1. Reminder: E-Dynamics in homogenous media and at interfaces

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1 0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods FDTD Plane-Wave Expansion T-Matrix, Scalar-Wave-Approximation, S-Matrix 2.5 Fabrication 2.6 Non-linear optics and Photonic Crystals 2.7 Quantumoptics 2.8 Chiral Photonic Crystals 2.9 Quasicrystals 2.10 Photonic Crystal Fibers Holey Fibers 3. Metamaterials and Plasmonics 3.1 Introduction 3.2 Background 3.2 Fabrication 3.3 Experiments

2 Plane-Wave Expansion We start with the wave equation for the magnetic field: 1 ω2 H ( r ) = 2 H ( r ) ε (r ) c The corresponding modes of a Photonic Crystal are Bloch states: H(r ) = ( ) i ( k + ) r H kn e Fourier expansion of a periodic function: 1 = ε (r ) ( ) i r κ e

3 Next, we substitute the expansions in the wave equation: ( ) i r κ e i ( k + ) r ω k2n = 2 H kn e c ( ) ( ) i ( k + ) r H kn e A straight forward calculation yields: ( ) i r i ( k + ) r ω k2n H kn e = 2 κ e i k + c ( ) ( )( ) ( ) i ( k + + ) r ω k2n = 2 κ i k + H kn e c, ( ) ( ) i ( k + ) r H kn e ( ) i ( k + ) r H kn e i ( k + + ) r ω k2n κ k + + k + H kn e = 2 c, i ( k + ) r H kn e ( )( ) ( ) ( ) ( )

4 Next, we substitute = + : )( ) ( ) i ( k + ) r ω k2n k + k + H kn e κ = 2 c, ( ( ) ( ) i ( k + ) r H kn e By comparison of coefficients we obtain: ( )( ) ( ) ω k2n κ k + k + H kn = 2 H kn c ( ) ( ) => eigenvalues and expansion coefficients can be computed

5 ( ) ( Additional constraint (Maxwell equations): H kn k + ( ) Ansatz:,1,2 H kn = hkn e,1 + hkn e, 2 where e,1, e, 2, k+ k+ form a right-handed set of vectors. )

6 After some calculation, we obtain: 2 ω i, j, j,i k n j = 1 M k hkn = c 2 hkn e, 2 e, 2 = κ k + k + e e,1, 2 2 with M ki, j ( ) ( ( ) ) e, 2 e,1 e,1 e,1 Some remarks: Partial differential equation -> algebraic eigenvalue equation with an infinite number of eigenvalues (sum over all ) Numerical computation: truncate after sufficiently large number N of reciprocal lattice vectors Problem: sometimes poor convergence due to discontinuities in dielectric function CPU time is proportional to N3

7 Band structure of an empty 2D square-lattice for TE polarization, calculated with plane-wave expansion choose k point choose appropriate base with respect to calculate the eigenvalues of the matrix choose next k point

8 How to implement frequency dependent dielectrics in plane wave expansion?

9 How to implement frequency dependent dielectrics in plane wave expansion? Cutting surface method, O. Toader and S. John, Phys. Rev. E 70, (2004)

10 0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods FDTD Plane-Wave Expansion T-Matrix, Scalar-Wave-Approximation, S-Matrix 2.5 Fabrication 2.6 Non-linear optics and Photonic Crystals 2.7 Quantumoptics 2.8 Chiral Photonic Crystals 2.9 Quasicrystals 2.10 Photonic Crystal Fibers Holey Fibers 3. Metamaterials and Plasmonics 3.1 Introduction 3.2 Background 3.2 Fabrication 3.3 Experiments

11 How to calculate spectra? Finite 1D Photonic Crystal sandwiched between two halfspaces ε air/substrate ε air/superstrate E0 r t ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2 a

12 Reflection and transmission at an interface n1 n2 x0 x E0 t E1 r Field and first derivative have to be continuous across interface: E0 e ik1 E0 e + re ik1 x0 = te ik1r e ik1 x0 = ik 2t e ik1 x0 ik1 x0 ik 2 x0 + E1 e ik 2 x0 ik 2 x0 ik 2 E1 e ik 2 x0

13 Transfer Matrix E0 e ik1 E0 e e ik x ke 10 1 ik1 x0 + re ik1 x0 = te ik1r e ik1 x0 = ik 2t e ik1 x0 ik1 x0 ik 2 x0 + E1 e ik 2 x0 ik 2 E1 e E0 e = ik1 x0 ik 2 x0 r k1e k2e e ik1 x0 ik 2 x0 ik 2 x0 Transfermatrix ik 2 x0 t ik 2 x0 k2e E1 e ik 2 x0 1 t E0 = M 1 M 2 r E1

14 Multiple interfaces Finite 1D Photonic Crystal with N-1 layers and N interfaces ε air/substrate ε air/superstrate E0 r t ε1 ε2 ε1 ε2 ε1 ε2 ε1 ε2 a t E0 = M x0 + a ( N -1) M x0 r E1

15 Transfer Matrix E0 m11 = r m21 m12 t m22 E1 Finally, t and then r can be calculated (E1=0!): E0 = m11t + m22 0 E0 t= ; r = m21t + m22 0 m11 Repeat calculation for each frequency to obtain spectrum.

16 How many periods do we need to obtain a Photonic Crystal?

17 0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods FDTD Plane-Wave Expansion T-Matrix, Scalar-Wave-Approximation, S-Matrix 2.5 Fabrication 2.6 Non-linear optics and Photonic Crystals 2.7 Quantumoptics 2.8 Chiral Photonic Crystals 2.9 Quasicrystals 2.10 Photonic Crystal Fibers Holey Fibers 3. Metamaterials and Plasmonics 3.1 Introduction 3.2 Background 3.2 Fabrication 3.3 Experiments

18 Spectra for 3D Photonic Crystals Usual experimental conditions: Spectra perpendicular to the surface Unpolarized light Finite thickness Samples on a substrate

19 Scalar wave approximation Unpolarized light => neglect polarization E (r ) = kb Ck e i ( k b ) r b Expansion into Bloch functions Assume infinitely extended, periodic (opal) samples i r ε (r ) = ε avg + U e f filling fraction ε avg = fε s + (1 f ) ε b = 2π/d R sphere radius 3f ( ε s ε b ) [ sin(r) R cos(r)] U = 3 ( R ) Rayleigh-ans

20 Scattering occurs along one direction E (r ) = kb Ckb eikb r + Ckb 111 ei ( kb 111 ) r ε (r ) = ε avg + U 111 e i111r Introduce this ansatz into the wave equation { } ω 2 E (r ) = 2 ε (r ) E (r ) c

21 and stay calm 2 2 ω ω 2 k 2Ck e ikr + ( k ) Ck e i ( k ) r 2 U Ck e i ( k ) r 2 U Ck e ikr = c c 2 ω2 ω = 2 ε avgck e ikr 2 ε avgck e i ( k ) r c c Compare coefficients 2 ω 2 ω 2 k ε avg 2 Ck 2 U Ck = 0 c c 2 ω 2 ω 2 2 U Ck + ( k ) ε avg 2 Ck = 0 c c

22 To be solvable, the determinant has to vanish: 2 2 ω k ε avg 2 c 2 ω 2 U c Ck = 0 2 ω Ck 2 ( k ) ε avg 2 c ω 2 2 U c This leads to the dispersion relation (bandstructure): 1 k (ω ) = ± 2 2 ω 2 + ε avg 2 4 c 2 4 ω ω 2ε avg 2 + U 2 4 c c imaginary k (gap) for radical < 0

23 Electric fields in- and outside k 2 ε avg k02 S= k02u e ik0 x + re ik0 x E ( x) = Ck + e ikx + Se i ( k ) x + Ck e ikx + Se i ( k ) x ik 0 x te ( ) ( ) in front inside behind evanescent modes inside the gap To generate heterostructures, or samples on substrates, or colloidal crystals with planar defects, we can now use the transfer-matrix method.

24 Bandstructure 1.2 photon energy (ev ) Γ bloch vectors L n Sphere diameter: 305 nm

25 Bandstructure 1.2 photon energy (ev ) Γ bloch vectors L n Sphere diameter: 305 nm

26 Bandstructure 1.2 photon energy (ev ) Γ bloch vectors L n Sphere diameter: 305 nm

27 Frequency dependent epsilon? 1,2 photon energy (ev ) 1,1 1,0 0,9 0,8 0,7 0,6 0,5 Γ bloch vectors L 2,8 3,0 3,2 3,4 3,6 3,8 4,0 4,2 n Numerical experiment

28 Frequency dependent epsilon Single resonance with resonance frequency ω0 Ω2 ε (ω ) = ε ω 0 ω 2 iγ ω Experimental analogue: two-level systems, e.g., atoms, quantum dots, dye molecules The small gap in inverse opals could be widened

29 Frequency dependent dielectrics photon energy (ev) Γ bloch vectors L n

30 Comparison to experiments Opals with different numbers of layers Transmittance along [111] direction (perpendicular to the surface) Higher bands are NOT included in SWA.

31 The Phase in TM- and SWA-calculations We know the field transmission coefficient: ik (ω ) L Etrans e i ( k ( ω ) k0 ) L t= = t ik0 L = t e Eref e Relation between phase and group velocity φ = ( k (ω ) k0 ) L φ k (ω ) = + k0 L 1 k (ω ) 1 φ 1 = = + vg ω L ω c0 group delay

32 and experiment Same samples as before. Phase can be measured interferometrically. Discrepancies are due to disorder and scattering.

1. Reminder: E-Dynamics in homogenous media and at interfaces

0. Introduction 1. Reminder: E-Dynamics in homogenous media and at interfaces 2. Photonic Crystals 2.1 Introduction 2.2 1D Photonic Crystals 2.3 2D and 3D Photonic Crystals 2.4 Numerical Methods 2.4.1