Supplementary Information
Active impedance metasurface with full 360 reflection phase tuning
Bo O. Zhu, Junming Zhao, & Yijun Feng
School of Electronic Science and Engineering, Nanjing University, Nanjing 210093,
China.
Correspondence and requests for materials should be addressed to B.O. Z. (bzhu@nju.edu.cn) or Y. F.
(yjfeng@nju.edu.cn)
1. Bandwidth of linear phase variation and considerations for beam
steering application
The linearity of the phase variation is an important issue in application of the impedance
metasurface, especially in beam steering reflectarray antenna. According to the
reflection coefficient formula   
jX s   /  jX s    , and assuming R  0 , the reflection
phase can be expressed as 2arctan( X s /  )   with the range between 0 to 2, where
X s is the imaginary part of the surface impedance. To achieve a linear phase function as
    t  b , it is required that
X s   tan(
t  b  
),
2
( S1)
where, t is the slope and b the coordinate on the phase axis. Equation (S1) indicates that
in order to realize linear reflection phase variation, X s should be similar to a tangent
function of the frequency. Observing Fig. 4(a), we find that the surface reactance within
the two poles and one zero in between behave quite similarly to a tangent function in one
period. Therefore, by optimizing the inductance and capacitance in the half unit cell,
linear reflection phase variation can be obtained approximately within
0 ~ 2
, as
shown by the circuit model results in Fig. S1(a). Outside this phase range, the linearity
can still extend for a certain frequency band at either side resulting in a total linear phase
range of more than
2
. The frequency bandwidth for the linear phase variation is
essentially determined by the separation between the two poles. By tuning the
capacitors in the half unit cell, the linear phase curve can be shifted without too much
distortion, as shown in Fig. S1(a). Furthermore, a frequency band with 360 o full phase
control can be produced when the phase curve shift is larger than its linear variation
frequency range, as marked by the green region in Fig. S1(a). Since the practical tuning
mechanism, such as utilizing varactor diode, always has a finite tuning range, there is a
trade off between the linear phase variation bandwidth and the full phase tuning
bandwidth. This property can be used to control the reflection phase of a certain band or
single frequency signal.
The situation in beam steering application is a little bit different, where linear reflection
phase variation over a certain bandwidth is preferred when the wave is normal incident.
With this property, the reflection phase distribution function over the antenna reflector
will change linearly with frequency. Thus the gradient of the phase distribution on the
reflector keeps unchanged so that the beam direction is not distorted over the signal
frequency band. Furthermore, the reflection phase should be tuned by 360o without the
distortion of linearity within this bandwidth so as to control the beam direction. This
tuning mechanism is illustrated schematically by the green region in Fig. S1(b). Hence,
the linear phase variation should be larger than 360o actually.
Observing the reflection phase depicted in Fig. S1(a), we find that the linearity of either
phase curve is not only within the frequency range delimited by the two poles (denoted
by the green and yellow points), but also extends outward at the two ends of this range.
Thus the linear phase range is larger than 360o, and phase variation is still linear around
the poles. This property can be applied in beam steering. As illustrated in Fig. S1(c), the
phase is linear around the second pole of the blue curve, and if we shift the blue curve
toward the red one through the gray one so that the first pole (green point) after tuning
aligns with the second pole (yellow point) before tuning, the frequency band marked by
the green color will experience a continuous 360o phase change while the linearity of the
phase is conserved in this band. Beam steering function can be realized within this band
without aperture phase gradient distortion. The parameters in the circuit model as given
in the caption of Fig. S1 are all reasonable and within the realizable range in practice.
(a)
(b)
(c)
Fig. S1. Linear phase variation and full phase tuning with constant phase
linearity. (a) Full phase control with linear phase variation by the circuit model. (b)
Sketch of linear phase tuning for beam steering application. (c) Linear phase tuning by
the circuit model. The parameters in the circuit model are L1 = 2.5 nH, L2 = 3.1 nH, L3 =
2 nH, L4 = 2.2 nH, C1 = 0.30 pF (blue) / 0.225 pF (red), C2 = 0.34 pF (blue) / 0.25 pF
(red).
2. Oblique incidence
For the proposed impedance metasurface, we have also investigated the reflectivity and
phase under oblique incident P-polarized and S-polarized waves. The dimensional
parameters of the half unit cell are shown in Fig. 3(a). The electric field is oriented in the
incidence plane for the P-polarized wave and perpendicular to the incidence plane for the
S-polarized wave, as illustrated in the inset of Fig. S2(a). The incident angle is chosen as
45o for the study. Due to the oblique incidence, there is phase delay between
neighboring unit cells so that the EM coupling is different from that under normal
incidence. This leads to the shift of the resonance frequency, as can be seen in Fig. S2.
Furthermore, the wave impedance of the P-polarized wave is smaller than 377 Ω and
that of the S-polarized wave is larger than 377 Ω. Hence, the absorption of the
P-polarized wave is reduced, while that of the S-polarized wave is increased at the poles.
Under both cases, the 360o full phase variation is achieved similar to that under normal
incidence, as shown in Fig. S2(b).
(a)
(b)
Fig. S2. Reflectivity and phase for oblique incidence. Full wave simulation results
for the (a) reflectivity and (b) reflection phase under normal (black), P-polarized (blue)
and S-polarized (red) oblique incidence. The inset illustrates the polarization of P- and Spolarized waves.