Transfer matrix formalism for negative multilayered structures with

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Transfer matrix formalism for negative multilayered structures with point sources J. B. Pendry and

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

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

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

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

(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

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

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)

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

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

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

Perspectives Coordinates transformations for: Perfect corner reflector Perfect cylindrical/spherical lenses Planar meta-waveguides (periodically loaded transmission lines with NRI)