Supplementary material for:
Contribution of the electric quadrupole resonance in optical
metamaterials
David J. Cho1, Feng Wang1, Xiang Zhang2 and Y. Ron Shen1, 3
1
Department of Physics, University of California at Berkeley, Berkeley, California
94720, USA
2
5130 Etcheverry Hall, Nanoscale Science and Engineering Center, University of
California at Berkeley, Berkeley, California 94720, USA
3
Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley,
California 97420, USA
Derivation of effective  and

from transmission and
reflection coefficients including the electric quadrupole
We can derive the form of effective  and  including the electric quadrupole
( Q ) contribution by directly considering the transmission and reflection coefficients
from a dielectric-metamaterial interface. Here, we consider the simple case of isotropic
metamaterial. The solution of the wave equation derived from Maxwell equations with
the proper boundary conditions yields the transmission and reflection coefficients in
terms of the refractive index ( n   ) and the impedance ( z   /  ), from which
we can uniquely deduce  and  .
We consider here p-polarized plane wave incident at an oblique angle. Figure 1 shows
the assumed propagation direction and the orientation of the fields. The refractive
indices are n1 and n2 for the dielectric medium and metamaterial, respectively. The
case for s-polarized plane wave yields the same end result.
n1 dielectric
n2 metamaterial
reflected wave
ko
Bry
x
y
Er
1
1
z
Ei
ko
transmitted wave
2
k2
Et
Bty
Biy
incident wave
Fig. 1. Linearly p-polarized plane wave incident on a dielectric-metamaterial interface.
The electric field is parallel to the incident plane. Orientations of the electric and
magnetic fields of incident, transmitted and reflected waves are shown.
1. Refractive index ( n2 ) of metamaterial
The wave equation is derived from Maxwell equations [1] including the magnetic
dipole and electric quadrupole terms. It has the form

4 i P

2 E  ( )2 E   2 [  c M    Q] .
c
c
t
t
With P   E E , M   M B , and Q  i Q kE , the wave equation becomes



k 2 E  ( ) 2 E  4 [( ) 2  E E  k 2  M E  k 2 ( ) 2  Q E ] .
c
c
c
The solution of this wave equation yields an expression for the wavevector k in the
medium.
We find
 2 2  n2 2 
k2

( )
c

2
1  4 E

1  4 M  4 ( )  Q
c
.
(1)
2
2. Impedance (z2) of metamaterial from Fresnel coefficients
We calculate the Fresnel coefficients for reflection and transmission. Care must be
taken in deriving the relations between the fields at the interface because the electric
quadrupole term alters the boundary conditions [2]. Continuity of tangential
components of the E field at the boundary is still valid.
( Ei cos1  Er cos1 )  Et cos 2
(2)
The tangential components of the B field, however, are not continuous at the
boundary in the presence of M and Q . Application of the Stoke’s theorem to the
Maxwell equation,  B 
1 E 4 P


(  c M    Q) , at the boundary yields
c t
c t
t
Biy  4 M iy  Bry  4 M ry  Bty  4 Mty  4 ikoQxz .
assuming that Qxz only exists in medium 2. Knowing that Qij  i Q (ki E j  k j Ei ) and
Bly  4 M ly  Bly /   nEl /  , we find with the relation between magnetic and electric
field as,

n1
( Ei  Er )  n2 [1  4 (  M  ( ) 2  Q ]Et .
1
c
(3)
Solution of Eqs. 2 and 3 gives the transmission and reflection amplitude coefficients.
The coefficients are
t
Et
2(n1 / 1 ) cos 1
2 z1 cos 1


,
Ei [1  4 (   ( ) 2  )]n cos   ( n /  ) cos 
z2 cos 1  z1 cos  2
M
Q
2
1
1
1
2
c

[1  4 (  M  ( ) 2 Q )]n2 cos 1  (n1 / 1 ) cos  2
Er
z cos1  z1 cos  2
c
,
r

 2

Ei [1  4 (   ( ) 2 )]n cos   (n /  ) cos 
z2 cos1  z1 cos  2
M
Q
2
1
1
1
2
c
with
z1  1 / 1 and
z2 
z2   2 /  2 [1, 3]. We immediately recognize that
2

 [1  4 (  M  ( )2 Q )]n2 .
2
c
(4)
3. Effective permittivity (  2 ) and permeability (  2 )
From Eqs. (1) and (4), we obtain
 2  (1  4 E ) ,

2 1  [1  4 (  M  ( ) 2  Q )] .
c
They have exactly the same expressions as those derived in the text. The derivation is
similar and the result holds true for light propagation along high symmetry direction in
a non-isotropic metamaterial. We arrive again at the conclusion that for these cases the
contribution of the electric quadrupole can be included in effective  so that the
metamaterial can be characterized simply by k -independent  and  .
References
[1]
J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1975).
[2]
[3]
V. M. Agranovich et al., Phys. Rev. B 69, 165112 (2004).
M. Dressel, and G. Gruner, Electrodynamics of Solids (Cambridge University
Press, 2002).