Optical properties of MetamaterialsBruno Gompf
1.Physikalisches Institut, Universität Stuttgart
Neumann-Curie Principle:“The symmetry group of a crystal is a subgroup of t he symmetry groups of all the physical phenomena which
may possibly occur in that crystal”
Franz Neumann (1841)
Photonic crystals and Metamaterials
Photonic crystals a≈λ
Metamaterials a<λ
Photonic crystals
One-dimensional photonic crystals: dielectric mirror and grating
K. Busch et.al. Physics Reports 444, 101 (2007)
Photonic crystals are a periodic arrangement of dielectric materials with different dielectric constants, or a regular arrangement of holes in a dielectric material.
The period is comparable to the wavelength, leading to band structure effects, as know from electrons in a periodic lattice.
Photonic crystals
Subwavelength hole arrays
Suppressed transmission through ultrathin subwavelength hole arrays
Julia Braun, Bruno Gompf, Georg Kobiela, Martin Dressel, Physical Review Letters 103, 203901 (2009)
Lattice is approximated by homogeneous layer of thickness Lz
with averaged effective dielectric constant ε1
The periodic structure is considered by folding the resulting dispersion relation into the first Brillouin zone
Empty lattice approximation
Dispersion of surface plasmonsfolded back into the first Brillouin zone
Metamaterials
Metamaterials are artificial periodic nanostructures with lattice constants smaller than the wavelength.
The “photonic atoms” are functional building blocks (mostly metallic) with tailered electromagnetic properties, for example, to realize electric as well as
magnetic dipoles.
Light averages over the nanostructure and “sees” a homogenous material with an effective neff
Metamaterials
V.G. Veselago, Sov.Phys. Usp. 10, 509 (1968)
What happens when:
Zero reflection:
Negative index of refraction
G. Dollinger et.al. Optics Express 14, 1842 (2006)
Realization of negative-index materials
K. Busch et.al. Physics Reports 444, 101 (2007)
K. Busch et.al. Physics Reports 444, 101 (2007)
Is it possible to describe a Metamaterialby effective optical parameters?
⇓
Back to the roots(first approach)
Temporal dispersion
)(0),( trkieEtrE ω−=
rrrrrNormal wave with electric field:
per definition (complex tensor) links E and D:ijijij i 21 εεε +=
jiji EDrr
ε=
simplest case: transparent media, small frequency range, large wavelength: εij=const.
→ FT → ),( kErr
ω
If the polarization and thereby the induction at a given time depends on the field strength at previous times:
Pr
PEDrrr
π4+=)(ωεε ijij =
Temporal dispersion
)(ωεε ijij =
in gerneral:
Choosing the unit cell axes as frame, for crystals with symmetry higher than
orthorhombic is diagonal:)(~ ωε ij
In crystals with symmetry lower than orthorhombic the principal axes do not coincide with the crystal axes and the
axes of ε1 and ε2 are not parallel anymore and may rotate with energy:
To obtain Kramers-Kronig consistent ε1i(ω) and ε2i(ω) along the crystallographic axes
an additional transformation T is necessary
Temporal dispersion
)(ωεε ijij =
In uniaxial (ε11= ε22≠ ε33) and biaxial (ε11≠ ε 22 ≠ ε33) crystals an incoming light beam is split into two
orthogonal linear polarized beams (birefringence). These two beams “see” two different optical constants.
Optical activity goes beyond this description and is therefore often treated in textbooks as separate phenomenon
Spatial dispersion
),(),(),( kEkkD jiji
rrrrrωωεω =
If the polarization at a given point in a medium depends on the field in a certain neighborhood a of this point:
)(kijij
rεε =
In terms of Fourier-components spatial dispersion indicates that εij depends on the wave vectork or the wavelength λ.
How strong this dependence is depend on the ratio a/λλλλ with a characteristic dimension of the medium (molecule, lattice constant, nanostructure etc.)
Example:
λ ≈ 1 µm; a ≈ 1nm; n ≈ 10
⇒ a/λ ≈ 10-2 weak spatial dispersion
⇒
k
jijkijij x
Egk
∂∂
+≈ )(),( ωεωεr
Spatial dispersion leads to gyrotropic effects (optical activity)
V.M. Agranovich, V.L. Ginzburg: ”Crystal Optics with Spatial Dispersion, and Excitons”, Springer-Verlag, Berlin 1984
Is it possible to describe a Metamaterialby effective optical parameters?
⇓
Back to the roots(second approach)
Constitutive Relations
HEcB
HcED
01
10
µµζγεε
+=
+=−
−
1=µ 0== ζγ purely dielectric
ζγµε ,,, scalars: bi-isotropic (sugar solution)
ζγµε ,,, tensors: bi-anisotropic
ζγµε ,,, in general complex and frequency dependent
Bi-anisotropy and spatial dispersion are uniquely related to each other*
Magneto-electric coupling and spatial dispersion can not be distinguished
In general these materials are gyrotropic and non-reciprocal
Only bi-isotropic media are optical active and reciprocal
(homogenous magnetic materials and sugar solutions)
HEcB
HcED
01
10
µµζγεε
+=
+=−
−
),(),(),( kEkkD jiji
rrrrrωωεω =⇔
*R.M. Hornreich und S. Shtrikman, Phys. Rev. 171, 1065 (1968).
Ellipsometry on Metamaterials
Reflection described by Fresnel equations
1
~N
2
~N
( ))(),(),(),(~~
22 ωζωγωµωεNN =
The polarization state: Stokes vektors
Presentation of polarization by the Poincare’ sphere
Mueller Matrix formalism
Mueller Matrices: Examples
Ideal linear polarizer Ideal circular polarizer Ideal depolarizer
Isotropic sample
Rotating Analyzer Ellipsometer
How can the Mueller-matrix be measured
monochromator
polarizer
sample
detector
analyzercompensator
Φa
n
ϕ
Visualization of Mueller Matrix Elements
),,( ωϕaijij MM Φ=
M. Dressel, B. Gompf, D. Faltermeier, A.K. Tripathi, J. Pflaum, M. Schubert, Optics Express 16, 19770 (2008)
Non-reciprocity
Reciprocity requires equivalence upon time reversal:
In frequency domain response: kkrr
−→
}1,1,1,1{
),,,( 3210
−−=−−==′
diagT
SSSSTSS
In ellipsometry this is equivalent to:
πϕϕ +→
If we define the matrices:
TMTM T )()(),( 1 πϕϕπϕϕ +−=+Σ −−
then for reciprocal (purely dielectric, no optical activity) samples:
0),( =+Σ− πϕϕ
Samples with combined optical anisotropy and chirality (optical activity) produce non-reciprocity
0),(31 ≠+Σ− πϕϕ
D. Schmidt, E. Schubert, M. Schubert, phys. stat. sol. 205 748 (2008)
Measured contour plots of 20nm Au/glass
Bianisotropic (uniaxial, =0, =0) Bianisotropic (biaxial, =0, =0)
Calculated contour plots
)(00
0)(0
00)(
ωεωε
ωε
z
x
x
)(00
0)(0
00)(
ωεωε
ωε
z
y
x
)(00
010
001
ωµ z
)(00
0)(0
00)(
ωµωµ
ωµ
z
y
x
Bianisotropic (biaxial, =0, =0)
⇒ Σ31=0, Σ21=0, β=0
Bianisotropic (biaxial, = 1, =1)
Calculated contour plots
)(00
0)(0
00)(
ωεωε
ωε
z
y
x
)(00
0)(0
00)(
ωµωµ
ωµ
z
y
x
=100
010
001
γ
=100
010
001
ζ
Bianisotropic (biaxial, =1, =1)
⇒ Σ31≠0, Σ21≠0, β≠0
Σ31 Σ21
β
Gyrotropic, non-reciprocal
P=300 nm; d=200 nm; t=20 nm
Eight-fold symmetry
Measured contour plots of hole array
No purely dielectric response
Summary
• In Metamaterials a/λ<<1 is not fulfilled ⇒ spatial dispersion ⇒
• Metamaterials show magneto-electric coupling
• both leads to a gyrotropic and non-reciprocal optical response
• Müller-Matrix contour plots allow to visualize complex optical behavior
),(),(),( kEkkD jiji
rrrrrωωεω =
HEcB
HcED
01
10
µµζγεε
+=
+=−
−
Neumann-Curie Principle:“The symmetry group of a crystal is a subgroup of t he symmetry groups of all the physical phenomena which
may possibly occur in that crystal”
Really?
Spatial dispersion in αααα-Quartz
The two mirror image crystal structures of left- and right handed quartz
Optic axesA plane linear polarized wave parallel to the optic
axis (no birefringence) split into two circularly polarized waves of opposite hand. The two waves
travels with different velocities nl and nr, but unchanged in form. Afterwards they interfere again
into a linear polarized wave rotated by:
)( rlo
nnd −=
λπφ
Although nl-nr ≈10-4 for 1 mm quartz φ=21.7°
In general the rotary power: n
G
d oλπφρ == jiij llgG =
Principle of superposition:222 )2( ρδ +=∆
)( ijεδδ = Phase shift due to birefringence
Phase shift due to optical activity)( ijgρρ =
Indicatrix of α-quartz:
Full curve: undistorted surface (birefringence)
Dashed curve: superposition of optical activity and birefringence
Transmitted polarization light microscopy
Orthoscopy:
Each pixel in image corresponds to a dot in the sample.
Conoscopy:
Each pixel in image corresponds to a direction in the sample.
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