Download A linear polarization converter with near unity efficiency in

A linear polarization converter with near unity efficiency in
microwave regime
Peng Xu, Shen-Yun Wang, and Wen Geyi
In this file, we demonstrate the derivation of the transmission line model for the presented unit cell
in the main article file.
Figure S1 Cut view of the unit cell
Figure S1 shows a cut view of the unit cell. The region V (bounded by the dashed rectangle) is
defined as the space between the input port surface S in and the metallic ground surface S pec
and the boundaries between unit cells, as illustrated in Fig. S1. By using the complex Poynting
theorem, we obtain
1
S  ds    J*c Edv  j 2  ( wm  we )dv
2
S  Sin  S p  S pec  Sout
V
V
Ò

(1)
where S  E  H* 2 is the Poynting vector, we   E  E* 4 is the electric field energy density,
wm  H  H* 4 is the magnetic field energy density, and J c is the induced conductivity current.
Ignoring the EM energy in the source region V0 , which also stands for the volume occupied by the
bended metallic wires, and is bounded by the red dashed lines, we obtain
1
S  ds    J*c Edv  j 2  ( wm  we )dv
2
S  Sin  S p  S pec  Sout
V0
V V0
Ò

(2)
Assume that the incident electrical field is polarized along x-axis and reflected field is polarized
along y-axis
Ex
Einc  x Ex e jkz Hinc  y Hy e jkz  y
0
e jkz Eref  y Ey e jkz Hinc  x Hx e jkz  x
Ey
0
e jkz (3)
If the input surface is far enough away from the resonator, there are only incident and reflective
fields on the incident port surface. Poynting vector on the input port surface is given by
2
2
Ex
Ey
1
1
1
Sinc  E  H*  Einc  H*inc  Eref  H*ref  z
z
2
2
2
20
20
(4)
where   0 /  0 is the wave impedance of free space. On condition that the incident port
surface Sin is far enough from the metallic resonator and there is no co-polarized reflective wave
on the incident port surface, we obtain
Ò

S  Sin  S p  S pec  Sout
S  ds   Sinc  ds   S  ds 
Sin
Sp
 S  ds   S  ds
S pec
(5)
Sout
where
2
2
2
 Ex 2

Ey
Ex
Ey
Sinc  ds  Px  Py   z
n  z
 n    Px  Py
 Px  Py
  Pinc  Pref (6)

 




Sin


It is reasonable to make the conclusion that there is no net power flow through the periodic
boundary surfaces Sp and the metallic ground plane surface Spec
 S  ds =0,  S  ds  0
Sp
(7)
S pec
Ignoring the loss of the very short coaxial line, the power outflow on the coaxial output port
surface is equal to the power inflow when the terminal port is in open state, so we obtain
1
 S  ds  z 2 I
2
port
Sout
R0  z
2
1
I port R0  0
2
(8)
where R0  50 is the characteristic impedance of the coaxial line. I port is the induced one
way current on the bended wires. So we get
Ò

S  Sin  S p  S pec  Sout
Thus left side of (1) is a real number. We have
S  ds   Pinc  Pref
(9)
 1

 Im   J*c Edv   2  ( wm  we )dv
V V0
 V0 2

(10)
Inspired by the definitions of the stored electric field and magnetic field energy densities [1], we
define the stored electric field and magnetic field energy densities by
we  we  weinc  weref , wm  wm  wminc  wmref
Considering (9) and (10), the stored electric field and magnetic field energy are only confined in
the volume V  V0 . So the total stored electric field energy and magnetic field energy are given by
We 
  w
e
 weinc  weref dv 
V V0
Wm 
 4   E  E
1
*
 Einc  E*inc  Eref  E*ref dv
V V0
inc
ref
  wm  wm  wm dv 
V V0
1
  H  H*  Hinc  H*inc  H ref  H*ref dv

V V0 4
(11)
Based on (11), the stored electrical and magnetic energy can be easily calculated. From the
calculated We and Wm , the total equivalent capacitance and inductance can be expressed by [2]
2Ceff 
IC
2
4 We
2
, 2 Leff 
4Wm
IL
2
(12)
2
where I C  I port  2 Pout R0  2Pinc R0 . Here, we have supposed that, in matching state,
2
the one way power outflow at the coaxial port surface is approximately equal to the power inflow
at the incident port surface
Pout  Pinc  Ex Sin 20
2
(13)
Figure S2 Equivalent circuit model of the unit cell (The inset is the full wave simulation model)
The transmission matrix of the equivalent capacitance and inductance, as indicated in Figure S2,
can be expressed as
1
T 
YL
where Z C   j
0  1 Z C  1 Z C   1
1  0 1  0 1  YL
ZC
0  1  2YL ZC



1   2YL (1  YL ZC ) 1  2YL ZC 
(14)
1
1
, YL   j
. Under matching condition, we have
C
L
Zin 
T11 R  T12

T21 R  T22
1
2
 Leff Ceff
2
j
2
Ceff



2 
1
2
j
1  2
 Z 0  1  2

 L   Leff Ceff 
  Leff Ceff 
 Z0
(15)
where Z 0  L0 C0  0  0  120 is the wave impedance of free space. Making use of (15),
the equation (12) becomes
Ceff

Ceff Z 02  Z 0 Ceff Z 0
Pinc

, Leff 
4 2We R0
2

2
4
(16)
With the calculated stored electrical energy within a unit cell, we get values of the equivalent
capacitance and inductance as Ceff  1.05 pF, and Leff  1.65 nH, respectively. To validate the
equivalent circuit model of the presented resonator, the reflectance of the circuit model (using
ADS) is compared against the full-wave simulation result (using HFSS), as shown in Fig. S3. It is
found that the results of circuit model and full-wave simulation have a reasonable agreement.
Fig. S3 Reflectance of the unit cell obtained from the equivalent circuit model and full-wave
simulation
References
[1] Collin, R. E. and S. Rothschild, “Evaluation of antenna Q”, IEEE Trans. Antennas and
Propagat., vol. AP-12, 23–27, Jan. 1964.
[2] W. Geyi, “A method for the evaluation of small antenna Q,” IEEE Trans. Antennas Propagat.,
vol. 51, pp. 2124–2129, 2003.