Transfer matrix formalism for negative multilayered structures with
- Slides: 11
Transfer matrix formalism for negative multilayered structures with point sources J. B. Pendry and S. Guenneau Condensed Matter Theory Group Imperial College London October 22, 2003
Geometry of the problem x 3 h. N N , N h h 2 h 1 2 = -1+ i , 2 = -1+ i 1 = 1 , 1 = 1 h 0 ● Point source P=P 1 e 1+P 2 e 2+P 3 e 3 Nota Bene : small absorption in negative layers: >0 x 1
Harmonic Maxwell’s system with loss and sources E = i H H = -i E +J ● m( , ) 0 and 0< <| , |< ● J (x) = -i P δ (x-x 0) , P C 3 (Maxwell) (causality) (source) 2< + (NRJ) ● dx | E , H | R 3 (NRJ) Fourier Transform in x 1 and x 2 G = (2 )-1 dx 1 dx 2 G exp(-ik 1 x 1) exp(-ik 2 x 2) R 2
New basis (TE/TM decoupling) e’ 1 = (k 12+k 22)-1/2 (k 1 e 1 + k 2 e 2) e’ 2 = (k 12+k 22)-1/2 (k 1 e 2 - k 2 e 1) e’ 3 = e 3 Remark: k’ 12=k 12+k 22 and k’ 2=0 ; If k 1=k 2=0 we take e’i=ei. (Maxwell) + FT For = , - 0 F/ x 3 = F + P δ(x 3 -h 0) 0 ( 2 -k’ 12)/ F E’ 2 = i H’ 1 P = 0 2 p’ F = 2 H’ 2 -i E’ 1 P = i p’ 1 - k’ 1 p 3/ (h 0)
(NRJ) + h-periodicity in x 3 Floquet-Bloch Decomposition in x 3 G = G (x 3+nh) exp(-2 i nk 3) ~ n Z ~ , k 3 [-1/2, 1/2] ~ F (k 3, x 3+h) = F (k 3, x 3) exp(2 i k 3) F/ x 3 = - ~ 0 ~ F ( 2 -k’ 12)/ 0 δ(x 3 -h 0 -nh) exp(2 i nk 3) +P n Z
Transfer matrix associated with a homogeneous layer m x 3 Tm x 3, m Layer m x 3, m-1 x 1 (m) F (x 3, m) 1 m T m 014 Cos( mhm) T = = 041 F 1 1 m-1 m sin( mhm) - m m-1 sin( mhm) Cos( mhm) (x 3, m-1) m = 1, …, M , m 2= 2 m m-k’ 12
Transfer matrix associated with a « point current source» in layer m 0 x 3 (m. O) m o’’ 2 0 : ● m o’ F 1 (h 0+ +nh) = Pn = exp(2 i nk 3) P P h 0 +nh I 4 Pn F 014 1 1 x 1 (h 0 - +nh) p (m. O)= (m O’’ ) p (m O’ )
Overall matrix for M layers and a point source = (M) … (m 0+1) (m 0 -1) … (1) = T 014 P(n) 1 with T=TM TM-1 … T 1 and P(n) =TM TM-1 … Tm 0+1 Tm 0’’ Pn Theorem: • If there is at least one number m ( = , and m=1, . . , M) with non zero imaginary part, then there exists a unique solution • F[(n+1)h] = TF(nh) + P(n) = exp(2 i k 3) F(nh) = [exp(2 i k 3) I 4 – T] -1 P(n) • Indeed, the matrix exp(2 i k 3) I 4 – T has an inverse for all real frequency
Numerical illustration for a point source P=(1, 0, 0) located at h 0=0. 5 in a layer of thickness 1 with =1, =1. A second layer of thickness 1 consists of meta-material i. e. =-1+i , =-1+i with =0. 001
Numerical illustration for a point source P=(1, 0, 0) located at h 0=0. 5 in a layer of thickness 1 with =1, =1. A second layer of thickness 1 consists of meta-material i. e. =-1+i , =-1+i with =0. 001
Perspectives Coordinates transformations for: Perfect corner reflector Perfect cylindrical/spherical lenses Planar meta-waveguides (periodically loaded transmission lines with NRI)
- Formalistic approach in literature
- Homologous structures example
- Waves are repeating disturbances that transfer
- Same sign add and keep
- Negative velocity negative acceleration
- Constant rightward velocity
- Transfer matrix quantum mechanics
- T matrix method
- Gram matrix style transfer
- Knowledge silo matrix
- Formuö
- Typiska drag för en novell