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  • Optical properties of MetamaterialsBruno Gompf

    1.Physikalisches Institut, Universitt Stuttgart

    Neumann-Curie Principle:The symmetry group of a crystal is a subgroup of the symmetry groups of all the physical phenomena which

    may possibly occur in that crystal

    Franz Neumann (1841)

  • Photonic crystals and Metamaterials

    Photonic crystals a

    Metamaterials a

  • 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 Lzwith averaged effective dielectric constant 1

    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 =rrrrr

    Normal wave with electric field:

    per definition (complex tensor) links E and D:ijijij i 21 +=

    jiji EDrr

    =

    simplest case: transparent media, small frequency range, large wavelength: ij=const.

    FT ),( kErr

    If the polarization and thereby the induction at a given time depends on the field strength at previous times:

    Pr

    PEDrrr

    4+=)( ijij =

  • Temporal dispersion

    )( ijij =

    in gerneral:

    Choosing the unit cell axes as frame, for crystals with symmetry higher than

    orthorhombic is diagonal:)(~ ij

    In crystals with symmetry lower than orthorhombic the principal axes do not coincide with the crystal axes and the

    axes of 1 and 2 are not parallel anymore and may rotate with energy:

    To obtain Kramers-Kronig consistent 1i() and 2i() along the crystallographic axes

    an additional transformation T is necessary

  • Temporal dispersion

    )( ijij =

    In uniaxial (11= 22 33) and biaxial (11 22 33) 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 jijirrrrr

    =

    If the polarization at a given point in a medium depends on the field in a certain neighborhood a of this point:

    )(kijijr

    =

    In terms of Fourier-components spatial dispersion indicates that ij depends on the wave vectork or the wavelength .

    How strong this dependence is depend on the ratio a/ with a characteristic dimension of the medium (molecule, lattice constant, nanostructure etc.)

    Example:

    1 m; a 1nm; n 10

    a/ 10-2 weak spatial dispersion

    k

    jijkijij x

    Egk

    + )(),( r

    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

    +=

    +=

    1= 0== purely dielectric

    ,,, scalars: bi-isotropic (sugar solution) ,,, tensors: bi-anisotropic ,,, 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

    +=

    +=

    ),(),(),( kEkkD jijirrrrr

    =

    *R.M. Hornreich und S. Shtrikman, Phys. Rev. 171, 1065 (1968).

  • Ellipsometry on Metamaterials

    Reflection described by Fresnel equations

    1

    ~N

    2

    ~N

    ( ))(),(),(),(~~ 22 NN =

  • 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

    a

    n

    Visualization of Mueller Matrix Elements

    ),,( aijij MM =

  • 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:

    +

    If we define the matrices:

    TMTM T )()(),( 1 +=+ then for reciprocal (purely dielectric, no optical activity) samples:

    0),( =+

  • Samples with combined optical anisotropy and chirality (optical activity) produce non-reciprocity

    0),(31 +

    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)(

    z

    x

    x

    )(00

    0)(0

    00)(

    z

    y

    x

    )(00

    010

    001

    z

    )(00

    0)(0

    00)(

    z

    y

    x

    Bianisotropic (biaxial, =0, =0)

    31=0, 21=0, =0

  • Bianisotropic (biaxial, = 1, =1)

    Calculated contour plots

    )(00

    0)(0

    00)(

    z

    y

    x

    )(00

    0)(0

    00)(

    z

    y

    x

    =100

    010

    001

    =100

    010

    001

    Bianisotropic (biaxial, =1, =1)

    310, 210, 0

    31 21

    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/

  • Neumann-Curie Principle:The symmetry group of a crystal is a subgroup of the symmetry groups of all the physical phenomena which

    may possibly occur in that crystal

    Really?

  • Spatial dispersion in -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 =

    Although nl-nr 10-4 for 1 mm quartz =21.7

    In general the rotary power: n

    G

    d o == jiij llgG =

    Principle of superposition:222 )2( +=

    )( ij = Phase shift due to birefringence Phase shift due to optical activity)( ijg =

  • Indicatrix of -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.