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

Transcription

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 Chiral Photonic Crystals 2.7 Photonic Crystal Fibers Holey Fibers 2.8 Quantumoptics 2.9 Quasicrystals 3. Metamaterials

2 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

3 Multiple interfaces Finite 1D Photonic Crystal with N-1 layers and N interfaces E0 t r 0 Fields on left side) = T-Matrix * Fields on right side)

4 Same problem different approach Finite 1D Photonic Crystal with N-1 layers and N interfaces E0 t r 0 Outgoing fields) = S-Matrix * Incident Fields) D.M. Whittaker et al., Phys. Rev. B 60, ) S.G. Tikhodeev et al., Phys. Rev. B 66, )

5 T-Matrix works fine? Why do we need another method? The T-Matrix formalism is inherently unstable since it deals with exponentially decaying and exponentially growing fields. Therefore, the T-Matrix formalism is not suited for more complicated problems e.g. 2D or 3D metallic structures). In contrast to this, exponentially growing fields do not appear in the S-Matrix formalism.

6 Finite extent in z-direction S-Matrix formalism for finite 3D Photonic Crystals Note: The electromagnetic field is a superposition of partial waves diffraction!) with wavevectors kg = Periodic in 2 2 k x + Gx ) k y + G y ) k x + Gx k y + Gy ω 2ε c2 xy-direction Outgoing fields) = S-Matrix * Incident Fields)

7 How to calculate the S-Matrix? 1.) Decompose the structure into layers which are homogenous in z-direction and periodic in xy-direction. 2.) Calculate the eigenmodes of each layer 2D Photonic Crystal problem!). 3.) Calculate transfer matrix for each layer and interface matrices for adjacent layers. 4.) Construct the S-Matrix of the whole 3D Photonic Crystal by an iterative procedure from the transfer- and interface matrices. D.M. Whittaker et al., Phys. Rev. B 60, ) S.G. Tikhodeev et al., Phys. Rev. B 66, )

8 How to calculate the S-Matrix? 1.) Decompose the structure into layers which are homogenous in z-direction and periodic in xy-direction. 2.) Calculate the eigenmodes of each layer 2D Photonic Crystal problem!). 3.) Calculate transfer matrix for each layer and interface matrices for adjacent layers. 4.) Construct the S-Matrix of the whole 3D Photonic Crystal by an iterative procedure from the transfer- and interface matrices.

9 Decompose the structure into layers which are homogenous in z-direction and periodic in xy-direction. Example: For each layer: 1 1 ig r η G ) = dr e S ε r ) Area of unit cell Define matrix η with elements: η G,G = η G G )

10 How to calculate the S-Matrix? 1.) Decompose the structure into layers which are homogenous in z-direction and periodic in xy-direction. 2.) Calculate the eigenmodes of each layer 2D Photonic Crystal problem!). 3.) Calculate transfer matrix for each layer and interface matrices for adjacent layers. 4.) Construct the S-Matrix of the whole 3D Photonic Crystal by an iterative procedure from the transfer- and interface matrices.

11 Ansatz for the eigenmodes magnetic field) in each layer: H n r, z, t ) = [Φ G +Φ x,n G) ex y,n G) ey 1 qn 1 qn ) )] e k x + Gx )ez k y + Gy )ez i k + G ) r eiqn z e iω t Out-off-plane propagation! Numerical simulations: truncate summation after check for convergence!!! Ng reciprocal lattice vectors!

12 Inserting the ansatz into Maxwell s equations yields the following matrix eigenproblem: η 0 1 Φ 2 ε 0 µ 0ω 1 Z K 0 η Φ ) 2 Φ = qn Φ y,n x,n y,n x,n Good exercise: derive this equation! D.M. Whittaker et al., Phys. Rev. B 60, )

13 Inserting the ansatz into Maxwell s equations yields the following matrix eigenproblem: Unit matrix η 0 1 Φ 2 ε 0 µ 0ω 1 Z K 0 η Φ ) Ky G,G = k y + G y ) δ G,G ) G,G = k x + Gx ) δ G,G Kx ) 2 Φ = qn Φ y,n x,n Kyη Ky Z= K η K x y y,n x,n Ky η Kx K x η K x D.M. Whittaker et al., Phys. Rev. B 60, )

14 Inserting the ansatz into Maxwell s equations yields the following matrix eigenproblem: K= KxKx KyKx KxK y K y K y η 0 1 Φ 2 ε 0 µ 0ω 1 Z K 0 η Φ ) Ky G,G = k y + G y ) δ G,G ) G,G = k x + Gx ) δ G,G Kx ) 2 Φ = qn Φ y,n x,n y,n x,n D.M. Whittaker et al., Phys. Rev. B 60, )

15 Inserting the ansatz into Maxwell s equations yields the Φ x,n G1 ) Φ G ) x,n 2 N g following matrix eigenproblem: η 0 1 Φ 2 ε 0 µ 0ω 1 Z K 0 η Φ ) 2 Φ = qn Φ y,n x,n y,n x,n Φ y,n G1 ) Φ G ) y,n 2 N g D.M. Whittaker et al., Phys. Rev. B 60, )

16 Inserting the ansatz into Maxwell s equations yields the following matrix eigenproblem: Eigenvector η 0 1 Φ 2 ε 0 µ 0ω 1 Z K 0 η Φ ) Eigenvalue 2 Φ = qn Φ y,n x,n y,n x,n Note: In contrast to normal band structure calculations, we solve for qn z-component of the wave vector) and not the frequency ω! D.M. Whittaker et al., Phys. Rev. B 60, )

17 How to calculate the S-Matrix? 1.) Decompose the structure into layers which are homogenous in z-direction and periodic in xy-direction. 2.) Calculate the eigenmodes of each layer 2D Photonic Crystal problem!). 3.) Calculate transfer matrix for each layer and interface matrices for adjacent layers. 4.) Construct the S-Matrix of the whole 3D Photonic Crystal by an iterative procedure from the transfer- and interface matrices.

18 Define for each eigenmode Φ,n n a 2Ng dimensional vector: Φ x,n G1 ) Φ x,n GN g ) = Φ y,n G1 ) Φ G ) y,n N g Important: Φ, n does not depend on z!

19 Define two 2Ng x 2Ngdimensional matrices: H = Φ,1,..., Φ, 2 N g ) The eigenvectors Φ, n form the columns of this matrix. and q1 0 κ = q q2 N g 0 0 The eigenvalues qn are the diagonal elements of this matrix.

20 The total magnetic field in each plane z can be written as a superposition of the eigenmodes, propagating along and opposite to the z-axis, i.e. exp iqn z-iω t) and exp -iqn z-iω t), respectively. Define 4Ng dimensional vector of amplitudes: + Amplitude of 2Ng modes A z) A z ) = propagating in z direction. A z) Amplitude of 2Ng modes propagating in z direction. Note: the vector of amplitudes depends only on z, while the eigenmodes only depend on x and y!

21 Alternatively, the total magnetic field can also be Fourierdecomposed into a sum of partial waves plane wave expansion). The following equation allows us to switch between the description of the total magnetic field in terms of amplitudes of the eigenmodes and the in-plane coefficients of the partial waves: H x z) H z ) = H z ) = H, H A z ) y where H x z ) = H x G1, z ) H x G2 N g, z ) ) and H y G1, z ) H y z) = H G, z ) y 2Ng

22 Analog, we can calculate the coefficients of the in-plane electric field components of the plane waves from the amplitudes of the eigenmodes by E y z) = E z ) = Ex z) = ε 0 µ 0ω 1 Z H κ, ε 0 µ 0ω 1 Z H κ A z ) ωε0 ωε0 where E x z ) = ) E x G1, z ) E x G2 N g, z ) and ) E y z) = E y G1, z ) E y G2 N g, z )

23 Next, we define the transfer matrix TL which connects the amplitudes at different planes z and z+l within one layer: A z + L) = TL A z ) It can be written as a diagonal matrix: expi κ L) 0 TL = 0 exp i κ L) Diagonal matrices with on the main diagonal. exp± i qi L), j = 1, 2,,2 N g

24 The in-plane components of the electric and magnetic field must be continuous across the interface between two layers a and b at z = za,b: E z ) = H z) za,b 0 E z ) H z ) z a,b + 0 In order to connect the amplitudes at the interfaces between adjacent layers, we define the interface matrix Tb, a : A z a,b + 0) = Tb,a A z a,b 0)

25 In order to calculate Tb, a, we define the material matrix F of the layer by: E z ) = F A z ) H z ) where ) ε 0 µ 0ω 1 Z H κ F= ωε0 H Thus, the interface matrix is given by: 1 Tb,a = Fb Fa ) ε 0 µ 0ω 1 Z H κ ωε0 H

26 How to calculate the S-Matrix? 1.) Decompose the structure into layers which are homogenous in z-direction and periodic in xy-direction. 2.) Calculate the eigenmodes of each layer 2D Photonic Crystal problem!). 3.) Calculate transfer matrix for each layer and interface matrices for adjacent layers. 4.) Construct the S-Matrix of the whole 3D Photonic Crystal by an iterative procedure from the transfer- and interface matrices.

27 We could now calculate the total through transfer matrix the whole 3D Photonic Crystal: Ttotal = Ts, a N TLN Ta N, a N 1 TL1 Ta1,v As = Ttotal Av However, the transfer-matrix calculations are numerically unstable exponentially growing fields for evanescent modes)!

28 The total scattering matrix S v, s connects the incident field with the outgoing field: + As Av = S v,s + Av As

29 The total scattering matrix S M + 1 of a system containing M+1 layers can be constructed from the total scattering matrix S M of the system with M layers S11 S M = S 21 S12 S 22 and the inverse transfer matrix T through the additional M+1)th layer T11 T12 T = T21 T22 For example, if we add an interface with interface matrix Tb,a, 1 then T = Ta,b = Tb,a.

30 The scattering matrix connects the amplitudes for the existing M layers: + AM S11 = Av S 21 S12 S 22 + Av AM The inverse transfer matrix connects the amplitudes between layers M and M+1: + + AM T11 T12 AM + 1 = AM T21 T22 AM + 1 This yields a system of equations for the amplitudes in layer M+1. Solving for the amplitudes in layer M+1 yields

31 after a lengthy calculation) the construction rule: QS11 SM + 1 = S 21 + S 22T21QS11 where P = S12T22 T12 ) QP S 22T21QP + S 22T22 and Q = T11 S12T21 ) Exercise: derive this formula for the 1D case! 1.

32 The total scattering matrix of the system can be calculated iteratively: We start from the obvious condition S v,v = 1. Next, we add the interface of the first layer and calculate the corresponding scattering matrix by the procedure described above. Then, we account for the first layer itself and calculate the new scattering matrix.... Finally, we add the interface between the last layer and the substrate and obtain the total scattering matrix.

33 And again: comparison to experiments Experimental spectra at different angles

34 And again: comparison to experiments Bandstructure expectations

35 experiment S-Matrix