Structured matrices in the computation of band spectra of photonic - PowerPoint PPT Presentation

Due giorni di Algebra Lineare Numerica Structured matrices in the computation of band spectra of photonic crystals Pietro Contu, Cornelis van der Mee, and Sebastiano Seatzu Universit` a degli Studi di Cagliari 17 Febbraio 2012

Outline 1 Photonic Crystals (PC) 2 2D PC 3 FDFD Method 4 FDFE Method 5 Numerical Results 6 3D Photonic Crystals 7 Conclusions

Outline 1 Photonic Crystals (PC) 2 2D PC 3 FDFD Method 4 FDFE Method 5 Numerical Results 6 3D Photonic Crystals 7 Conclusions

Outline 1 Photonic Crystals (PC) 2 2D PC 3 FDFD Method 4 FDFE Method 5 Numerical Results 6 3D Photonic Crystals 7 Conclusions

Outline 1 Photonic Crystals (PC) 2 2D PC 3 FDFD Method 4 FDFE Method 5 Numerical Results 6 3D Photonic Crystals 7 Conclusions

Outline 1 Photonic Crystals (PC) 2 2D PC 3 FDFD Method 4 FDFE Method 5 Numerical Results 6 3D Photonic Crystals 7 Conclusions

Outline 1 Photonic Crystals (PC) 2 2D PC 3 FDFD Method 4 FDFE Method 5 Numerical Results 6 3D Photonic Crystals 7 Conclusions

Outline 1 Photonic Crystals (PC) 2 2D PC 3 FDFD Method 4 FDFE Method 5 Numerical Results 6 3D Photonic Crystals 7 Conclusions

Photonic Crystal: What is it? R 3 , satisfies the periodicity condition R 3 , where m 1 , m 2 Photonic crystals are dielectric media whose dielectric constant ε ( x ), with x ∈ ε ( x + m 1 a 1 + m 2 a 2 + m 3 a 3 ) = ε ( x ) for certain linearly independent vectors a 1 , a 2 , a 3 ∈ and m 3 are arbitrary integers. The periodicity of the dielectric constant ε ( x ) causes optical properties which are similar to the electronic properties for semiconductor crystals with a periodic potential. Photonic crystals exhibit frequency intervals where incident light can propagate ( bands ) and frequency intervals in which incident light cannot propagate ( band-gaps ).

Physical Assumptions R 3 → R . In order to study photonic crystals we have to refer to Maxwell’s equations and cast them into the photonic crystals frame. Isotropy and linearity yield: D = ε ( r ) E , B = µ ( r ) H . (1) Magnetic permeability constant ( µ ( r ) ≃ 1): B = H . Lossless media: ε ( r ) : In a photonic crystal we don’t have free charge ( ρ = 0) and free current ( J = 0). We seek time-harmonic modes: H ( r , t ) = H ( r ) e i ω t , E ( r , t ) = E ( r ) e i ω t . (2)

Physical Assumptions R 3 → R . In order to study photonic crystals we have to refer to Maxwell’s equations and cast them into the photonic crystals frame. Isotropy and linearity yield: D = ε ( r ) E , B = µ ( r ) H . (1) Magnetic permeability constant ( µ ( r ) ≃ 1): B = H . Lossless media: ε ( r ) : In a photonic crystal we don’t have free charge ( ρ = 0) and free current ( J = 0). We seek time-harmonic modes: H ( r , t ) = H ( r ) e i ω t , E ( r , t ) = E ( r ) e i ω t . (2)

Maxwell’s Equations for photonic crystals Maxwell equations, which govern light transmission in photonic crystals, reduce to the following system of equations: ∇ · [ ε E ] = 0 , [Coulomb’s law] ∇ × H − i √ ηε E = 0 , [Amp` ere’s law] ∇ × E + i √ η H = 0 , [Faraday’s law] ∇ · H = 0 , [Absence of free magnetic poles] where √ η = ω c . We apply Bloch’s theorem : E ( x ) = e i k · x E ( x ), H ( x ) = e i k · x H ( x ), where E ( x + m 1 a 1 + m 2 a 2 + m 3 a 3 ) = E ( x ) and H ( x + m 1 a 1 + m 2 a 2 + m 3 a 3 ) = H ( x ), we get: ∇ · [ ε ( x ) E ( x )] + i k · [ ε ( x ) E ( x )] = 0 , ∇ × H ( x ) + i [ k × H ( x )] − i √ ηε ( x ) E ( x ) = 0 , ∇ × E ( x ) + i [ k × E ( x )] + i √ ηε ( x ) H ( x ) = 0 , ∇ · [ H ( x )] + i k · H ( x ) = 0 .

Maxwell’s Equations for photonic crystals Maxwell equations, which govern light transmission in photonic crystals, reduce to the following system of equations: ∇ · [ ε E ] = 0 , [Coulomb’s law] ∇ × H − i √ ηε E = 0 , [Amp` ere’s law] ∇ × E + i √ η H = 0 , [Faraday’s law] ∇ · H = 0 , [Absence of free magnetic poles] where √ η = ω c . We apply Bloch’s theorem : E ( x ) = e i k · x E ( x ), H ( x ) = e i k · x H ( x ), where E ( x + m 1 a 1 + m 2 a 2 + m 3 a 3 ) = E ( x ) and H ( x + m 1 a 1 + m 2 a 2 + m 3 a 3 ) = H ( x ), we get: ∇ · [ ε ( x ) E ( x )] + i k · [ ε ( x ) E ( x )] = 0 , ∇ × H ( x ) + i [ k × H ( x )] − i √ ηε ( x ) E ( x ) = 0 , ∇ × E ( x ) + i [ k × E ( x )] + i √ ηε ( x ) H ( x ) = 0 , ∇ · [ H ( x )] + i k · H ( x ) = 0 .

2D: TE and TM Modes When k z = 0, the modes of every two-dimensional photonic crystal can be classified into two distinct polarizations: either ( H x , H y , E z ) or ( E x , E y , H z ). Figura: TM mode: the magnetic Figura: TE mode: the electric field field is confined to the xy plane. is confined to the xy plane.

2D: TE and TM Eigenvalue Equations In the TM mode we have to study spectral eigenvalue problem for the Helmholtz equation � ∂ 2 ψ ∂ 2 x + ∂ 2 ψ � − = ηε ( x , y ) ψ (3) ∂ 2 y and in the TE mode we have to solve the following � � 1 − ∇ · ε ( x , y ) ∇ ψ = ηψ (4) (where ε ( x , y ) = n 2 ( x , y )). In (3) the electric field is given by (0 , 0 , ψ ( x , y )) T , whereas in (4) the magnetic field is given by (0 , 0 , ψ ( x , y )) T . The main goal is to find the eigenvalues η .

Photonic Crystals in 2 Dimensions Basically, we study two numerical methods for the following two cases: As an example, in the connected case:

Prevailing Numerical Methods Time Domain Methods 1) Plane Wave Expansion ( PWE ) Method; 2) Finite Difference Time Domain ( FDTD ) Method. Frequency Domain Methods 1) Finite difference frequency domain ( FDFD ) method; 2) Fourier expansion ( FE ) method; 3) Finite element frequency domain ( FEFD ) method.

FDFD Method We get the 2-D (modified) Helmholtz equations for TE modes ε k · ∇ φ + � k � 2 � 1 � � 1 � − i 1 − ∇ · ε ∇ φ − i ∇ · ε k φ φ = ηφ, (5) ε and for TM modes − ∇ 2 φ − 2 i k · ∇ φ + � k � 2 φ = ηεφ, (6) under the following periodicity conditions φ ( x , 0) = φ ( x , b ) , φ (0 , y ) = φ ( a , y ) , ∂φ ∂ y ( x , 0) = ∂φ ∂φ ∂ x (0 , y ) = ∂φ ∂ y ( x , b ) , ∂ x ( a , y ) .

FDFD Method Let us introduce the grid points � � ja n , lb x j , l = , m where j = 0 , 1 , . . . , n , n + 1 and l = 0 , 1 , . . ., m , m + 1 . Then finite differencing Eq. (5) (TE modes) and Eq. (6) (TM modes) yields, for h x = a / n and h y = b / m , � 1 � 1 + 1 � � − 1 � 1 � � − 1 � − ik x + ik x 1 φ j +1 , l + 1 + φ j − 1 , l 2 h 2 2 h 2 ε j +1 , l ε j , l h x ε j , l ε j − 1 , l h x x x � 1 � 1 + 1 � � − 1 � 1 � � − 1 � − ik y + ik y + 1 φ j , l +1 + 1 + φ j , l − 1 2 k 2 2 h 2 ε j , l +1 ε j , l h y ε j , l ε j , l − 1 h y y y � � 2 � � 1 + 2 1 � 1 + k 2 + + x 4 h 2 ε j +1 , l ε j , l ε j − 1 , l x � � 2 � �� 1 + 2 1 + k 2 + 1 + φ j , l = ηφ j , l , (7) y 4 h 2 ε j , l +1 ε j , l ε j , l − 1 y

FDFD Method − φ j +1 , l − 2 φ j , l + φ j − 1 , l − φ j , l +1 − 2 φ j , l + φ j , l − 1 h 2 h 2 x y φ j +1 , l − φ j − 1 , l φ j , l +1 − φ j , l − 1 + [ k 2 x + k 2 − 2 ik x − 2 ik y y ] φ j , l 2 h x 2 h y = ηε j , l φ j , l , (8) Equations (8) and (17) can both be written in the form ( C − η D )Ψ = 0 Modi TM Modi TE C two-index sparse circulant C positive semidefinite matrix; sparse hermitian matrix; D diagonal matrix with D identity matrix of order positive entries. mn .

FDFD Method − φ j +1 , l − 2 φ j , l + φ j − 1 , l − φ j , l +1 − 2 φ j , l + φ j , l − 1 h 2 h 2 x y φ j +1 , l − φ j − 1 , l φ j , l +1 − φ j , l − 1 + [ k 2 x + k 2 − 2 ik x − 2 ik y y ] φ j , l 2 h x 2 h y = ηε j , l φ j , l , (8) Equations (8) and (17) can both be written in the form ( C − η D )Ψ = 0 Modi TM Modi TE C two-index sparse circulant C positive semidefinite matrix; sparse hermitian matrix; D diagonal matrix with D identity matrix of order positive entries. mn .

FDFD Method ¯ α β 0 β γ 0 0 0 0 0 0 0 γ ¯ 0 0 0   ¯ β α β 0 0 γ 0 0 0 0 0 0 0 ¯ γ 0 0   ¯  0 β α β 0 0 γ 0 0 0 0 0 0 0 γ ¯ 0    ¯  β 0 β α 0 0 0 γ 0 0 0 0 0 0 0 ¯ γ    ¯  γ ¯ 0 0 0 α β 0 β γ 0 0 0 0 0 0 0    ¯  0 γ ¯ 0 0 β α β 0 0 γ 0 0 0 0 0 0    ¯   0 0 γ ¯ 0 0 β α β 0 0 γ 0 0 0 0 0   ¯   0 0 0 ¯ γ β 0 β α 0 0 0 γ 0 0 0 0   C = , ¯   0 0 0 0 ¯ γ 0 0 0 α β 0 β γ 0 0 0    ¯  0 0 0 0 0 γ ¯ 0 0 β α β 0 0 γ 0 0    ¯  0 0 0 0 0 0 γ ¯ 0 0 β α β 0 0 γ 0    ¯  0 0 0 0 0 0 0 γ ¯ β 0 β α 0 0 0 γ     ¯ γ 0 0 0 0 0 0 0 ¯ γ 0 0 0 α β 0 β     ¯ 0 γ 0 0 0 0 0 0 0 ¯ γ 0 0 β α β 0     ¯ 0 0 γ 0 0 0 0 0 0 0 γ ¯ 0 0 β α β   ¯ 0 0 0 γ 0 0 0 0 0 0 0 γ ¯ β 0 β α