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# FDTD Presentation

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### FDTD Presentation

1. 1. FDTD Simulations of Metamaterials in Transformation Optics Presentation by Reece Boston March 7, 2016
2. 2. What is Transformation Optics? Transformation Optics (TO) is the theoretical prediction of material parameters, ˆ, ˆµ for a medium that effects any desired transformation in the paths of light rays, by association between Maxwell’s equations in a material with Maxwell’s Equations in a curved space. Theoretical underpinnings in differential geometry, with tie-ins to GR. Materials predicted are, in general, inhomogenous and bi-anisotropic. D = ˆE + ˆγ1H B = ˆµH + ˆγ2E In most simple applications, ˆγ1 = ˆγ2 = 0.
3. 3. Why Does Anyone Care About Transformation Optics? Using this procedure, we have near-perfect control over the movement of light. Most famously used to design Pendry’s Cloak of Invisibility. Not magic, but real science! −→ \$\$\$ from military. Useful in antenna design, focusing incoming radio waves more efﬁciently. Design of optical devices in systems where GR effects are relevant, such as satellites in orbit. Studying cosmological models in the laboratory. Creating anything our imagination desires! I’ll show you!
4. 4. Example: Beam Turner Start in ﬂat empty optical space, (x , y ). Perform transformation to curved physical space, (x, y): x −→ x = x cos πy 2R2 y −→ y = x sin πy 2R2 This transformation will turn straight lines in to either rays or circular arcs
5. 5. Example: Beam Turner Require distances to be conserved between two spaces: ds 2 = dx 2 + dy 2 = gij dxi dxj We take Jacobian matrix of transformation, invert to ﬁnd dx = dx cos θ + dy sin θ dy = 2R2 πr (−dx sin θ + dy cos θ) Leads to gij = cos2 θ + A2 sin2 θ (1 − A2) sin θ cos θ (1 − A2) sin θ cos θ sin2 θ + A2 cos2 θ where A = 2R2/πr. This gij is metric of curved physical space. Light rays “see” optical space, we “see” curved space.
6. 6. Interlude So far, we have the “Transformation” part of “Transformation Optics”. Transformation step generates curved space metrics, gij . Useful for clearly specifying how rays should be distorted. Not strictly necessary. Now that we have a curved space, we’d like to examine Maxwell’s Equations in curved space, to get the “Optics” part. This requires a slight detour as we discuss spatial derivatives in curved space. We shall now learn everything we need about Differential Geometry in four short slides.
7. 7. Curved Space Derivatives The Physics Major’s Dream: ·F(r, θ, φ) = ∂ ∂r , ∂ ∂θ , ∂ ∂φ ·(Fr , Fθ, Fφ) = ∂Fr ∂r + ∂Fθ ∂θ + ∂Fφ ∂φ Wouldn’t it be great! But why isn’t it? (Using = ∂ ∂r , 1 r ∂ ∂θ , 1 r sin θ ∂ ∂φ isn’t any better. ) Vector operators , ·, and × are formally deﬁned in terms of volume and area elements: T( ) = lim V →0 1 V ∂V T(ˆn)dS where T is a linear expression, e.g. T(a) = a · F
8. 8. Curved Space Derivatives, cont. In spherical coordinates, volume and area elements are dV = r2 sin θ dr dθdφ, dSr = r2 sin θ dθdφ, dSθ = r sin θ dr dφ, dSφ = r drdθ Taking sides of a cuboid, T( ) = 1 r2 sin θdrdθdφ [T(r2 ˆr)+ − T(r2 ˆr)−] sin θdθdφ +[T(sin θˆθ)+ − T(sin θˆθ)−]rdrdφ +[T(ˆφ)+ − T(ˆφ)−]rdrdθ = 1 r2 sin θ ∂T(r2 sin θˆr) ∂r + ∂T(r sin θˆθ) ∂θ + ∂T(r ˆφ) ∂φ = 1 r2 ∂T(r2ˆr) ∂r + 1 r sin θ ∂T(sin θˆθ) ∂θ + 1 r sin θ ∂T(ˆφ) ∂φ This is correct answer in spherical coordinates for any linear expression T.
9. 9. But Wait, There’s More! Switch from orthonormal basis to coordinate basis, ˆr = er , ˆθ = reθ , ˆφ = r sin θeφ . Then gij =   1 0 0 0 r2 0 0 0 r2 sin2 θ  , and det gij = r2 sin θ. Then, T( ) = 1 r2 sin θ ∂T(r2 sin θˆr) ∂r + ∂T(r sin θˆθ) ∂θ + ∂T(r ˆφ) ∂φ = 1 r2 sin θ ∂T(r2 sin θer ) ∂r + ∂T(r2 sin θeθ) ∂θ + ∂T(r2 sin θeφ) ∂φ = 1 det gij ∂T( det gij ea) ∂xa This is general expression in any coordinate system, in any curved space, for any linear expression T.
10. 10. Divergence and Curl in Curvilinear Coordinates Maxwell’s Equations require divergence and curl. Divergence is easy: T(a) = a · F · F = 1 √ g ∂( √ gea · F) ∂xa = 1 √ g ∂( √ gFa) ∂xa Curl is slightly more difﬁcult: T(a) = a × F × F = 1 √ g ∂( √ gea × F) ∂xa = 1 √ g ∂( √ gea) ∂xa × F + ea × ∂F ∂xa = ea × eb ∂Fb ∂xa = εabc ec ∂Fb ∂xa Now let’s go back to Beam Turner and put these in Maxwell’s Equations.
11. 11. Back to the Beam Turner We found gij = cos2 θ + A2 sin2 θ (1 − A2) sin θ cos θ (1 − A2) sin θ cos θ sin2 θ + A2 cos2 θ as metric tensor of curved space. Maxwell’s Equations in curved space (without sources, c = 1)) are 1 √ g ∂ ∂xi ( √ gEi ) = 0 1 √ g ∂ ∂xi ( √ gBi ) = 0 1 √ g [ijk] ∂Ek ∂xj = − ∂Bi ∂t 1 √ g [ijk] ∂Bk ∂xj = ∂Ei ∂t , Rearranging slightly, using H = B in free space ∂ ∂xi ( √ ggil El ) = 0, ∂ ∂xi ( √ ggil Hl ) = 0 [ijk] ∂Ek ∂xj = − ∂( √ ggil Hl ) ∂t [ijk] ∂Hk ∂xj = ∂( √ ggil El ) ∂t ,
12. 12. Beam Turner, cont. Maxwell Equation in our curved space (c = 1): ∂ ∂xi ( √ ggij Ej ) = 0, ∂ ∂xi ( √ ggij Hj ) = 0 [ijk] ∂Ek ∂xj = − ∂( √ ggij Hj ) ∂t [ijk] ∂Hk ∂xj = ∂( √ ggij Ej ) ∂t , Maxwell Equations in a medium: ∂ ∂xi Di = ∂ ∂xi ( ij Ej ) = 0, ∂ ∂xi Bi = ∂ ∂xi (µij Hj ) = 0 [ijk] ∂Ek ∂xj = − ∂Bi ∂t = − ∂(µij Hj ) ∂t [ijk] ∂Hk ∂xj = ∂Di ∂t = ∂( ij Ej ) ∂t , Now we transform again, form curved space to ﬂat space, by means of a medium, called “Transformation Medium”: jk = µjk = √ ggjk = A cos2 θ + 1 A sin2 θ (A − 1 A ) sin θ cos θ (A − 1 A ) sin θ cos θ 1 A cos2 θ + A sin2 θ . Light can’t tell the difference between gik and jk, µjk.
13. 13. Beam Turner Wrap Up To track what we did: 1. We started in a ﬂat optical space, where light moves along straight lines. 2. We performed a transformation to a curved space to cause light rays to bend. 3. Then we undid the curvature in the space with an equivalent medium. This medium we found to be anisotropic and inhomogenous. This is the general procedure for any desired spatial curvature.
14. 14. Now What? We want to verify theoretical result before wasting time and money building it. Simulate light impinging upon a material with , µ as given above. In E&M simulations, two main methods are: Finite Element Method (FEM). Finite Difference Time Domain (FDTD). FEM discretizes functional solution space, approximates solution as sum of basis functions. Solves for steady-state (inﬁnite time) solution. FDTD discretizes spatial and temporal grid, ﬁnd ﬁeld values at grid points. Marches forward in the time domain. In our calculations, we used a 2-dimensional FDTD calculation for Transverse Magnetic case.
15. 15. Simulation of Beam Turner Quater-ring shaped device with jk = µjk = 2R2 πr cos2 θ + πr 2R2 sin2 θ (2R2 πr − πr 2R2 ) sin θ cos θ (2R2 πr − πr 2R2 ) sin θ cos θ πr 2R2 cos2 θ + 2R2 πr sin2 θ .
16. 16. The Update Procedure Consider the Maxwell-Ampere Equation for D, ∂D ∂t = × H. Discretize time in to timestep ∆t, space by ∆x. Then Dn+1/2 − Dn−1/2 ∆t = 1 ∆x ˜ × Hn ( ˜ ×F)z = Fx (i, j+1, k)−Fx (i, j, k)−Fy (i+1, j, k)+Fy (i, j, k). Rearranging Dn+1/2 = Dn−1/2 + ∆t ∆x ˜ × Hn Likewise, for Maxwell-Faraday Equation, Bn+1 = Bn − ∆t ∆x ˜ × En+1/2 In between these two, we perform En+1/2 = 1 o −1 Dn+1/2 , Hn+1 = 1 µo µ−1 Bn+1
17. 17. The Yee Cell E and B ﬁelds are staggered in time and space. This process tends to even out errors due to grid approximation. In 1D: Taken from Understanding the Finite Difference Time-Domain Method, John Schneider, www.eecs.wsu.edu/~schneidj/ufdtd
18. 18. The Yee Cell E and B ﬁelds are staggered in time and space. This process tends to even out errors due to grid approximation. In 2D: Taken from Understanding the Finite Difference Time-Domain Method, John Schneider, www.eecs.wsu.edu/~schneidj/ufdtd
19. 19. The Yee Cell E and B ﬁelds are staggered in time and space. This process tends to even out errors due to grid approximation. In 3D:
20. 20. The FDTD Method in Summary Divide space and time according to Yee cell. Specify , µ over entire spatial domain. Introduce source ﬁeld by altering E value at some point(s). Propagate source ﬁeld through space and time by leap-frog algorithm: 1. Dn+1/2 = Dn−1/2 + ∆t ∆x ˜ × Hn 2. En+1/2 = 1 o −1 Dn+1/2 . 3. Bn+1 = Bn − ∆t ∆x ˜ × En+1/2 4. Hn+1 = 1 µo µ−1 Bn+1 Continue iterating in time, as long as you wish.
21. 21. Example: Cloak of Invisibility Long a staple of fantasy and science ﬁction: How do we make it a reality? Transform the single point of the origin in to a circle of radius R1 r → r = R1 + r R2 − R1 R2 , θ → θ = θ , z → z = z
22. 22. Example: Cloak of Invisibility Long a staple of fantasy and science ﬁction: How do we make it a reality? Transform the single point of the origin in to a circle of radius R1 r → r = R1 + r R2 − R1 R2 , θ → θ = θ , z → z = z
23. 23. Example: Cloak of Invisibility Long a staple of fantasy and science ﬁction: How do we make it a reality? Transform the single point of the origin in to a circle of radius R1 r → r = R1 + r R2 − R1 R2 , θ → θ = θ , z → z = z This leads to gij = R2 R2 − R1 2    cos2 θ + α2 sin2 θ (1 − α2) sin θ cos θ 0 (1 − α2) sin θ cos θ sin2 θ + α2 cos2 θ 0 0 0 R2−R1 R2 2    where α = r−R1 r . As above, an equivalent medium is given by ij = µij = √ ggij , ij = µij =    α cos2 θ + 1 α sin2 θ (α − 1 α ) sin θ cos θ 0 (α − 1 α) sin θ cos θ 1 α cos2 θ + α sin2 θ 0 0 0 R2 R2−R1 2 α    .
24. 24. Simulation of Cloak of Invisibility
25. 25. Optical ‘Bag of Holding’ Transform spatial distances inside a cylinder so that the inside is bigger than the outside.
26. 26. Optical ‘Bag of Holding’ Transform spatial distances inside a volume so that the inside is bigger than the outside. This means ds2 = B2(dx2 + dy2 + dz2) for some scale factor B. gij =   B2 0 0 0 B2 0 0 0 B2   ⇒ ij = √ ggij =   B 0 0 0 B 0 0 0 B   This is similar normal dielectric, = B, like glass or plastic! Inside dielectric, phase velocity v < c. Same speed in optical space −→ slower speed in physical space. We also have ij = µij : this is non-scattering condition. Truly non-glare glasses.
27. 27. Ilustratration of Velocity Distotion
28. 28. Example: Schwarzschild Black Hole Don’t need transformation; we already know the metric. ds2 = − 1 − R∗ r dt2 + 1 − R∗ r −1 dr2 +r2 dθ2 +r2 sin2 θdφ2 . More complicated, as now we have spacetime curvature. Equations for this ﬁrst discovered by Plebanski, listed without derivation: ij = µij = − |g| g00 gij (γT 1 )ij = (γ2)ij = −[ijk] g0j g00 , where Di = ij Ej + γij 1 Hj , Bi = µij Hj + γij 2 Ej . Using this, we ﬁnd, for 2D case: ij = µij = r r − R∗    1 − R∗ r x2 r2 −R∗ r xy r2 0 −R∗ r xy r2 1 − R∗ r y2 r2 0 0 0 1    .
29. 29. Illustration of Gravitational Lensing
30. 30. Further Work Simulations analyzing metamaterial periodic elements for actual construction. Reduced cloaks for broadband cloaking. Perfect black body layer, based off black hole metric. Makes perfect ‘one-way mirror’, possibly for solar panels. Anti-telephonic device that I have affectionately named “Galadriel’s Mirror.” Possibilities are limited only by our imagination.
31. 31. We are the Masters of Time and Space! Light bends to our whim and caprice! Questions?