Negative index of refraction

University of Ljubljana
Faculty of Mathematics and Physics
Department of Physics
Negative index of refraction
Seminar
Author: Matej Bobnar
Advisor: prof. dr. Martin Čopič
May 2006
Abstract
First, I present theory behind the negative refraction, which was introduced
by Victor Veselago in 1968. I describe how negative permeability
and permittivity lead to negative refractive index and some consequent
phenomena: reversed Doppler and Cerenkov effect, different types of lens.
Next I propose how materials with negative permittivity and permeability
can be made. Finaly, I describe some experiments and present measurements
with such materials. They verify theoretical predictions.
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Contents
1 Introduction 3
2 Refraction at the boundry of a substence with ɛ < 0 and
µ < 0 3
2.1 Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Cerenkov effect . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Lens with the negative index of refraction . . . . . . . . . . 6
3 The search for materials with negative index of refraction 7
3.1 Materials with negative ɛ or µ . . . . . . . . . . . . . . . . . 7
3.2 Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Experimental verification of a negative index of refraction 11
5 Conclusion 14
2

1 Introduction
Victor Veselago, in a paper published in 1968 [1], pondered the consequences
for electromagnetic waves interacting with a hypothetical material
for which both the electric permittivity ɛ and the magnetic permeability
µ were simultaneously negative. Because no naturally occurring material
or compound has ever been demonstrated with negative ɛ and µ, Veselago
wondered whether this apparent asymmetry in material properties was
just happenstance or perhaps had a more fundamental origin. He concluded
that not only should such materials be possible, but if ever found,
they would exhibit remarkable properties unlike those of any known materials.
He termed these materials ”left-handed” (LHM). Among many
properties of LHMs are index of negative refraction, reversed Doppler and
Cerenkov effect and other.
In the last decade the fabrication and measurement of structured metamaterial
having a range of frequencies over which the refractive index was
predicted to be negative for one direction of propagation were reported
[2], [3]. These structures use split ring resonators [4] to produce negative
magnetic permeability over a particular frequncy region and wire elements
to produce negative electric permittivity in an overlapping frequency region.
One must choose the negative root of the index of refraction given
by n = ± √ ɛµ. The experiments [2], [3] confirm that LHMs do indeed
exhibit negative refraction.
2 Refraction at the boundry of a substence
with ɛ < 0 and µ < 0
Figure 1: Passage of a ray through the boundry between two media of different
handedness.
At first it seems as if the refractive index should not change if both
the electric permittivity and the magnetic permeability are both less than
zero. But at closer look, we see that we have to take the minus sign from
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the square root of ɛ and µ. We conclude this, not from the wave equation,
but rather directly from Maxwell equations and the constitutive relations:
rotE = − 1 ∂B
,
c ∂t
(1)
rotB = 1 ∂D
,
c ∂t
(2)
B = µH, (3)
D = ɛE. (4)
For a plane monochromatic wave in which all quantities are proportional
to e ı(kz−ωt) the upper expressions reduce to:
k × E = ω
µH, (5)
c
k × H = − ω
ɛE. (6)
c
E, H and k form a right-handed triplet of vectors if ɛ > 0 and µ > 0 and
if ɛ < 0 and µ < 0 they form a left-handed set.
Independently of the handedness of the two media, the rays passing
from one medium to another must satisfy the boundry conditions [5]:
Et1 = Et2, Ht1 = Ht2,
ɛ1En1 = ɛ2En2, µ1Hn1 = µ2Hn2.
From the first two follows that the x and y components of the fields E
and H in the refracted ray maintain their directions whatever the handedness
of the two media. However the z components change sign, if the
handednesses are diffrent. This happens because in this case the vectors
E and H undergo a reflection relative to the interface of the two media.
They also change in magnitude if ɛ and µ differ in their absolute value.
The energy flux carried by the wave is determined by the Poynting
vector S:
S = c
E × H. (8)
4π
The energy can not pile up on the boundry and is therefore flowing
ahead into the medium. Therefore the Poynting vector is directed into
the medium and because the fields E and H are reflected at the boundry
it also lies on the same side of the z axis as before refraction.
The paralel phase (the paralel component of the wave vector, k||) must
also remain the same at the boundry. From that fact and the direction of
the Poynting vector we conclude that the wave vector is directed in the
opposite direction as is the Poynting vector. Effectivly, this means that
the wave vector is also reflected across the boundry of the two media.
If all three vectors are reflected at the same time, then they form a
left-handed set. The path of the refracted ray is shown in Fig. 1. So when
the second medium is left-handed the refracted ray lies on the opposite
side of the z axis as it would in the normal medium. The direction of
the reflected ray however is always the same. When the handedness of
medium 1 and 2 are diffrent, we have to correct the usual Snell’s law:
sin α p2
= n1,2 =
sin β p1
ɛ2µ2
ɛ1µ1
(7)
, (9)
where the quantity p = 1 for a right-handed medium and p = −1 for
a left-handed medium. From above equation we can see that the index
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of refraction can be negative if the handednesses of the two media are
diffrent. The amplitudes of reflected and refracted light are found by
applying Fresnel’s formulas [6]. A special case is that of a ray passing
from the medium with ɛ1 > 0 and µ1 > 0 into the medium with ɛ2 = −ɛ1
and µ2 = −µ1. We see that the ray is refracted at the boundry, but there
is no reflection.
From that we can see that for right-handed substances S and k are
in the same direction and for left-handed substances they are in opposite
directions. By their directions are also defined group and phase velocities,
vector k is in the direction of the phase velocity and S in the direction
of group velocity. This reversal of phase and group velocity in a medium
implies a simply stated but profound consequence: The sign of the refractive
index, n , must be taken as negative. So the statement: ”left-handed
material” is equivalent to: ”material with the negative index of refraction.”
2.1 Doppler effect
Figure 2: a) Doppler effect in a right-handed substance; b) Doppler effect in a
left-handed substance. A - source of radiation; B - the receiver [1].
First we look at the Doppler shift in the left-handed substance. It is
reversed. Suppose that a detector of radiation situated in the left-handed
material moves relative to a source (for example it is approaching to a
source) which emits a frequncy ω0. Since the phase velocity is reversed
the detector will see the phase of radiation as if it would be coming from
the other side, as is shown in Fig. 2. The frequency received by the
detector will thus be smaller then ω0, not larger like it would be in a
normal medium. Upgraded formula for the Doppler shift is now:
ω = ω0(1 − p v
), (10)
c
where the velocity of the detector v is positive when it is receding from
the source.
2.2 The Cerenkov effect
Just like the Doppler effect, the Cerenkov effect will also be reversed. A
particle moving in a medium with velocity v in a straight line will emit
according to the law e ı(kzz+krr−ωt) . The wave vector of the radiation will
be given by k = kz/cosθ and is in the general direction of the velocity v.
However kr will be different in different media:
kr = p| k2 2 − kz|. (11)
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Figure 3: a) The Cerenkov effect in a right-handed substance; b) The same
effect in a left-handed substance [1].
In a left-handed medium the vector kr will be directed toward the trajectory
of the particle because the energy moves away from the particle to
infinity. The cone of radiation will be directed backward relative to the
motion of the particle. The angle between v and S will be:
c2 cos θ = p
v2n2
. (12)
2.3 Lens with the negative index of refraction
Figure 4: Paths of rays through a) convex and b) concave lenses made of lefthanded
substances, situated in vacuum [1].
For the lenses made of negative index material the paths of rays
through them are shown in Fig. 4. It is seen that the concave and convex
lenses have changed places, since the convex lens has a diverging effect
and the concave lens a converging effect. Also for thin lenses, geometrical
optics - valid for either positive or negative index - gives the result
that the focal length f is related to the lens’s radius of curvature, R, by
f = R/(n−1). The denominator in the focal-length formula implies an inherent
distinction between positive- and negative-index lenses: A material
with n = +1 does not refract electromagnetic fields, whereas a material
with n = −1 does. The result is that negative-index lenses can be more
compact.
With negative refractive index material we can also achieve a focusing
effect by a simple slab of it [1]. Negative refraction by a slab of material
(n = −1) bends a ray of light back toward the axis and thus has a focusing
effect at the point where the refracted rays meet the axis which turns out
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Figure 5: Passage of rays of light through through a plate of thickness d made
of a left-handed substance. A - source of radiation, B - detector of radiation
to be twice. This is revealed by a simple ray diagram. Light transmited
through a slab of thickness l, located a distance d1 from the source comes
to a second focus when,
d2 = l − d1. (13)
The other relevant quantity, the impedance of the medium,
Z =
µ
Z0, Z0 =
ɛ
µ0
ɛ0
(14)
retains its positive sign, so that when both ɛ = −1 and µ = −1, the
medium is a perfect match to free space and interfaces show no reflection.
This can also be seen from the following relation for reflectivity:
r =
Z
Z0
Z
Z0
− 1
+ 1
2
, (15)
where Z0 is the impedance of the vacuum. At the far boundry there
is again an impedance match and the light is perfectly transmited into
vacuum. This however is not a lens in the usual sense because it will not
focus at a point a bundle of rays coming from infinity.
3 The search for materials with negative
index of refraction
3.1 Materials with negative ɛ or µ
First, we take a look at a Drude-Lorentz model [5], [10] of a material (Fig.
6), which conceptually replaces the atoms and moleculs of a real material
by a set of harmonically bound electron oscillators resonant at some frequency
ω0. At frequencies far below ω0, an applied electric field induces
a polarization in the same direction as the applied field. At frequencies
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Figure 6: Typical behaviour of the real and imaginary parts of the dielectric
constant for a Lorentz model of a material.
near resonance, the induced polarization becomes very large, as is typical
in resonance phenomena; the energy is accumulated over many cycles, so
that a considerable amount of energy is stored in the resonator relative
to the driving field. As the frequency of the driving electric field is swept
through the resonance, the polarization flips from in-phase to out-of-phase
with the driving field, and the material exhibits a negative response. If
instead of electrons the material response were due to harmonically bound
magnetic moments, then a negative magnetic response would exist.
Negative materials are quite easy to find. Materials with negative ɛ
include metals (such as silver, gold, and aluminum) at optical frequencies.
Materials with negative µ include resonant ferromagnetic or antiferromagnetic
systems.
However, since negative material parameters occur near a resonance,
there are three important consequences:
1. Negative material parameters will exhibit frequency dispersion. In
fact, it can be seen from the relation
w = ɛɛ0E 2 + µµ0H 2
(16)
that when there is no frequency dispersion nor absorption we cannot
have ɛ < 0 and µ < 0, since in that case the energy density would
be negative[1], [7]. When there is frequency dispersion, the above
relation must be replaced by
w = ∂(ɛω)
∂ω ɛ0E2 + ∂(µω)
∂ω µ0H2 . (17)
In order for the energy w to be positive it is reqired that
∂(ɛω)
∂ω
> 0,
∂(µω)
∂ω
> 0. (18)
2. There will always be absorption. If ɛ ′ and µ ′ are negative (real part
of the permittivity and permeability) than, according to Kramers-
Kronig relations [5], the imaginary part of the permittivity (ɛ ′′ ) and
of the permeability (µ ′′ ) are non-zero in the left-handed material.
3. The usable bandwidth of negative materials will be relatively narrow
compared with positive materials.
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This is why materials with ɛ and µ both negative are not found in
nature. In existing materials, the resonances that give rise to electric
polarizations typically occur at very high frequencies - in the optical for
metals, and at least in the terahertz-to-IR region for semiconductors and
insulators. On the other hand, resonances in magnetic systems typically
occur at much lower frequencies and usually tail off toward the THz and
IR region [9].
3.2 Metamaterials
Figure 7: Sinusni val
Metamaterials are artificial structures. To form an artificial material,
we start with a collection of repeated elements designed to have a strong
response to applied electromagnetic fields. As long as the size and spacing
of the elements are much smaller than the electromagnetic wavelengths of
interest, incident radiation cannot distinguish the collection of elements
from a homogeneous material. We can thus conceptually replace the inhomogeneous
composite by a continuous material described by material
parameters ɛ and µ. At lower frequencies, conductors are excellent candidates
from which to form artificial materials, because their response to
electromagnetic fields is large.
To construct the material with negative ɛ we use a periodic array of
thin conducting wires (Fig. 8(a)). The wire medium can be characterized
by frequency dependent permittivity having the Drude-Lorentz form [3]
ɛ(ω) = 1 −
ω 2 ep
ω 2 − ω 2 e0
+ ıγω , (19)
for which ωep is the electric plasma frequency, ωe0 is a resonant frequency
and γ is a damping factor. In naturally occuring metals, ωep is typically
in visible-UV range, being determined by the inherent properties of the
conduction charge carriers. In the thin wire medium, however, ωep is determined
by radii, spacing and self-inductance of the wires, and can be
scaled to virtually any frequency below which the permittivity is negative
with relatively low damping. Structures are often designed with continuous
wires so that ωe0 = 0.
To build a medium that would respond magnetically to electromagnetic
radiation, a periodic array of conducting split ring resonators (SRRs)
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(a) A periodic array of thin conducting
wires.
(b) A periodic array of conducting split
ring resonators - SRRs.
Figure 8: The basic building blocks of LHM metamaterials [3].
were introduced [4]. A SRR (Fig. 8(b)) is made of two concentric conducting
rings, cut on opposite sides. They are like miniature circuits: A
time-varying magnetic field induces an electromotive force in the plane of
the element, driving currents within the conductor. A gap in the plane
of the structure introduces capacitance into the planar circuit and gives
rise to a resonance at frequency set by the geometry of the element. The
effective permeability of the SRR medium is of the form [4]:
µ(ω) = 1 −
F ω 2
ω 2 − ω 2 m0
+ ıγω
(20)
in which F is constant between zero and unity and ω 2 m0 is a magnetic
resonant frequency.
A metamaterial formed by combining the wire and SRR media possesses
unexpected and unique electromagnetic properties, including negative
index of refraction.
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4 Experimental verification of a negative
index of refraction
Figure 9: Planar wave, launched down the rectangular waveguide, interacts with
a metamaterial sample placed at the center of the circular plates. The detector,
confined to the radius by a bearing mounted arm, sweeps around circumference
of the plates. Absorber (black) is placed inside the plates to minimize reflective
scattering [2].
The design of the experimental apparatus is shown in the Fig.
9. It is based on a planar waveguide geometry, with a central chamber
formed by two semicircular aluminum plates, joined to an extended channel
formed by two rectangular aluminum plates. The samples occupy a
small region near the center of the central chamber and are exited by a
beam launched from the extended chamber.
Figure 10: A unit cell of LHM on the left and on the right a negative index
material formed by SRRs and wires deposited on opposite sides lithographically
on standard circuit board [9].
The left-handed material used in experiments is usually a two or
one dimensional periodic array of copper SRRs and wires on fiber glass
circuit board material. They are assembled and than cut into desired
shape, e.g. a prism for a beam deflection experiments. Fig. 10 depicts a
one unit cell and many of them assembled to form a 2D array.
Transmisson measurement [3] uses rectangular slabs of three types:
wires only; SRRs only; wires and SRRs together. Forward transmittance
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(a) Transmited power versus frequency
through a metamaterial composed of
wires (solid black) and a metamaterial
composed of SRRs (dashed black).
The raw data is superimposed as grey
curves. This results were obtained on
a one-dimensional array of SRRs and
wires.
(b) Measured power through metamaterial
composed of SRRs and wires. For
comperison, the spectra for SRRs only
and wires only is also shown.
Figure 11: The results of the measurement of the transmission through a slab
of LHM [3].
is measured as a function of frequency. From Fig. 11(a) we see that as expected
the transsmited power for SRRs only and wires only is substantially
lowered around the resonance region. However, for both media combined
we see on Fig. 11(b) that where the individual resonances overlap we get
increased transmission.
Refractive properties of the metamaterial sample are investigated
with a wedge fashioned from the SRR and wire unit cells shown in Fig.
12. The angular power distribution from the empty chamber, a Teflon
Figure 12: Schematic of the one-dimensionally isotropic wedge design based on
a 5mm unit cell.
wedge control sample and the SRR/wire metamaterial is compared on Fig.
13. The wedge angle for the Teflon was 34 ◦ . Using this angle with the
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measured peak angle corresponding to the Teflon sample (17.5 ◦ ) yieldes
a value for the refractive index of n = 1.4 [3].
Over the frequency band corresponding to the transmission peak in
Fig. 11(b), the angular power distribution coresponding to the metamaterial
was peaked at angles negative with respect to the surface normal;
that is, the metamaterial indeed behaved as if it had a negative refractive
index. This metamaterial sample, the 3-over-1 cut corresponds to 18.4 ◦
for the average wedge angle. The two peaks shown in Fig. 13 correspond
to two different frequencies within the band, showing the dispersive nature
of the negative index. This angle with the measured peak angle allowes an
Figure 13: Detected power versus angle for the empty chamber (grey), a Teflon
wedge (right hand curve), and the negative index metamaterial sample wedge
(left hand curves) [3].
effective refractive index to be computed for the metamaterial, which is
shown on the Fig. 14 . Because this metamaterial sample was anisotropic,
it is the square root of ɛzµx that is actually being determined.
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Figure 14: The refractive index of the metamaterial sample as a function of
frequency (solid line). The measured peak power of the refracted beam (dashed
line) [3].
5 Conclusion
Negative refraction is a subject with constant capacity for surprise: Innocent
assumptions lead to unexpected and sometimes profound consequences.
This new field has generated great enthusiasm but also controversy,
yet even the controversies have had the positive effect that key
concepts have been critically scrutinized. In the past years, experimental
data have been produced that validate the concepts. As a result, there
is a firm foundation on which to build. Many groups are already moving
forward with applications. The microwave area has naturally been most
productive, because the metamaterials required are easier to fabricate.
In addition to microwave lenses, novel waveguides and other devices are
under consideration. We are not yet done with theory, because the assumption
of negative refraction has many ramifications that are still being
explored and are sure to cast more light on this strange but fascinating
subject.
References
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Design and measurement of anisotropic metamaterials that exhibit
negative refraction, IEICE TRANS. ELECTRON., vol. E87-C, NO.3,
359, (2004).
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[4] D.R. Smith, et al., Composite medium with simultaneously negative
permeability and permittivity, Phys. Rev. Lett. Vol.84, No.18, (2000)
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