Using transformation media to manipulate waves. C.T. Chan Hong Kong University of Science and Technology

Transcription

1 Using transformation media to manipulate waves C.T. Chan Hong Kong University of Science and Technology

2 Collaborators Key contributor: Huanyang (Kenyon) Chen Ho Bou, Prof. WJ Wen s group: experimental work Jiaotung U (Prof. H. Ma), Institute of Microsystem and Information Technology Prof. X. Jiang, Fudan U (Prof. Z Jian)

3 Invisibility cloak for EM waves: Transformation media Maxwell s equations at fixed frequency J. B. Pendry, D. Schurig, and D. R. Smith, Science 312, 1780 (2006). G. W. Milton, M. Briane and J. R. Wills, New J. Phys 8, 248 (2006).

4 Metamaterials (a << λ) Negative refractive index Man-made materials with small Cloaking, invisibility Superlens, perfect resolution embedded artificial resonators which gives nearly any value of ε,µ

5 Forward vs reverse Photonic crystals, metamaterials Define the structure first, see the effect on wave propagation later Design requires iteration, search in a large parameter space Transformation media Define the path of light first, obtain the material specifications directly Strange material properties If there were no metamaterials, task would be impossible

6 Invisibility Cloaking without cloak with cloak

7 EM Cloaking at Microwave Frequency Metamaterials provide the spatial profile Reduced material parameters used Proof-of-concept experiment D Schurig, et. al., Science 314, 977 (2006).

8 Some What if s What if the transformation is angular instead of radial? What if we have other kinds of wave? Can transformation media cloak hide everything inside? Can we extend the bandwidth?

9 Invisibility cloak is a radial transform What will happen if we perform a angular transform?

10 An invisible cloak that rotate fields Rotation mapping: r < a: r' = r, z' = z, θ' = θ + θ ; a< r < b: f ( b) f( r) r' = r, z' = z, θ' = θ + θ0 ; f ( b) f( a) r > b: r' = r, z' = z, θ' = θ. 0 b a Space

11 t = θ 0 rf '( r) f( b) f( a) Define auxiliary angle

12 2D Field Rotator t ε = ε ε xx xy ε ε xy yy For TE wave: (Ex,Ey, Hz) ε ε θ τ ε θ τ 2 2 xx = u cos ( + / 2) + v sin ( + / 2) ε = ( ε ε )sin( θ + τ / 2)cos( θ + τ / 2) xy u v ε ε θ τ ε θ τ 2 2 yy = u sin ( + / 2) + v cos ( + / 2) μ = z 1 If f () r = ln(), r t = θ0 ln( b/ a) εu εv μzτ are constants

13 Rotation Transformation: Cloak rotates field observers inside/outside the rotation coating would see a rotated world with respect to each other. Cloak is invisible Appl. Phys. Lett. 90,

14 Inner radius a Inner radius a=0. Magnetic-field distribution in the vicinity of the rotation coating. Power-flow lines in white show the flow of EM power.

15 Rotation Cloaking without cloak With cloak: Object apparently rotated to external observer

16 Same functionality for any kind of source Point source apparently rotated to inner observer

17 Simpler constructions Layered system composing of isotropic materials

18 Shifting by t units x' = x, z' = z and y' = y (for x< x ), x' = x, z' = z and y' = y+ t( x x ) (for x < x< x ), x' = x, z' = z and y' = y+ t( x x ) (for x> x ), t ε t μ t t tt 1 T ' = det( Λ) ΛεΛ, t t tt 1 T ' = det( Λ) ΛμΛ, For x1 < x < x2 t ε ' 1 0 ε a 0 1 t = t 1 0 ε b 0 1

19 Oblique layered system: each layer isotropic Layer thickness << λ Effectively anisotropic t ε normal 2εε 1 2 ε + ε = 0 ε1+ ε 2 2 t ε oblique cosα sinα εu 0 cosα sinα = sinα cosα 0 ε v sinα cosα Comparing equations, there is a correspondence between an oblique layered system and the transformation media

20 Correspondence between an oblique layered system and the transformation media: ε v = + t t t (2 4) / 2 ε u = + t + t t (2 4) / 2 α = τ /2 cosτ = t 2 t + 4 sinτ = t An oblique layered system is mathematically identical to a transformation media with a shift of coordinates TE polarized (magnetic field along the z-direction) H.Y. Chen, C.T. Chan, Phys. Rev. B 78, (2008).

21 Oblique layer as optical element: It shifts a beam laterally (b)the beam passes through the alternative oblique layer system. (c) The beam passes through the corresponding transformation media. The black dotted-dash lines denote the boundary of the shifters. (They should be identical in the effective medium limit) H.Y. Chen, C.T. Chan, Phys. Rev. B 78, (2008). See also, M. Rahm, et al., Phys. Rev. Lett. 100, (2008).

22 Finite element simulation results of the magnetic field distribution near the center of the oblique incident Gaussian beam. (a) The beam is in free space. (b) The beam passes through the wave shifter with a positive shift parameter. (c) The beam passes through the wave shifter with a negative shift parameter. The black dotted-dashed lines indicate the boundaries of the shifters.

23 Finite element simulation results of the magnetic field distribution near the center of the incident Gaussian beam(s). (a) The beam is normally incident propagating through the wave splitter. (b) The beam is obliquely incident propagating through the wave splitter. (c) Two beams are normally incident propagating through the wave combiner. The black dotted-dashed lines indicate the boundaries of the shifters.

24 one dimensional cloak: Reduced cross section Cloak reduces the cross section of the diamondshaped domain to that of a thin-plate (independent of what you put inside the diamondshaped domain) If there is no cloak Similar ideas: see e.g. Jensen Li and J. Pendry,

25 Use layered media as (invisible) wave rotator H.Y. Chen, C.T. Chan, Phys. Rev. B 78, (2008). (a) perfect rotation cloak (transformation media). (b) Layered rotation cloak with 72 layers. (c) with 36 layers. (d) with 18 layers.

26 Experiment: Reduced rotator 166 rectangular Al 6x1 mm 2 plates Split ring measures Hz

27 Measurements at 8 GHz

28 Using a dipole antenna Note the broad band width rotation functionality

29 Acoustic cloak Can we make an object invisible to sound?

30 2D EM cloaking 2D acoustic cloaking: r' = a + r( b a)/ b, φ ' = φ, z'=z 2 r r a ( b a) r, φ, 2 ρr = ρ = λ = r a r b r a S. A. Cummer and D. Schurig, New J. Phys 9, 45 (2007).

31 DC conductivity equations: ( σ ( x) V( x)) = f ( x) ' ( σ '( x ') ' V '( x')) = f '( x ') = f '( x') = f( x) / det A T σ '( x ') Aσ( x) A / det A Acoustic eqns at fixed frequency : 2 1 ω ( p( x)) = p( x) ρ( x) λ( x) 2 1 ω '( ' p '( x')) = p'( x') ρ'( x') λ'( x') 1 1 T = A A /deta ρ'( x') ρ( x) A. Greenleaf, M. Lassas and G. Uhlmann, Physiol. Meas. 24, 413 (2003). G. W. Milton, M. Briane and J. R. Wills, New J. Phys 8, 248 (2006). 2 1 [ V ( x ), ( x ), f ( x )] [ p ( x ω σ ),, ( )] ρ( x) λ( x) p x 2 2 ω ω = /deta λ'( x') λ( x) λ '( x') = λ( x) det A H. Chen and C. T. Chan Appl. Phys. Lett. 91, (2007).

32 Extension to acoustic wave: 3D acoustic cloaking r' = a + r( b a)/ b, θ ' = θ, ϕ'= ϕ b a r b a ( b a) r ρr =, ρ = ρ =, λ = b ( r a) b b ( r a) 2 θ ϕ 3 2 H. Chen and C. T. Chan, APL (2007). See, also S. A. Cummer et al., Phys. Rev. Lett. 100, (2008).

33 Acoustic metamaterials Man made material with embedded subwavelength mechanical resonators, which give effectively any value of density tensor and modulus Negative, small, big values are allowed Anisotropy allowed J. Li and C. T. Chan, Phys. Rev. E 70, (2004).

34 Simple ball-and-spring model Mechanical: K ka ω = sin ( ) M 2 EM: ( )( ) k = εμ ε μ ω Positive M and K What if K and M are negative?

35 Resonance gives negative mass Gap 2 4K 2 ka ω = sin ( ) M eff 2 where M eff = M 2 mω0 + 2 ω ω 2 0 and ω 0 = 2G m Negative M by resonance

36 Standard effective medium for spherical particles composite 1 f 1 f = + κ κ κ eff s 0, { ρ, κ } s s ρeff ρ0 ρs ρ0 = f 2ρ + ρ 2ρ + ρ eff 0 s 0. { ρ, κ } eff eff { ρ, κ } 0 0 J. G. Berryman, J. Acoust. Soc. Am (1980). Effective modulus and density in a composite cannot be negative from standard theory

37 Effective medium with resonances The standard effective medium equations are correct only in linear dispersion regime. New theory needed to take care of resonances: κ 1+ = f κ 3i eff ( kr 0 ) 1+ D s 0 ρeff ρ 3i D = f 2ρ + ρ 1+ eff D ( kr 0 s ) D1 k = ω ρ / κ Dl = Mie scattering coefficient f = volume-filling ratio With resonance: arbitrary values of modulus and density can be realized Jensen Li, CT Chan, Phys. Rev. E 70, (R) (2004)

38 Anisotropy By arranging isotropic resonators in an anisotropic lattice (e.g. rectangle), we can get anisotropic response. For example, the effective density is different along X and Y direction.

39 If If EM cloak can make an object invisible to radar, the acoustic cloak can make an submerged object invisible to sonar

40 Anti-cloak Can transformation media cloak hide everything inside? r r a b r c a b r Cloak What is this?

41 r Anti-cloak c a b Continuation of the line: This gives zero cross section This gives back some cross section? r Material properties: Between [c,a] the ε,µ are negative They cancel some effect of the perfect cloak There exist something that transformation cloak cannot hide

42 2D cylindrical system Two dimensional situations, consider E field along z axis: 11 r k E = 0 Transformation: z εz r r μθ r εz r θ μr θ ( f ( r), θ, z) μ θ μ 0 μr μ 0 ε z ε 0 = = = f r f f rf 1 r ' ff ' ' Either f or f =df/dr is negative implies negative index material

43 Anti-cloak Can we avoid dealing with a negative radius? Avoid the divergence as the function passes through zero Bend the function up

44 TM: (Hx, Hy, Ez) Anticloak folded geometry See, e.g. Leonhardt et al, Milton et al. (a) The coordinate transformation of the cloak and anti-cloak: [a,b] is the cloak [c,a] is the anti-cloak. They are in direct contact. Note that anti-cloak region has negative parameters H.Y. Chen, X. Luo, H. Ma, C.T. Chan Opt. Express 16, (2008)

45 A tiny PEC cylinder with a radius r o A PEC cylinder with a radius d Cloaking shell Anti-Cloak A PEC cylinder with a radius a wearing a partial cloak, reducing the Cross section to r o PEC cylinder with a radius c wearing a partial cloak in contact with an anti-cloak. Cross section is same as d

46 Cloak and Anti-cloak

47 Large cross section Tao Yang et. al. Yu Luo et al.

48 More general transformations b a b a a c r'( r, θ ) = ( r b) + b= r+ b, b c b c b c θ' = θ, 0 θ < 2π, z' = z, a= ρ ( θ), b= ρ ( θ), c= ρ ( θ), μ rr = r' r' ( ) + ( ) r r θ r' r' r r 2 2, μ rθ r ' = μ r θr = θ, r ' r μ θθ r ' = r r ' r, ε zz 1 = r' r' r r.

49 Mirage cylinders : Transformation media can make observer see a cylinder with a different size, different shape, at a different position c ρ ( θ) x cosθ a x cos θ, = = + + A circle centered at x o PEC cylinder with radius a and centered at x o A cylinder with a different radius at origin, coated with the transformation media shell

50 Some subtle properties of cloaks

51 Energy transport velocity of 3D EM cloak b r a b r a 2[(( ) ( ) cos θ) + (( ) ( )sin θ) ] ve = b a r b a r c b dε dμ b a dω dω b b r a 2 b b r a 2 + ( cos θ ) + ( ) b a b a r b a b a r /2 2 r 2 r 2 ( sin θ) [( εr ω )(cos ϕ) ( μr ω )(sin ϕ) ]. the energy transport velocity near the inner boundary is very small

52 Time delay Ray tracing for beams incident at different positions A beam pointing at the origin of the cloak will take infinite time to pass through the cloak H.Y. Chen, C.T. Chan J. Appl. Phys. 104, (2008).

53 Extension to broader frequency range Transformation media equations can be realized at one single frequency, but it cannot be made to be compatible with the causality over an extended range of frequencies By a simple adaptation, the form transformation media equations can be made to be compatible with the causality requirements, which then leads to a simple way of designing a reduced-cross-section cloak for a finite range of frequencies. H.Y. Chen, Z. Liang, P. Yao, X. Jiang, H. Ma, C.T. Chan, PRB 76, (R) (2007).

54 Summary Don t believe your eyes or ears (if they react to only one frequency) If you don t see anything, it may still be there If don t hear a sound, it does not mean it is not there If I am facing you, I may not be It may be bigger (smaller) than it looks If it is here, it may actually be there

55 Thank You

56 Invisibility cloaking t ε = ε ε xx xy ε ε xy yy ε = ε θ + ε θ μ xx z = r 2 2 cos ( ) θ sin ( ) ε = ( ε ε )sin( θ)cos( θ) xy yy r r θ ε = ε θ + ε θ 2 2 sin ( ) θ cos ( ) b r a b a r 2 ( ) ( ) Rotation cloaking ε ε θ τ ε θ τ 2 2 xx = u cos ( + / 2) + v sin ( + / 2) ε = ( ε ε )sin( θ + τ / 2)cos( θ + τ / 2) xy u v ε ε θ τ ε θ τ 2 2 yy = u sin ( + / 2) + v cos ( + / 2) μ = z 1