Optical properties of Metamaterials Bruno Gompf 1.Physikalisches Institut, Universität 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≈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

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. Subwavelength hole arrays Suppressed transmission through ultrathin subwavelength hole arrays

Julia Braun, Bruno Gompf, Georg Kobiela, Martin Dressel, Physical Review Letters 103, 203901 (2009) Empty lattice approximation

Lattice is approximated by homogeneous layer of thickness Lz ε with averaged effective dielectric constant 1 The periodic structure is considered by folding the resulting dispersion relation into the first Brillouin zone Dispersion of surface plasmons folded 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

What happens when:

V.G. Veselago, Sov.Phys. Usp. 10, 509 (1968)

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 Metamaterial by effective optical parameters?

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Back to the roots (first approach) Temporal dispersion

Normal wave with electric field: r r r rr r r = i( rk −ωt) → → ω E(r,t) E0e FT E( ,k ) per definition ε = ε + iε (complex tensor) links E and D: r r ij 1ij 2ij = ε Di ij E j

ε simplest case: transparent media, small frequency range, large wavelength: ij =const. r r r r If the polarization P and thereby the induction D = E + 4 πP at a given time ε = ε ω depends on the field strength at previous times: ij ij ( ) Temporal dispersion ε = ε ω ij ij ( )

in gerneral: In crystals with symmetry lower than orthorhombic the principal axes do not coincide with the crystal axes and the ε ε axes of 1 and 2 are not parallel anymore and may rotate with energy:

Choosing the unit cell axes as frame, for crystals with symmetry higher than ε~ ω orthorhombic ij ( ) is diagonal:

ε ω To obtain Kramers-Kronig consistent 1i ( ) ε ω and 2i ( ) along the crystallographic axes an additional transformation T is necessary Temporal dispersion ε = ε ω ij ij ( )

ε ε ≠ ε ε ≠ ε ≠ ε 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

If the polarization at a given point in a medium depends on the field in a certain neighborhood a of this point: r r r r r r ε = ε ω = ε ω ω ij ij (k ) ⇒ Di ( ,k ) ij ( ,k )E j ( ,k ) ε In terms of Fourier-components spatial dispersion indicates that ij depends on the wave vector k or the wavelength λ. How strong this dependence is depend on the ratio a/ λλλ 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 r ∂E ε (ω,k ) ≈ ε (ω) + g j ij ij ijk ∂ xk 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 Metamaterial by effective optical parameters?

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Back to the roots (second approach) Constitutive Relations

= εε + −1γ D 0 E c H = −1ζ + µµ B c E 0 H

µ = 1 γ = ζ = 0 purely dielectric

ε,µ,γ ,ζ scalars: bi-isotropic (sugar solution)

ε,µ,γ ,ζ tensors: bi-anisotropic ε,µ,γ ,ζ in general complex and frequency dependent D = εε E + c −1γH r r r r r 0 ⇔ D (ω,k ) = ε (ω,k )E (ω,k ) = −1ζ + µµ i ij j B c E 0 H

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)

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

Ellipsometry on Metamaterials

~ N1

~ N2

Reflection described by Fresnel equations

~ = ~ (ε ω µ ω γ ω ζ ω ) N2 N2 ( ), ( ), ( ), ( ) 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 Visualization of Mueller Matrix Elements monochromator detector

n ϕ

polarizer Φ a analyzer compensator

sample = Φ ϕ ω M ij M ij ( a , , ) 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: r r In frequency domain response: k → −k

′ = = − − S TS (S0 , S1, S2 , S3 ) T = diag ,1,1{ − ,1 − }1 In ellipsometry this is equivalent to: ϕ → ϕ +π

If we define the matrices: −1 T Σ− (ϕ,ϕ +π ) = M (ϕ) −T M (ϕ +π )T then for reciprocal (purely dielectric, no optical activity) samples:

Σ− (ϕ,ϕ +π ) = 0 Samples with combined optical anisotropy and chirality (optical activity) produce non-reciprocity

Σ ϕ ϕ +π ≠ −31 ( , ) 0

D. Schmidt, E. Schubert, M. Schubert, phys. stat. sol. 205 748 (2008) Measured contour plots of 20nm Au/glass Calculated contour plots Bianisotropic (uniaxial, =0, =0) Bianisotropic (biaxial, =0, =0)

ε (ω) 0 0  ε (ω) 0 0   x   x  ε ω Bianisotropic (biaxial, =0, =0) ε ω  0 x ( ) 0   0 y ( ) 0     0 0 ε (ω) 0 0 ε (ω)  z  ⇒ Σ Σ β  z  31 =0, 21 =0, =0    µ (ω) 0 0  1 0 0  x    µ ω 0 1 0   0 y ( ) 0   µ ω   µ ω   0 0 z ( ) 0 0 z ( ) Calculated contour plots

Bianisotropic (biaxial, = 1, =1) Bianisotropic (biaxial, =1, =1)

Σ Σ 31 21

ε (ω) 0 0  1 0 0  x    ε ω γ =  0 y ( ) 0  0 1 0  ε ω     0 0 z ( ) 0 0 1   β  µ (ω) 0 0  1 0 0  x    µ ω ζ = 0 1 0  0 y ( ) 0  ⇒ Σ ≠ Σ ≠ β≠   31 0, 21 0, 0  µ ω  0 0 1  0 0 z ( ) Gyrotropic, non-reciprocal Measured contour plots of hole array

P=300 nm; d=200 nm; t=20 nm

Eight-fold symmetry

No purely dielectric response Summary

λ • In Metamaterials a/ <<1 is not fulfilledr r r r r ⇒ ⇒ ω = ε ω ω spatial dispersion Di ( ,k ) ij ( ,k )E j ( ,k )

• Metamaterials show magneto-electric coupling = εε + −1γ D 0 E c H = −1ζ + µµ B c E 0 H • both leads to a gyrotropic and non-reciprocal optical response

• Müller-Matrix contour plots allow to visualize complex optical behavior 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 ααα-Quartz

A plane linear polarized wave parallel to the optic Optic axes axis (no birefringence) split into two circularly polarized waves of opposite hand. The two waves

travels with different velocities nl and n r, but unchanged in form. Afterwards they interfere again into a linear polarized wave rotated by: πd φ = (n − n ) λ l r o

Although n -n ≈10 -4 for 1 mm quartz φ=21.7° The two mirror image crystal structures of l r left- and right handed quartz φ π ρ = = G G = g ll In general the rotary power: λ ij i j d on

Principle of superposition: ∆2 = δ 2 + 2( ρ)2 δ = δ ε ( ij ) Phase shift due to birefringence ρ = ρ (gij ) Phase shift due to optical activity 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.