microwave
by measurements using sliding loads. We attribute this distortion to the
imbalance of the IQ mixer and model it in the sequel to refine the analysis.

###  III-B Normalized Vector Reflection

Let us consider the ideally reflected complex voltage $v$ in the near-field of
the tested surface. As said it should be constant over the tiny frequency
range considered because our materials are far from any resonance. The voltage
$v$ thus undergoes only a phase shift $\exp j\phi$ due to the electrical
length before being transformed by the subharmonic IQ mixer to produce
$\widetilde{v}$. We assume that mixer imbalance is twofold: first, it affects
the gain by a factor $\gamma$ possibly different from unity that we apply only
to the imaginary part, and second, the imperfect quadrature is accounted for
by an additional angle $\theta$ added to the $\pi/2$ phase shift of the Q
channel. This is written

$\widetilde{v}=\Re(v\exp j\phi)+j\gamma\Im(v\exp j(\phi+\theta)).$ (3)

If we now divide the vector voltage measured on the sample
$\widetilde{v}_{S}$, for instance the GaAs wafer of Fig. 4 by the vector
voltage measured on a reference $\widetilde{v}_{R}$, namely the gold mirror in
the same figure, which is highly conductive at mmW and supposed to produce the
highest possible reflection, we obtain

$\frac{\widetilde{v}_{S}}{\widetilde{v}_{R}}=\rho\frac{\cos(\phi+\varphi)+j\gamma\sin(\phi+\varphi+\theta)}{\cos\phi+j\gamma\sin(\phi+\theta)},$
(4)

where $\rho=\left\lvert{v_{S}}\right\rvert/\left\lvert{v_{R}}\right\rvert$ and
$\varphi=\angle v_{S}-\angle v_{R}$. As in reflectometry, the normalized
vector reflection $\rho\exp j\varphi$ is thus intended to be the near-field
mmW reflection coefficient $\Gamma$ of the tested material, provided that the
reference is a perfect metal.

Eq. 4 is easily identified as a circle in the complex plane parametrized by
$\phi$. It is completely described if $\phi\in[0,\pi]$ and collapses to a
single point with a perfect IQ mixer ($\gamma=1$ and $\theta=0$). Solving a
set of 3 equations for 3 different points such as $\phi=0$, $\phi=\pi/2$ and
$\phi=-\theta$ which must belong to the same circle
$(X-X_{c})^{2}+(Y-Y_{c})^{2}-R^{2}=0$, provides the link with our parameters
in Eq. (4). After some lengthly developments, we end up with the definition of
the circle

$\displaystyle X_{c}$ $\displaystyle=\rho\cos\varphi$ (5a) $\displaystyle
Y_{c}$ $\displaystyle=\frac{\rho(1+\gamma^{2})\sin\varphi}{2\gamma\cos\theta}$
(5b) $\displaystyle R$ $\displaystyle=\zeta Y_{c}$ (5c)

with

$\zeta=\sqrt{1-\frac{4\gamma^{2}\cos^{2}\theta}{\left(1+\gamma^{2}\right)^{2}}}.$
(6)

Interestingly the term $\zeta$ is completely independent of both the sample
and reference. It is thus some kind of measure of the IQ mixer ideality within
our experiment. The gain imbalance $\gamma$ and the quadrature error $\theta$
are mixed in $\zeta$ and cannot be solved separately because we are missing a
fourth variable. Order of magnitude of $\zeta$-values are given in Fig. 5 as a
function of $\gamma$ and $\theta$. For gain errors lower than 3 dB and phase
errors lower than 30° it does not exceed 0.6 which is nonetheless significant
compared to its limit value of 1 obtained when $\gamma$ or $1/\gamma\to\infty$
or $\theta\to\pm\pi/2$. We must keep in mind this limit value of 1 which
cannot be exceeded under any circumstances in our experiments. It is a
criterion of suitability of the measurements and their exploitation that will
be implemented in the following. Values on the order of 0.4 and less, however,
seem acceptable.

Figure 5: Ideality factor $\zeta$ as a function of gain mismatch $\gamma$ and
quadrature phase mismatch $\theta$.

Go back to measurements to calculate and plot the ratio
$\widetilde{v}_{S}/\widetilde{v}_{R}$ from data of Fig. 4. As shown in Fig. 6
we end up with a nearly perfect circle, just like predicted above. We have
limited here the frequency range previously considered to $[60.690-60.815]$
GHz in order to describe the circle without overlap. Since the points are not
evenly distributed on the circle, there is no simple criteria to immediately
access the position of its center, so a least-squares fit was used to
determine the triplet $(X_{c},Y_{c},R)$. A secondary result of the fit that
minimizes the distance between the measured points and the circle is the error
distance accumulated for measurement points to the circle. It is used in the
sequel to estimate the uncertainties on each of the inferred values using
Student’s t-test with 95% confidence level and classical propagation of errors
at the first level on the equations. The error is only $\approx 6\,10^{-4}$
per point in this example, as could be expected given the excellent agreement
between the best circle, represented by a solid line in the Fig. 6, with the
measurement points. The position of the center is noted by a filled square.

The module and phase of the near-field reflection are then straightforwardly
calculated from Eqs. (5) by

$\displaystyle\rho$ $\displaystyle=\sqrt{X_{c}^{2}+Y_{c}^{2}-R^{2}}$ (7a)
$\displaystyle\varphi$ $\displaystyle=\arccos\frac{X_{c}}{\rho}.$ (7b)

Figure 6: Ratio $\widetilde{v}_{S}/\widetilde{v}_{R}$ in the complex plane for
the GaAs sample and the gold mirror reference of Fig. 4. Dots are
measurements, the solid line is the corresponding best circle fit whose center
is the filled square, and the filled diamond is the deduced $\Gamma=\rho\exp
j\varphi$ normalized vector reflection.

In the case of Fig. 6 this deduced point is represented by a diamond. It is
obviously different from the center, even if according to Eq. (5a) their real
parts are identical. In essence, we end up here with the normalized vectorial
reflection $\Gamma=v_{S}/v_{R}=(0.482+j0.225)\pm 0.004$ with an ideality
factor $\zeta=0.436\pm 0.005$, which illustrate the effectiveness of the
entire measurement operation process, including error estimates. In the ideal
case where our reference is perfect, it is expected that $\Gamma$ is the true
reflection coefficient in near-field at mmW.

##  IV Influence of the mechanical modulation

In scattering-type scanning near-field optical microscopy (s-SNOM) the
mechanical probe-to-sample distance modulation is always sinusoidal since it
is provided by a quartz resonance [25, 12, 17, 28]. Unconventional modulation
waveforms are possible in our experiment since the mechanical displacement is
performed by a piezoelectric actuator controlled by a low-frequency generator.
The actuator is optimized for 40 Hz but has a cut-off frequency of about 100
Hz, so we can use any arbitrary waveform validating these conditions.
Measurements have been performed as test cases with our two samples: the GaAs
wafer and the gold mirror.

We try successively sinusoid, square and triangle modulations at constant 40
Hz fundamental frequency. In each case a measure is done on the reference and
reproduce in exactly the same conditions for position, modulation and
detection for the GaAs sample. Initial position of the probe is always the
minimum distance $h=10$ µm above the surface before applying the modulation
whose amplitude is intended to be $\delta h=20$ µm, as calibrated for
sinusoid. Depending on the modulation waveform, the effective $h$ may be
smaller, especially with square modulation where the transfer function of the
piezoelectric actuator exhibits an overshoot. Although the displacement
approximately follows the command, the rise and fall times are not
instantaneous and the effective distance between the probe and the sample is
distorted. This occurs also to a lesser extent with triangular modulation.

TABLE I: Deduced normalized reflection coefficient $\Gamma$ of GaAs versus the modulation waveform and the harmonic rank. Rank | Sinus Modulation | Square Modulation | Triangle Modulation
---|---|---|---
$\Gamma$ | $\zeta$ | $\Gamma$ | $\zeta$ | $\Gamma$ | $\zeta$
1 | $(0.482+j0.225)\pm 0.004$ | $0.436$ | $(0.489+j0.226)\pm 0.004$ | $0.435$ | $(0.487+j0.226)\pm 0.005$ | $0.436$
2 | $(0.655+j0.280)\pm 0.006$ | $0.408$ | $(0.679+j0.270)\pm 0.009$ | $0.440$ | $(0.687+j0.283)\pm 0.006$ | $0.410$
3 | — | $5.6$ | $(0.511+j0.231)\pm 0.005$ | $0.436$ | $(0.528+j0.234)\pm 0.004$ | $0.431$
4 | — | $1.00$ | $(0.728+j0.289)\pm 0.005$ | $0.414$ | $(0.735+j0.283)\pm 0.021$ | $0.418$
5 |  |  | — | $2.9$ | — | $2.9$
6 |  |  | $(0.760+j0.263)\pm 0.017$ | $0.418$ |  |
7 |  |  | $(0.411+j0.173)\pm 0.010$ | $0.452$ |  |
8 |  |  | $(0.779+j0.276)\pm 0.026$ | $0.428$ |  |
9 |  |  | $(0.355+j0.146)\pm 0.021$ | $0.47$ |  |

The experimental procedure and the determination of the vectorial voltage
described in Fig. 4 were reproduced for the three modulations with increasing
harmonic ranks selected on the lock-ins. This was done as long as the detected
signal emerged from the noise. As can be seen from the average
$\left\lvert{\widetilde{v}}\right\rvert$ voltages plotted in Fig. 7, the
detected signal decreases rapidly with harmonic rank, but it also depends a
lot on the waveform of the modulation. Nevertheless,
$\left\lvert{\widetilde{v}}\right\rvert$ is very similar for all waveforms at
the first two harmonics. Beyond that, the decrease in intensity is huge for
sinusoidal modulation, which generates those high harmonics only by the
nonlinear probe-to-sample interaction caused by the rapid decrease of
intensity in the near-field. This peculiarity of probe-to-sample interaction
is often used to improve the spatial resolution [30, 31, 24]. For other
modulation waveforms, higher harmonics are less attenuated, especially in the
case of square modulation. Actually they tend to follow the intensities that
could be computed by the Fourier transform of the initial waveforms, in
particular the high attenuation of even harmonics. For such waveforms we can
no more argue for an improvement in resolution by detecting high order
harmonics.

Figure 7: Average module of the voltage detected
$\left\lvert{\widetilde{v}}\right\rvert$ for $f\in[60.690,60.815]$ GHz versus
harmonic rank and modulation waveform: $\bullet$ are for sinus, $\blacksquare$
for square and $\blacklozenge$ for triangle modulations.

The results were interpreted using Eqs. (7) applied to the best fitted circles
on the experiments. We used $\zeta$ as a determinant: if its value exceeds 1,
which according to the discussion in §III-B is impossible in the framework of
our subharmonic IQ mixer model, then we rejected the measure as a whole. The
practical reasons for such a mismatch can be multiple, such as non-
reproducible positioning over our test surfaces, weak or noisy signals… The
factor $\zeta$ is therefore an effective quality criterion for the control of
our measurements. In the case of triangle modulation, the Fig. 8 illustrates
the measurement points with the 4 circles obtained for the first 4 harmonics.
Although the measurement points are all well located on circles, it is obvious
that the even and odd harmonics are grouped together, but the two groups do
not merge even if they overlap. As a result the extracted reflection
coefficient with even harmonics shows real and imaginary parts which both have
higher values.

Figure 8: Normalized reflection measured for the four first harmonic ranks
with a triangle modulation waveform. Dots are measurements and circles their
corresponding best fits.

The complete results for all harmonics of all waveforms are given in the Table
I. Rejected cases occurring because $\zeta\geq 1$ are represented by a dash
and this occurs especially for all harmonics 5. We believe that this is
because this frequency of 200 Hz is the only one that is also a harmonic of
the power supply frequency, which inevitably produces a large increase of
noise in the detection circuit. As soon as
$\left\lvert{\widetilde{v}}\right\rvert$ is high enough in all other cases,
except for the higher harmonics of the sinusoidal modulation, we obtain
$\zeta\approx 0.4$ and the results do not seem to be biased.

The data of Table I are plotted in Fig. 9. The superposition of the results
for all waveforms highlights the experimental independence in relation to this
setting. On the contrary, the alternation of the values between even and odd
harmonics is conspicuous. While the real parts have values that increase with
the rank of even harmonics, the opposite occurs for odd harmonics, and the
same trend is practically true for the imaginary part. Nevertheless the most
notable thing is the dichotomy between even and odd harmonics. The most
surprising fact of these results is that for a given harmonic rank the
normalized reflection value determined is the same regardless of the
modulation shape, and this is verified within the limit of the error bars for
the first two harmonics and still very close for next two one. Note that
increasing the harmonic rank rapidly decreases the signal and therefore
mechanically increases the error.

Figure 9: Real (left) and imaginary parts (right) of the normalized reflection
$\Gamma$ versus harmonic rank and modulation waveform: $\bullet$ are for
sinus, $\blacksquare$ for square and $\blacklozenge$ for triangle modulations.

Obtaining different $\Gamma$ values from one rank to another is deeply
counter-intuitive, as is the fact that even and odd harmonics seem to behave
identically. Actually, if only the material tested in the near-field matters,
the reflection should be expressed from the dielectric constant $\epsilon$ of
the material as predicted by the Fresnel coefficients [32],
$\Gamma=(\epsilon-1)/(\epsilon+1)$. For instance with sinusoidal modulation at
rank 1, it leads to $\epsilon=2.863+j1.581$, _i.e._ a refractive index of
$1.751+j0.451$, which is far from the tabulated value of $3.606+j0.0012$ at
300 GHz [29].

Therefore, the quantities determined here must not be regarded as reflection
coefficients in the usual sense of microwaves and it is not realistic to
extract any $\epsilon$ from them. Although we use the oscillation of the
probe-to-sample distance, which concentrates the useful signal on the near-
field, we had already shown by approach curves with these bow-tie probes that
a longer distance interaction also exists [24], which certainly enters here in
the final quantitative result. The probable origin of this interaction was
identified and attributed to the mandatory openings between the two triangles
constituting the bow-tie, and we had also noticed that such a long-distance
contribution may be favored if a large dielectric support is used to support
the two triangles. For the present probe all precautions had been taken to
minimize these effects. Nonetheless, the quantities extracted are not
reflection coefficients even if we used the usual $\Gamma$ notation of
microwaves. They seem however characteristic of the materials placed in the
near-field of our bow-tie probe and are obtained with a very good accuracy.

##  V Measurement of subharmonic IQ mixer ideality

Since the $\zeta$ factor is a measure of the ideality of our instrumentation,
it is worth characterizing it over the entire available bandwidth. Under
conditions of sinusoidal modulation with $h=10$ µm and $\delta h=20$ µm and
detection on the fundamental frequency, we therefore measured both our gold
mirror reference and our GaAs sample from 55 to 65 GHz with a 10 MHz stepsize.
We then proceed to the extraction defined in §III to obtain $\zeta$ and
$\Gamma$ using a 120 MHz sliding window which allows the application of Eqs.
(5-7).

Figure 10 shows the evolution of $\zeta$ as a function of frequency over the
entire operating band of the subharmonic IQ mixer. Obviously, $\zeta$ varies a
lot in the whole band with really high values above 61 GHz which exceed unity
around 64GHz. This is in full agreement with the observation of a drastic drop
of the measured $\widetilde{v}$ at such frequencies for both gold and GaAs. It
is attributed to the conjunction of a power drop of the two multipliers,
especially the one dedicated to the LO which provides 3dBm less than below,
and an increase in the conversion losses of the subharmonic IQ mixer, leading
to an almost $\widetilde{v}$ cancellation at lock-in outputs. Figure 10 shows
that the best frequency range for our system is around 59 GHz, but if we
assume a maximum value of $\approx 0.4$ it is still quite usable from about 56
GHz to 61 GHz. We set out this criterion here because although the frequency
around 60.75 GHz used in §III and §IV is at the limit of this useful band, we
saw that the results were quite satisfactory and accurate.

Figure 10: Ideality factor $\zeta$ versus frequency.

It should be noted here that due to a handling error, the spacing between the
probe tips has been modified since the recordings in §III. If this spacing was
previously 11 µm, it is now 15 µm due to a too abrupt touch that exceeded the
tungsten elasticity limits. Nevertheless the probe has been repaired and kept
its global shape with only this new spacing parameter changed. A fundamental
consequence of this probe crash and repair is that $\zeta$ is independent of
this spacing since we obtain here in Fig. 10 $\zeta=0.433\pm 0.024$ at 60.75
GHz whereas it was determined in §IV at $\zeta=0.436\pm 0.005$ (see Table I).
The parameter $\zeta$ is therefore a characteristic that is essentially
associated with the mixer imbalance but which nevertheless includes any
possible variations in LO power that affect its operating point.

The determination of $\zeta$ over the whole frequency band also yields
$\Gamma$ thanks to Eqs. (7). It is plotted in Fig. 11 where frequencies
leading to $\zeta\geq 1$, for instance around 64 GHz, have been rejected. If
$\Gamma$ really represented the reflection coefficient only related to the
interaction of a dipole with a homogeneous material, we should expect a value
corresponding to the Fresnel reflection coefficient,
$\Gamma=(\epsilon-1)/(\epsilon+1)$, which should be constant since both gold
and GaAs are not dispersive at mmW. This is not the case here, even if in the
whole range 56-60 GHz the values vary very little. Since, on the other hand,
$\zeta$ encompasses all the imperfections of the experiment involving the
subharmonic mixer and its LO, this variation can only be attributed here to
the interaction of the probe and the sample which does not seem to follow a
simple model. In truth this is confirmed by the fact that we have here at
60.75 GHz $\Gamma=(0.400+j0.152)\pm 0.014$ instead of
$\Gamma=(0.482+j0.225)\pm 0.004$ found previously before the probe was
modified.

Figure 11: Real and imaginary parts of the normalized reflection $\Gamma$
versus frequency.

We can therefore conclude that our near-field measurement remains dependent on
the probe used, even if the uniqueness of $\zeta$ has shown that all other
sources of variability related to the experiment have been compensated for.
This is clearly a drawback as we cannot trace back to universal material
characteristics but it is probably still possible to distinguish near-field
materials after learning with each probe used. In future work, we will try to
demonstrate and illustrate this behavior with other probes and materials.

##  VI Conclusion

We presented the realization of a new measurement bench for near-field
reflectometry in mmW. A detailed analysis of the operation of this bench has
been carried out, making it possible to take advantage of the electrical
length that exists between the test and measurement paths. We illustrate its
use to extract the near-field contribution that is independent of the mmW
board from measurements distorted by systematic errors. A spin-off of this
theoretical analysis of the operation of the experiment is the definition of a
coefficient of ideality $\zeta$ to which the IQ mixer imbalance and the gain
variations of the mmW sources mainly contribute.

We applied this technique to the near-field measurement of a GaAs substrate
and a gold mirror. At fixed frequency the study shows a result that is
substantially independent of the waveform of the mechanical modulation but not
of the order of the harmonic used for detection. Curiously the even and odd
harmonics lead to two groups of results that are quite consistent with each
other but remain unexplained.

A study of our measurement system over the entire 55-65 GHz band has allowed
us to qualify $\zeta$. For reasons related to our equipment, the latter one
turns out to be rather bad in high frequency but allows very reliable
measurements between 56 and 61 GHz. Moreover, $\zeta$ appears independent of
the probe, which is not the case of the normalized vectorial voltage $\Gamma$
measured in the near-field. It is clear that our probe does not have a trivial
interaction with the sample and is responsible for this unexpected variation.
In the past, we had already observed similar effects with a long range
detection effect from the lateral radiation of our probes [24]. However, here
we have taken great care to minimize it by eliminating any protruding
dielectric material used for mounting the probe. The values obtained for
$\Gamma$ and its evolution with frequency cannot be explained with a simple
model of the probe-sample interaction. Linking $\Gamma$ to the material
parameters of the sample under test in the near-field will therefore require a
further study, which can certainly only be carried out through full 3D
electromagnetic modeling of the coupled probe-sample system.

Experimentally, the deduced $\Gamma$ values remain reproducible and therefore
characteristic of the material for a given probe. In future work, they will be
used to distinguish materials in the near field, which probably implies prior
calibration and learning procedures. The sub-wavelength spatial resolution
that has been demonstrated for intensity measurement should also be studied in
this context.

## References

  * [1] X. Chen, D. Hu, R. Mescall, G. You, D. Basov, Q. Dai, and M. Liu, “Modern scattering-type scanning near-field optical microscopy for advanced material research,” Advanced Mater., vol. 31, no. 24, p. 1804774, 2019.
  * [2] A. Imtiaz, T. M. Wallis, and P. Kabos, “Near-field scanning microwave microscopy,” IEEE Microw. Mag., pp. 52–64, 2014.
  * [3] A. J. L. Adam, “Review of near-field terahertz measurement methods and their applications,” Int. J. Infrared Millim. Waves, vol. 32, pp. 976–1019, 2011.
  * [4] S. Kharkovsky and R. Zoughi, “Microwave and millimeter wave nondestructive testing and evaluation–Overview and recent advances,” IEEE Instrumentation & Measurement Magazine, vol. 10, no. 2, pp. 26–38, 2007.
  * [5] C. Plassard, E. Bourillot, J. Rossignol, Y. Lacroute, E. Lepleux, L. Pacheco, and E. Lesniewska, “Detection of defects buried in metallic samples by scanning microwave microscopy,” Phys. Rev. B, vol. 83, no. 12, p. 121409, 2011.
  * [6] K. Haddadi, D. Glay, and T. Lasri, “A 60 GHz scanning near-field microscope with high spatial resolution sub-surface imaging,” IEEE Microw. Wirel. Compon. Lett., vol. 21, no. 11, pp. 625–627, 2011.
  * [7] K. Moon, H. Park, J. Kim, Y. Do, S. Lee, G. Lee, H. Kang, and H. Han, “Sub-surface nanoimaging by broadband terahertz pulse near-field microscopy,” Nano Lett., vol. 15, pp. 549–552, 2014.
  * [8] C. Gao and X.-D. Xiang, “Quantitative microwave near-field microscopy of dielectric properties,” Review Scientific Instruments, vol. 69, no. 11, pp. 3846–3851, 1998.
  * [9] A. Imtiaz, T. M. Wallis, S.-H. Lim, H. Tanbakuchi, H.-P. Huber, A. Hornung, P. Hinterdorfer, J. Smoliner, F. Kienberger, and P. Kabos, “Frequency-selective contrast on variably doped p-type silicon with a scanning microwave microscope,” J. Appl. Phys., vol. 111, no. 9, p. 093727, 2012.
  * [10] T. Monti, O. B. Udoudo, K. A. Sperin, C. Dodds, S. W. Kingman, and T. J. Jackson, “Statistical description of inhomogeneous samples by scanning microwave microscopy,” IEEE Transactions on Microw. Theory Techniques, vol. 65, pp. 2162–2170, June 2017.
  * [11] H. Huber, I. Humer, M. Hochleitner, M. Fenner, M. Moertelmaier, C. Rankl, A. Imtiaz, T. Wallis, H. Tanbakuchi, P. Hinterdorfer, et al., “Calibrated nanoscale dopant profiling using a scanning microwave microscope,” J. Appl. Phys., vol. 111, no. 1, p. 014301, 2012.
  * [12] S. Berweger, J. C. Weber, J. John, J. M. Velazquez, A. Pieterick, N. A. Sanford, A. V. Davydov, B. Brunschwig, N. S. Lewis, T. M. Wallis, et al., “Microwave near-field imaging of two-dimensional semiconductors,” Nano Letters, vol. 15, no. 2, pp. 1122–1127, 2015.
  * [13] M. Horibe, S. Kon, and I. Hirano, “Measurement capability of scanning microwave microscopy: Measurement sensitivity versus accuracy,” IEEE Transactions on Instrumentation Measurement, vol. 68, pp. 1774–1780, June 2019.
  * [14] A. Oladipo, M. Kasper, S. Lavdas, G. Gramse, F. Kienberger, and N. Panoiu, “Three-dimensional finite-element simulations of a scanning microwave microscope cantilever for imaging at the nanoscale,” Appl. Phys. Letters, vol. 103, no. 21, p. 213106, 2013\.
  * [15] Z. Wei, Y. Cui, E. Y. Ma, S. Johnston, Y. Yang, R. Chen, M. Kelly, Z. Shen, and X. Chen, “Quantitative theory for probe-sample interaction with inhomogeneous perturbation in near-field scanning microwave microscopy,” IEEE Transactions on Microw. Theory Techniques, vol. 64, pp. 1402–1408, May 2016.
  * [16] B.-Y. Wu, X.-Q. Sheng, R. Fabregas, and Y. Hao, “Full-wave modeling of broadband near field scanning microwave microscopy,” Scientific Reports, vol. 7, no. 1, p. 16064, 2017.
  * [17] G. Dai, G. Geng, X. Zhang, J. Wang, T. Chang, and H. Cui, “W-band near-field microscope,” IEEE Access, vol. 7, pp. 48060–48067, 2019.
  * [18] B. Rosner, J. Peck, and D. van der Weide, “Near-field antennas integrated with scanning probes for thz to visible microscopy: scale modeling and limitations on performance,” IEEE Transactions on Antennas Propag., vol. 50, pp. 670–675, May 2002.
  * [19] L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics, vol. 5, pp. 83–90, 2011.
  * [20] T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Letters, vol. 11, no. 3, pp. 1009–1013, 2011.
  * [21] T. Nozokido, J. Bae, and K. Mizuno, “Scanning near-field millimeter-wave microscopy using a metal slit as a scanning probe,” IEEE Transactions on Microw. Theory Techniques, vol. 49, pp. 491–499, March 2001.
  * [22] J.-P. Guillet, L. Chusseau, R. Adam, T. Grosjean, A. Penarier, F. Baida, and D. Charraut, “Continuous-wave scanning terahertz near-field microscope,” Microw. Optical Technology Letters, vol. 53, pp. 580–582, 2011.
  * [23] R. Omarouayache, P. Payet, J. Raoult, and L. Chusseau, “Millimeter-wave near-field imaging with bow-tie antennas,” Opt. Express, vol. 23, pp. 12144–12151, May 2015.
  * [24] L. Chusseau, P. Payet, and J. Raoult, “Optimization of near-field image capture with millimeter-wave bow-tie probes,” IEEE Transactions on Instrumentation Measurement, vol. 66, pp. 61–68, Jan 2017\.
  * [25] H. G. von Ribbeck, M. Brehm, D. W. van der Weide, S. Winnerl, O. Drachenko, M. Helm, and F. Keilmann, “Spectroscopic THz near-field microscope,” Opt. Express, vol. 16, no. 5, pp. 3430–3438, 2008.
  * [26] F. Keilmann, A. J. Huber, and R. Hillenbrand, “Nanoscale conductivity contrast by scattering-type near-field optical microscopy in the visible, infrared and thz domains,” J. Infrared, Millimeter, Terahertz Waves, vol. 30, no. 12, pp. 1255–1268, 2009.
  * [27] M. C. Giordano, S. Mastel, C. Liewald, L. L. Columbo, M. Brambilla, L. Viti, A. Politano, K. Zhang, L. Li, A. G. Davies, E. H. Linfield, R. Hillenbrand, F. Keilmann, G. Scamarcio, and M. S. Vitiello, “Phase-resolved terahertz self-detection near-field microscopy,” Opt. Express, vol. 26, pp. 18423–18435, Jul 2018.
  * [28] C. Liewald, S. Mastel, J. Hesler, A. J. Huber, R. Hillenbrand, and F. Keilmann, “All-electronic terahertz nanoscopy,” Optica, vol. 5, no. 2, pp. 159–163, 2018.
  * [29] E. D. Palik, Handbook of optical constants of solids, vol. 3.  Academic press, 1998.
  * [30] P. Adam, J. Bijeon, G. Viardot, and P. Royer, “Analysis of the influence of the tip vibration in the formation of images in apertureless scanning near-field optical microscopy,” Opt. Commun., vol. 174, no. 1–4, pp. 91–98, 2000.
  * [31] B. Knoll and F. Keilmann, “Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy,” Opt. Commun., vol. 182, no. 4, pp. 321–328, 2000.
  * [32] L. Novotny and B. Hecht, Principles of Nano-Optics.  Cambridge: Cambridge University Press, 2006.