Supplemental Material for
Rapid microwave phase detection based on a solid state spintronic device
B. M. Yao1,2, L. Fu1,2, X. S. Chen2, W. Lu2, L. H. Bai1, Y. S. Gui1 and C.-M. Hu1
1
Department of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2 and
2
National Laboratory for Infrared Physics, Chinese Academy of Sciences, Shanghai 200083,
People's Republic of China
1. Principle of microwave rapid phase detection using solid state devices
If a solid state device has a non-linear relationship between current (I) and voltage (V),
it can convert alternating current (AC) to direct current (DC) according to the
trigonometric relation of cos2(t)=[1+cos(2t)]/2. This so-called rectification effect is
widely used to down-convert GHz microwave fields to a low-frequency or DC signal;
where a linear dependence between the produced voltage signal and the microwave
power can be found [as shown in Fig. 1(c)].
Based on the principle of rectification, we present a new approach of using a lock-in
amplifier to detect the magnitude and phase of a microwave field. In our experiment,
microwaves are generated, then divided into two paths, and finally coupled at the
sensor. Through the interference of the two microwaves and the rectification effect,
the GHz microwaves are down-converted to a low-frequency signal carrying both the
magnitude and phase information of the microwave field being tested.
As mentioned in the article, the microwaves eT cos(t  ) in path A have an
amplitude of
eT
and a phase of  , while the reference microwaves
eR cos(t  V t   0 ) in path B have an amplitude of eR and a phase of  0 , as well
as a periodic-time-dependent phase shift of 𝜔𝑉 𝑡(𝜔𝑉 <<𝜔). Although these GHz
signals cannot be directly measured by a lock-in amplifier SR830, which works only
up to 100 kHz, the lock-in amplifier can measure the low-frequency signal produced
by the nonlinear effect via the wave mixing of eT cos(t  )  eR cos(t  V t  0 ) .
The complete expression of the voltage signal Vsig produced by the microwave
field’s quadratic term due to the nonlinear effect can be written as:
Vsig  [eT cos(t   )  eR cos(t  V t   0 )] 2
 eT cos 2 (t   )  eR cos 2 (t  V t   0 )  2eT eR cos(t   ) cos(t  V t   0 )
2
2
eT  eR
2
 eT eR cos(V t   0   )
2
2

 eT
2
cos( 2t  2 )
2 cos( 2t  2V t  2 0 )
 eR
 eT eR cos( 2t  V t     0 ).
2
2
(S1)
Vsig includes a DC voltage signal  ( eT  eR ) / 2 , a low frequency signal
2
2
 eT eR cos(V t   0  ) , and second harmonic terms with frequencies of 2
(which are undetectable using conventional instruments). If there is not such a
periodic-time-dependent phase shift ( V  0 ) in path B, Eq. (S1) has a simple form as


Vsig  eT2  eR2  2eT eR cos(0  ) / 2 after neglecting the undetectable 2 term.
This is the situation discussed in Ref. S1, S2 and S3, where the phase of  can be
determined by tuning  0 through a series of values and performing a computer fitting
of the measured alternating fringes of rectified voltage.
The significance of the present approach is introducing a periodic-time-dependent
phase shift term of 𝜔𝑉 𝑡. The mixing of the two microwave paths, A and B, results in
a periodic-time-dependent voltage signal of Vsig  eT eR cos(V t  0  ) . Thus by
comparing with a standard low-frequency wave of cos(𝜔𝑉 𝑡), both the amplitude eT
and phase shift    0 of the microwaves in path A are directly determined by a
lock-in amplifier.
2. Experimental verification of the interference of two microwave beams
Microwave beams from either path A or B can be separately rectified into a DC
voltage signal according to the first term on the right side of Eq. (S1) by an MTJ
microwave sensor. Comparing with a standard low-frequency wave of cos(𝜔𝑉 𝑡), the
lock-in amplifier acts as a filter removing unwanted frequency components from the
signal and leaving only the components with a frequency of 𝜔𝑉 . In this sense, the
resultant VLI should be zero if only path A or B is on. If both path A and B are on, the
situation is different: Vsig includes a term of eT eR cos(V t   0   ) due to the
interference of the two microwave beams and thus the phase shift,    0 , can be
measured .
Figure S1. (a) Schematic graph for reflection experiment setup. (b) Voltage signal for
three different cases: both path A and B connected, path A (horn path) disconnected
and path B (sensor path) disconnected. The voltage signals are normalized for each
frequency for clarify because the absolute value of VLI depends on microwave
frequency.
Now we experimentally demonstrate that the interference of the two microwaves is
the dominating contribution to the measured VLI. The experimental setup is shown in
Figure S1 (a), a horn and sensor were installed closely for emitting and receiving
microwaves, with a copper board was fixed in front. Other configurations are the
same as those in Fig. 1(a) of the paper. For comparison, we measure the signals by
connecting both microwave paths, disconnecting microwave path A, and
disconnecting microwave path B. Since microwave fields depend on the microwave
frequency, we normalize the measured VLI for each frequency for comparison. The
voltage signals as a function of frequency for both path A and B connected (black),
horn path (A) disconnected (red), and sensor path (B) disconnected (blue) are plotted
in Figure S1 (b). The voltage signal due to the interference of the two microwave
beams is at least one order of magnitude greater than the voltage signals produced by
the microwaves from a single path. This confirms that the proposed approach indeed
detects the interference of microwaves instead of the microwaves from the signal
path.
3. Experimental verifications of the detection of microwave phase
In the second experiment, we demonstrate that the phase we detect is the phase
difference between the two paths. The setup is shown in Figure S2 (a), where the
copper board was installed on an adjustable stage. For the path A, the microwaves
are emitted from the horn antenna, reflected by the copper board, and received by the
sensor. Therefore the phase of path A,  , depends on the position of the copper
board, while the phase  0 of path B (microwave directly injected to the sensor)
should be insensitive to the position of the copper board placed in the far-field range.
Figure S2 (a) Schematic graph for reflection experiment setup. Copper board can be
moved forward and backward along the stage to adjust the microwave travelling
distance between the 2 paths. (b) Linear dependence of phase detected as a function of
the copper board’s displacement.
In the experiment, the copper board was initially placed 24cm away from horn (in the
far-field range) and then its position is varied by x . A typical result is shown in
Figure S2(b) at 10 GHz. The measured lock-in phase shows a linear dependent with
x , exactly following the expected relation of   (2x  f / c)  360 , where c is
the speed of light in a vacuum. This experiment clearly indicates the physical meaning
of the phase LI measured by the lock-in amplifier.
4. References
[S1]
A. Wirthman, X. Fan, Y. S. Gui, K. Martens, G. Williams, J. Dietrich, G. E.
Bridges, and C. -M. Hu, Phys. Rev. Lett. 105, 017202 (2010)
[S2]
X. Fan, S. Kim, X. Kou, J. Kolodzey, H. Zhang, and J. Q. Xiao, Appl. Phys.
Lett. 97, 212501 (2010)
[S3]
Z. X. Cao, M. Harder, L. Fu, B. Zhang, W. Lu, G. E. Bridges, Y. S. Gui, and C.
-M. Hu, Appl. Phys. Lett. 100, 252406 (2012).