DoD MURI on Metamaterials

Transcription

1 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 1 of 11 DoD MURI on Metamaterials Report of the Imperial College London group to the 7 March teleconference JB Pendr & SA Ramakrishna, The Blackett Laborator Imperial College, London SW7 2BU, UK

2 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 2 of 11 The original lens: Generalising the Lens A slab of n = 1 of thickness d focuses over a distance 2d. It is as if the slab cancels out or compensates for an equal thickness of the vacuum. The slab and the vacuum are complementar to each other. Question: are there other instances of complementar media one cancelling the optical effects of the other?

3 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 3 of 11 More Lenses The original perfect lens envisaged a slab of material with ε= 1, µ= 1. However focussing can be shown to occur under much more general conditions. We shall show that a sstem for which, ε =+ε,, µ =+µ,, z < 0 ) ) ε 2 = ε,, µ 2 = µ,, z > 0 will show focussing. just like a uniform slab in vacuum.

4 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 4 of 11 proof: We start from Mawell s equations,, ),, ) E=+ωµ i µ H H=+ωε i ε E 0 0 and decompose the fields into Fourier components, E E ) z E z,, = ep + ik z ep ik + ik, z < ) z E z,, = ep ik z ep ik + ik, 0< z < d 2 2 where we have assume the Bloch theorem, but need to prove that the proposed solution meets the boundar conditions at z = 0. Substituting into Mawell gives:

5 ,, ), ) k1z E 1 k k ˆ + E1 k k ˆ + kˆ + kˆ E1z k k k ) + k E 1 k, k + E1 k, k k, ;,, ˆ, ) k k k H k k H k k = ωµ µ ˆ ˆ ˆ ˆ = ωµ µ k, ;,, ) k k k H z k k 0 1 1, ) ˆ+ H ) 1 k, k + k + k H1z k, k k H k k 1z 1 k ) k H1 k, k H1 k, k k, ;,, ˆ, ) k k k E k k E k k =+ωε ε ˆ + ˆ ˆ+ ˆ + =+ωε ε k, ;,, ) k k k E z k k DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 5 of 11 ˆ ˆ ˆ ˆ ˆ z ˆ

6 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 6 of 11 Net substitute fields with reversed z- components, E = E and the reversed values of k, εµ,, After some rearrangement: ε ) 2 1, ) =, ) E k k E k k 2 1, ) =, ) E k k E k k 2z 1z, ) =, ) H k k H k k 2 1, ) =, ) H k k H k k 2 1, ) =, ) H k k H k k z 2z 1z 2z 1z k, ;,, ;, ) k k k = ε k k k k 2 1 k, ;,, ;, ) k k k k k k k 2 1 k = k µ = µ

7 ,, ), ) k2z E 2 k k ˆ + E2 k k ˆ kˆ + kˆ E2z k k k ) + k E 2 k, k + E2 k, k k, ;,, ˆ, ) k k k H k k H k k =+ωµ µ ˆ ˆ ˆ ˆ = ωµ µ k, ;,, ) k k k H z k k z ˆ, ) + H ) 2 k, k k + k H2z k, k k H k k 2z 2 k ) + k H2 k, k + H2 k, k k, ;,, ˆ, ) k k k E k k E k k ˆ ˆ ˆ ˆ k, ;,, ) k k k E z k k DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 7 of 11 ˆ ˆ ˆ ˆ ˆ ˆ z = ωε ε + =+ωε ε ˆ

8 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 8 of 11 i.e. the new fields solve Mawell s equations for the new ε 2, µ 2. The fields in the two regions also match across the boundar since the parallel components of E and H are defined to be continuous, and the z components equal and opposite. Therefore we have, ) E z,, =+ a = E, ep ik a = E z,, = a, a > 0 z and therefore the fields outside the lens are repeated inside the lens in inverse order eactl as was the case inside our original uniform slab. In general the fields will be sums over k z, but the result holds for each separatel and therefore for an summation. k z

9 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 9 of 11 Consider the lens we have just made: Even More General Lens + = Opticall speaking, equal amounts of negative and positive space cancel to give a zero space: one side of the sstem is opticall identical to the other

10 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 10 of 11 We can prove that the following smmetrical arrangement will focus light: + + +

11 DoD MURI on metamaterials filed as C:\word\serc\darpa\JBP_teleconf_ doc at 7 March 2003 page 11 of 11 Conclusion If we take an arbitrar spatiall varing εµ, for z < 0, ) ε =+ε z,,, µ =+µ z,,, z < and make a sign reversed and mirror cop of it in z > 0, ) ε = ε,, z, µ = µ,, z, z > then b eliminate successive pairs of matching elements at z ± we can shown that this sstem is equivalent to a null space. Therefore light incident on one side emerges from the other side as if there were no space in between i.e. the sstem constitutes a lens. The lens is also perfect.