Recent Advances on Mathematical Analysis and Simulations of - PowerPoint PPT Presentation

Recent Advances on Mathematical Analysis and Simulations of Invisibility Cloaks with Metamaterials Jichun Li University of Nevada Las Vegas (UNLV) Jichun Li (UNLV) June 27, 2018 1 / 62

Introduction to electromagenetic cloaking with metamaterials 1 Cylindrical cloak in time-domain 2 Arbitray star-shaped cloak 3 Carpet cloak model 4 Other Applications 5 Summary 6 Jichun Li (UNLV) June 27, 2018 2 / 62

Introduction to electromagenetic cloaking with metamaterials 1 Cylindrical cloak in time-domain 2 Arbitray star-shaped cloak 3 Carpet cloak model 4 Other Applications 5 Summary 6 Jichun Li (UNLV) June 27, 2018 3 / 62

Invisibility cloak with metamaterials • Science, vol.312 (June 23, 2006): “Controlling Electromagnetic Fields” (by J.B. Pendry, D. Schurig, D.R. Smith) [Cited 4510 times as 2/25/15; 6115 times as 4/8/17; 6404 times as 8/26/17; 6785 times as 3/15/18] • Science, vol.312 (June 23, 2006): “Optical Conformal Mapping” (by Ulf Leonhardt). [Cited 2369 times as 2/25/15; 3110 times as 4/8/17; 3250 times as 8/26/17; 3401 times as 3/15/18] • Science, vol.314 (Nov. 10, 2006): “Metamaterial Electromagnetic Cloak at Microwave Frequencies” (by Schurig, Mock, etc.) [Cited 3689 times as 2/25/15; 5037 times as 4/8/17; 5289 times as 8/26/17; 5632 times as 3/15/18] • Science, vol.328 (Apr. 16, 2010): “Three-Dimensional Invisibility Cloak at Optical Wavelengths” (by Ergin, Stenger, Brenner, Pendry, Wegener) [Cited 918 times as 3/15/18] • Greenleaf, Lassas and Uhlmann (2003) for conductivity equations; Approx/near cloaking: Milton, Nicorovici (May 3, 2006), Bouchitte, Schweizer (2010), Ammari etc (2013), Kohn, Weinstein etc (2008, 2014), G. Bao, J. Zou, H.Y. Liu, J.Z. Li, ... • Numer. methods: J. Li, Huang, Yang (2011, 2012,...), Z. Xie, C. Chen, etc (CiCP2016), Li-Lian Wang etc (CiCP2015, CMAME2016), D. Liang etc (JSC2016), Brenner, Gedicke, Sung (JSC2016, M2AS2017), D. Liang etc (JSC2016), J. Liu (OP2014), ... Jichun Li (UNLV) June 27, 2018 4 / 62

Figure 1 : (A) The simulation of the cloak with the exact material properties, (B) the simulation of the cloak with the reduced material properties, (C) the experimental measurement of the bare conducting cylinder, and (D) the experimental measurement of the cloaked conducting cylinder. Source: D. Schurig et al, Science, V.314, Nov. 2006, 977-980. Invisible to an incident plane wave at 8.5 GHz. Jichun Li (UNLV) June 27, 2018 5 / 62

Figure 2 : 2D microwave cloaking structure (background image) with a plot of the material parameters that are implemented. Source: D. Schurig et al, Science, V.314, Nov. 2006, 977-980. Reduced parameters: b − a ) 2 , µ r = ( r − a b r ) 2 , µ θ = 1. Exact parameters: ε z = ( b − a ) 2 r − a b r , µ r = r − a r ε z = ( r , µ θ = r − a Jichun Li (UNLV) June 27, 2018 6 / 62

Short math history • IMA Hot Topics Workshop on ”Negative Index Materials” during Oct.2-4, 2006. • CSCAMM at U. of Maryland’s workshop “Electromagnetic Metamaterials and Their Approximations: Practical and Theoretical Aspects”, Sept. 22-25, 2008. (Organized by E. Tadmor, G. Uhlmann, M. Vogelius etc) • UCLA’s IPAM workshop on ”Metamaterials: Applications, Analysis and Modeling”, Jan. 25-29, 2010. (Organized by S. Brenner, M.-C. Calderer, R. Kohn, J. Li, G. Milton, C.-W. Shu, etc) • Tsinghua Sanya International Mathematics Forum (TSIMF)’s workshop ”Mathematical Analysis of Metamaterials and Applications”, Dec.5-9, 2016. (Organized by J. Li, P . Monk, Y. Huang) • Books on DNG: Caloz, Itoh (2005); Eleftheriades, Balmain (2005); Engheta, Ziolkowski (2006), Shalaev (2007), Krowne, Zhang (2007), Marques, Martin, Sorolla (2008), Markos, Soukoulis (2008), Y. Hao, R. Mittra (2008), W. Cai, V. Shalaev (2009), Cui, Smith, Liu (2010), U. Leonhardt, T. Philbin (2010), R.V. Craster, S. Guenneau (2012), D. H. Werner, D.-H. Kwon (2013), ... Jichun Li (UNLV) June 27, 2018 7 / 62

Form invariant property for Maxwell’s equations Theorem 1 Under a coordinate transformation x ′ = x ′ ( x ) , the Maxwell’s equations ∇ × E + j ωµ H = 0 , ∇ × H − j ωε E = 0 , (1) keep the same form in the transformed coordinate system: ∇ ′ × E ′ + j ωµ ′ H ′ = 0 , ∇ ′ × H ′ − j ωε ′ E ′ = 0 , (2) where all new variables are given by E ′ ( x ′ ) = A − T E ( x ) , H ′ ( x ′ ) = A − T H ( x ) , A = ( a ij ) , a ij = ∂ x ′ i , (3) ∂ x j and µ ′ ( x ′ ) = A µ ( x ) A T / det ( A ) , ε ′ ( x ′ ) = A ε ( x ) A T / det ( A ) . (4) Jichun Li (UNLV) June 27, 2018 8 / 62

Introduction to electromagenetic cloaking with metamaterials 1 Cylindrical cloak in time-domain 2 Arbitray star-shaped cloak 3 Carpet cloak model 4 Other Applications 5 Summary 6 Jichun Li (UNLV) June 27, 2018 9 / 62

Cylindrical cloak in time domain The cloak modeling: ∂ B B B Faraday ′ s Law : ∂ t = − ∇ × E E E , ∂ D D D Ampere ′ s Law : ∂ t = ∇ × H H H , constitutive relations : D D D = ε E E E , B B B = µ H H H , E H E and H E H : electric and magnetic fields, D D D and B B B : electric and magnetic flux densities, ε and µ : cloak permittivity and permeability. Cylindrical cloak: Pendry et al (Science 2006): ε r = µ r = r − R 1 r , ε φ = µ φ = , r − R 1 r � � 2 r − R 1 R 2 ε z = µ z = , R 2 − R 1 r R 1 and R 2 : inner and outer radius of the cloak. Jichun Li (UNLV) June 27, 2018 10 / 62

Cylindrical cloak in time domain Transforming the polar coordinate system to the Cartesian coordinate system, and using the Drude model for the permittivity: ω 2 p ε r ( ω ) = 1 − ω 2 − j ωγ , we obtain � ∂ 2 � ∂ t 2 + γ ∂ ∂ t + w 2 ε 0 ε φ E E E p � ∂ 2 � � ∂ 2 � ∂ t 2 + γ ∂ ∂ t 2 + γ ∂ ∂ t + w 2 = M A D D D + ε φ M B D D D , p ∂ t where we denote D = ( D x , D y ) ′ and � � � � sin 2 φ cos 2 φ − sin φ cos φ sin φ cos φ M A = , M B = . sin 2 φ cos 2 φ − sin φ cos φ sin φ cos φ Jichun Li (UNLV) June 27, 2018 11 / 62

Cylindrical cloak in time domain Permeability using the Drude model: � � ω 2 R 2 pm µ z ( ω ) = A 1 − , A = , ω 2 − j ωγ m R 2 − R 1 ω pm > 0 and γ m ≥ 0: magnetic plasma and collision frequencies. � ∂ 2 � � ∂ 2 � ∂ ∂ ∂ t + ω 2 ∂ t 2 + γ m B z = µ 0 A ∂ t 2 + γ m H z . pm ∂ t In summary, ∂ B B B ∂ t = − ∇ × E E E , (5) ∂ D D D ∂ t = ∇ × H H H , (6) � ∂ 2 � � ∂ 2 � � ∂ 2 � ∂ t 2 + γ ∂ ∂ t 2 + γ ∂ ∂ t 2 + γ ∂ ∂ t + w 2 ∂ t + w 2 E D D ε 0 ε φ E = E M A D D + ε φ M B D D , (7) p p ∂ t � ∂ 2 � � ∂ 2 � ∂ ∂ ∂ t + ω 2 ∂ t 2 + γ m B z = µ 0 A ∂ t 2 + γ m H z . (8) pm ∂ t Jichun Li (UNLV) June 27, 2018 12 / 62

Li, Huang, Yang: Math Comp, 2015 Assume that γ = γ m . µ 0 A ε 0 ε φ ( E t 3 + γ E t 2 + ω 2 p E t ) µ 0 A ( M A + ε φ M B ) ∇ × ( H t 2 + γ H t )+ µ 0 A ω 2 = p M A ∇ × H . (9) To simplify the notation, we denote H = H z , M = M A + ε φ M B . Also we have µ 0 A ( H t 2 + γ H t + ω 2 pm H ) = − ∇ × E t − γ ∇ × E . (10) Taking curl of (10) and substituting into (9), we have µ 0 ε 0 A ε φ ( E t 3 + γ E t 2 + ω 2 p E t )+ M ∇ × ∇ × E t + γ M ∇ × ∇ × E − µ 0 AM ∇ × ( ω 2 pm H )+ µ 0 A ω 2 = p M A ∇ × H . (11) Jichun Li (UNLV) June 27, 2018 13 / 62

Analysis of the model: cont’d Lemma 1 Matrix M A is symmetric and non-negative definite, and M is SPD. Lemma 2 For matrix M C = ( M A + ε φ M B ) − 1 , M C · M A = M A holds true. Weak formulation: For any φ ∈ H 0 ( curl ;Ω) , ψ ∈ L 2 (Ω) , ε 0 µ 0 A [( ε φ M C E t 3 , φ )+ γ ( ε φ M C E t 2 , φ )+( ω 2 ( i ) p ε φ M C E t , φ )] +( ∇ × E t , ∇ × φ )+ γ ( ∇ × E , ∇ × φ ) − µ 0 A ( ω 2 pm H , ∇ × φ )+ µ 0 A ( ω 2 = p M C M A ∇ × H , φ ) , (12) � � ( H t 2 , ψ )+ γ ( H t , ψ )+( ω 2 ( ii ) µ 0 A pm H , ψ ) = − ( ∇ × E t + γ ∇ × E , ψ ) . (13) Jichun Li (UNLV) June 27, 2018 14 / 62

Analysis of the model: cont’d Theorem 1 For the solution of (12)-(13), the following stability holds true: ε 0 µ 0 A [( ε φ M c E t 2 , E t 2 )( t )+( ω 2 p ε φ M c E t , E t )( t )]+( ∇ × E t , ∇ × E t )( t ) +( ∇ × E , ∇ × E )( t )+ A ( ω 2 p ε φ M c E , E )( t ) + µ 0 A ( || H t || 2 0 + || ω pm H || 2 0 )( t ) ≤ CF ( 0 ) , (14) where F ( 0 ) depends on initial conditions ∇ × E ( 0 ) , ∇ × E t ( 0 ) , E ( 0 ) , E t ( 0 ) , E t 2 ( 0 ) , H ( 0 ) , ∇ × H ( 0 ) , H t ( 0 ) and D ( 0 ) . Jichun Li (UNLV) June 27, 2018 15 / 62

Numerical results: Li, Huang, Yang, Math Comp (2015) Use R 1 = 0 . 1 m , R 2 = 0 . 2 m , γ = γ m = 0 in our Drude model. A plane wave source: specified by H z = 0 . 1sin ( ω t ) , where ω = 2 π f with operating frequency f = 2 . 0 GHz. A point source wave (same H z ) at a point (0:078; 0:4). Simulation with 65536 triangles, 28672 rectangles, and time step τ = 0 . 2ps. Jichun Li (UNLV) June 27, 2018 16 / 62

Numerical results (a) (b) Figure 3 : (a): The cloak modeling setup; (b): A coarse mesh. Jichun Li (UNLV) June 27, 2018 17 / 62

Numerical results: plane wave source (a) (b) (c) Figure 4 : E y at (a) t = 0 . 8 ns ( 4000 steps ) ; (b) t = 1 . 6 ns ; (c) t = 3 . 2 ns . (a) (b) (c) Figure 5 : E y at (a) t = 4 . 0 ns ; (b) t = 6 . 0 ns ; (c) t = 8 . 0 ns ( 40 , 000 steps ) . Jichun Li (UNLV) June 27, 2018 18 / 62