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.