Compact Metamaterial-Enclosed Wireless Sensors with
Subtle Perception of Internal Physical Events
S1 Mie Scattering Theory for Cloaked Radiators
Consider the electromagnetic plane-wave scattering problem in Fig. 1(b). In the presence of
spherical symmetry, the total fields can be decomposed into orthogonal transverse electric (TE)
and transverse magnetic (TM) fields, accordingly associated with the magnetic vector potential A
and the electric vector potential F [S1-S4]:
E r  
E0
1
 A 
F
j  r    r 
 r 
E
1
H r   0   A +
F
 r 
j  r    r 
(S1)
A and F are solution to the Helmholtz equation for spherical harmonics centered at the origin of
the core-shell configuration,   r  is the permittivity in each medium, and   r   0 . In the
spherical coordinates  r, ,   . The theory could also be extended to an object of arbitrary
geometry. Due to the spherical symmetry and the electrical nature of the problem, the radial
magnetic and electric vector potentials, A = Ar rˆ and F = Fr rˆ , dominating the scattering, can be
expressed as a superposition of orthogonal spherical harmonics:
Ar  E 0
Fr  E 0
cos 

sin 
i

n
j
n 1

n
 j
n 1
2n  1
ki r nTM (r ) P1n (cos  )
n(n  1)
2n  1
ki r nTE (r ) P1n (cos  )
n(n  1)
, where
1
(S2)
anTM jn  k1r   bnTM hn(2)  k1r  if

 nTM (r )  d nTM jn  kc r   enTM yn  kc r  if

TM (2)
if
 jn  k0 r   cn hn  k0 r 
anTE jn  k1r 

 nTE (r )  d nTE jn  kc r   enTE yn  kc r 
 TE
TE (2)
 jn  k0 r   cn hn  k0 r 
0r a
a  r  ac
(S3)
r  ac
if
0r a
if
a  r  ac
if
r  ac
(S4)
and ki   0 i and i  0 /  i are the wave number and characteristic impedance of each
medium, jn  and yn  are the spherical Bessel functions of the first and second kind of order
(2)
n , and hn
 is the spherical Hankel function of the second kind of order n. For the mantle cloaking
case in Fig. 1(c), the racial function is written as:
anTM jn  k1r   bnTM hn(2)  k1r  if 0  r  ac
(r )  
TM (2)
if r  ac
 jn  k0 r   cn hn  k0 r 
(S5)
TE
TE (2)

an jn  k1r   bn hn  k1r  if 0  r  ac
(r )  
TE (2)
if r  ac

 jn  k0 r   cn hn  k0 r 
(S6)

TM
n

TE
n
The scattered fields may now be expanded into multipole fields with scattering coefficients cnTM
and cnTE of order n. As the dipolar radiation can be described as TM1 wave, the coefficient d nTM
describing fields re-radiated by a small dipole [S4], which is correlated to the coefficient describing
the local field, anTM , as: d1TM   n a1TM , where  n   j1n  k03 / 61 1  anTM ,  is the electric
polarizability of the dipole antenna with its expression described in the main text, and  nm is the
Kronecker’s delta function. We note that the electric dipole TM1 harmonic is the only nonzero
scattering contribution from the loaded dipole, i.e. d nTM  0 for n  1, since j1 ()  1 and jn ()  0 for
n  1. By enforcing the proper continuity of the tangential electric and magnetic fields at the
2
interfaces r  a and r  ac , the scattering coefficients can be compactly written as:
TM
n
c
U nTM
U nTE
TE
  TM
, cn   TE
,
U n  jVnTM
U n  jVnTE
(S7)
For the cloak cloaking case [Fig. 1(b)], U nTM , VnTM , U nTE and VnTE are given by
jn  k1a    n h1(2)  k1a 
U
TM
n
TM
n
V
jn  kc a 
yn  k c a 
0
 k2 a jn  kc a    c
 k2 a yn  kc a    c
0
jn  kc ac 
yn  kc ac 
jn  k0 ac 
0
 k2 ac jn  kc ac    c
 k2 ac yn  kc ac    c
 k2 ac jn  k0 ac   0
jn  k1a    n h1(2)  k1a 
jn  kc a 
yn  k c a 
0
 k2 a jn  kc a    2
 k2 a yn  kc a    c
0
jn  kc ac 
yn  kc ac 
yn  k0 ac 
 k2 ac jn  kc ac    2
 k2 ac yn  kc ac    c
 k2 ac y n  k0 ac    0


(2)

  k1a jn  k1a     n  k1a hn  k1a    1



0


(2)

  k1a jn  k1a     n  k1a hn  k1a    1



0
0
,
,
(S8)
where the wavenumbers in different regions are described by the respective constitutive parameters as
k1   10 , kc    c 0 , and k0    0 0 . The scattering coefficients cn can be similarly
TE
obtained by electromagnetic duality, but setting  n  0.
For the mantle cloaking case [Fig. 1(c)], we assume that a patterned metallic surface composed of
subwavelength periodic structures may be modeled as an isotropic and homogenous averaged
surface impedance Z s  RS  iX S , relating the averaged tangential electric field at the surface E tan
to the averaged induced surface current J s as: Etan  Z s J s , where J s  rˆ   H tan  H tan  is
induced by the discontinuity of tangential magnetic fields at the surface. Consider the geometry in
the inset of Fig. 4(d), the expressions of the scattering coefficients for a dipole antenna embedded
3
in the concentric, multilayered spherical structure (i.e. dielectric core, cladding layer and mantle
cloak) are given in the similar notation of (S8) by:
U nTM 
jn  k1a    n h1(2)  k1a 
jn  k 2 a 
yn  k 2 a 
0


(2)

  k1a jn  k1a     n  k1a hn  k1a    1


 k 2 a jn  k 2 a    2
k 2 a yn  k 2a    2
0
0
jn  k2 ac    k2 ac jn  k2 ac   j 2 ac Z s
yn  k 2 ac    k 2 ac yn  k 2 ac   j 2 ac Z s
jn  k 0ac 
0
 k2 ac jn  k2 ac    2
 k2 ac yn  k2 ac    2
 k2 ac jn  k0 ac    0
jn  k1a    n h1(2)  k1a 
jn  k 2 a 
yn  k 2 a 
0


(2)

  k1a jn  k1a     n  k1a hn  k1a    1



 k 2 a jn  k 2 a    2

k 2 a yn  k 2a    2
0
0
jn  k2 ac    k2 ac jn  k2 ac   j 2 ac Z s
yn  k 2 ac    k 2 ac yn  k 2 ac   j 2 ac Z s
yn  k0 ac 
0
 k2 ac jn  k2 ac    2
 k2 ac yn  k2 ac    2
 k2 ac yn  k0 ac    0
,
and
U nTM 
.
(S9)
TE
The scattering coefficients cn can be similarly obtained by electromagnetic duality, but setting
 n  0.
S2 Electromagnetic Cross-Sections
According to the optical theorem [S1], as the scatter interacts with the incident fields, the power
depleted from the incident fields is the sum of the absorbed and scattered fields, i.e. Pext  Pa  Ps .
This power balance is fundamentally restricted by the energy conservation followed by the
causality. The absorbed power is the power entering the scatter (radiative dipole and cloak). The
time-averaged absorbed power can be obtained from the surface integral of the inward flowing
flux of the Poynting vector of the total fields through the surface S surrounding the scatter as:
Pabs  
1
*
Re  Ei  Es    H i  H s    nˆ ds



2 S
(S10)
where n̂ is the unit vector outward pointing the surface normal. Similarly, the scattered power,
which intuitively represents the re-radiated power from the scattering and the overall visibility) is
4
the surface integral of the outward flowing flux of the Poynting vector of the scattered fields, with
its time averaged expression:
Psca 
1
Re Es  H*s   nˆ ds

2 S
(S11)
The total incident power flowing through the surface S is zero:
Pi 
1
Re Ei  H*i   nˆ ds  0

2 S
(S12)
The total extracted power Pext and the time-averaged stored electric and magnetic energy Wext can
be written as a surface integral:
Pext  j 2Wext  
1
Ei  H*s  E*s  Hi   nˆ ds

2 S
(S13)
Therefore, one may obtain:
Pext  
1
Re Ei  H*s  Es  H*i   nˆ ds

2 S
2Wext  
(S14)
1
Im Ei  H*s  Es  H*i   nˆ ds

2 S
(S15)
Naturally, the total extracted power (power extinction) given by Pext  Psca  Pabs . As long as S does
not enclose the sources of the incident field, (S11)-(S15) are valid, independent of the surface
enclosing the scattering object. The electromagnetic cross-section of a scatter, which could be
much larger than its physical cross-section, may provide a quantitative measure for the ability of
an object to scatter or absorb incoming irradiation. The scattering cross-section  sca can be defined
5
as the ratio of total scattered power to the radiant power density incident on surface of the scatter,
i.e.  sca  Psca / Si where Si is the intensity (flux density) of the incident radiation. Similarly, the
absorption and extinction cross-sections are:  abs  Pabs / Si and  sca  Pext / Si . From the Mie
scattering theory calculation, the multipole-based scattering fields can be calculated, and the
scattering, extinction, and absorption cross-sections can be expressed as multipole-based series
expansions [S1-S4]:
 sca
2
 2
k0
 ext 
2
k02
 abs 
2
k02
 (2n  1)  c

n 1
TM 2
n

 (2n  1) Re c
n 1
TM
n

 (2n  1) Re c
n 1
,
(S16)
 cnTE ,
(S17)
 cnTE
TM
n
2
2
2
 cnTE  cnTM  cnTE ,

(S18)
where cnTM and cnTE are scattering coefficients of order n, as described in the section S1.
S2 Electromagnetic Cross-Sections
References
[S1]
C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles
(John Wiley and Sons, New York 1998).
[S2]
A. Alù, and N. Engheta, Phys. Rev. E 72, 016623 (2005).
[S3]
P. Y. Chen, and A. Alù, Phys. Rev. B 84, 205110 (2011).
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[S4]
R. Fleury, J .Soric, and A. Alù, Phys. Rev. B 89, 045122 (2014).
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