Electromagnetic Waves The Finite-Difference Time Domain (FDTD) - PowerPoint PPT Presentation

Title Prob Maxwell Algor Implementation Assess Electromagnetic Waves The Finite-Difference Time Domain (FDTD) Algorithm Rubin H Landau Oregon State University Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II 1 / 10

Title Prob Maxwell Algor Implementation Assess Problem: Determine E & H Fields for All Times Given: Space 0 ≤ z ≤ 200 E x ( z , t = 0 ) = 0 . 1 sin 2 π z H y ( z , t = 0 ) = 0 . 1 sin 2 π z 100 , 100 x E x H y t z H y y E x E&M practical import New: coupled fields FDTD ∆ z , ∆ t = step New: vector fields, 3-D 2 / 10

Title Prob Maxwell Algor Implementation Assess Theory: Maxwell’s Equations in Free Space E = E x , H = H y , S = E × H = S z ⇒ 3-D ∂ E x ( z , t ) � ∇ · E = 0 ⇒ = 0 (Tranverse) (1) ∂ x ∂ H y ( z , t ) � ∇ · H = 0 ⇒ = 0 (Tranverse) (2) ∂ y ∂ E ∂ t = + 1 ∂ E x = − 1 ∂ H y ( z , t ) � ∇ × H ⇒ (3) ǫ 0 ∂ t ǫ 0 ∂ z ∂ H ∂ t = − 1 ∂ H y = − 1 ∂ E x ( z , t ) � ∇ × E ⇒ (4) µ 0 ∂ t µ 0 ∂ z x E x H y t z y H y E x 3 / 10

Title Prob Maxwell Algor Implementation Assess Finite Difference Time Domain (FDTD) Algorithm Central-Difference Derivatives ⇒ = E k , n + 1 / 2 = H k + 1 / 2 , n E z , t H z , t , x x y y ≃ E ( z , t + ∆ t 2 ) − E ( z , t − ∆ t ∂ E ( z , t ) 2 ) , (1) ∂ t ∆ t ≃ E ( z + ∆ z 2 , t ) − E ( z − ∆ z 2 , t ) ∂ E ( z , t ) (2) ∂ z ∆ z Substitute into Maxwell , rearrange for t stepping ∆ t E k , n + 1 / 2 = E k , n − 1 / 2 H k + 1 / 2 , n − H k − 1 / 2 , n � � − , (3) x x y y ǫ 0 ∆ z ∆ t H k + 1 / 2 , n + 1 = H k + 1 / 2 , n E k + 1 , n + 1 / 2 − E k , n + 1 / 2 � � (4) − y y x x µ 0 ∆ z 4 / 10

Title Prob Maxwell Algor Implementation Assess Displaced E x , H y Space-Time Lattices = E k , n + 1 / 2 = H k + 1 / 2 , n E z , t H z , t , x x y y 2 - 1/2 / 1 1 + + k k k k n- 1/2 n n+ 1/2 H y n+1 E x t Space variation H y ⇒ time variation E x Space variation E x ⇒ time variation H y 5 / 10

Title Prob Maxwell Algor Implementation Assess Alternate Formulation: Even & Odd Times Double Index Values ∆ t � � E k , n = E k , n − 2 H k + 1 , n − 1 − H k − 1 , n − 1 , k even, odd , (1) − x x y y ǫ 0 ∆ z ∆ t � � H k , n = H k , n − 2 E k + 1 , n − 1 − E k − 1 , n − 1 , k odd, even . (2) − y y x x µ 0 ∆ z E : even z , odd t H : odd z , even t 6 / 10

Title Prob Maxwell Algor Implementation Assess Normalized Algorithm; Stability Analysis E With Same Dimension as H , ˜ ˜ � E = ǫ 0 /µ 0 E � � E k , n + 1 / 2 ˜ = ˜ E k , n − 1 / 2 H k − 1 / 2 , n − H k + 1 / 2 , n + β (1) x x y y � � H k + 1 / 2 , n + 1 = H k + 1 / 2 , n E k , n + 1 / 2 ˜ − ˜ E k + 1 , n + 1 / 2 + β (2) y y x x c 1 β = ∆ z / ∆ t , c = (light) (3) √ ǫ 0 µ 0 β = light/grid speed Courant Stability: β ≤ 1 / 2 ω wave ⇒ t scale Smaller ∆ t ↑ precision λ wave ⇒ z scale Smaller ∆ t ↑ stability > 10 points/ λ Smaller ∆ z ⇒ smaller ∆ t 7 / 10

Title Prob Maxwell Algor Implementation Assess Implementation FDTD.py Initial conditions (0 ≤ z ( k ) ≤ 200): E x ( z , t = 0 ) = 0 . 1 sin 2 π z H y ( z , t = 0 ) = 0 . 1 sin 2 π z 100 , 100 Discrete Maxwell equations: Ex [ k , 1 ] = Ex [ k , 0 ] + beta ∗ ( Hy [ k − 1 , 0 ] − Hy [ k + 1 , 0 ]) Hy [ k , 1 ] = Hy [ k , 0 ] + beta ∗ ( Ex [ k − 1 , 0 ] − Ex [ k + 1 , 0 ]) 0 = old time, 1 = new time Spatial endpoints via periodic boundary conditions: 8 / 10

Title Prob Maxwell Algor Implementation Assess Assessment Impose BC such that fields vanish on boundaries 1 Show effect of these BCs 2 Test Courant stability condition 3 Solve with inserted dielectric slab 4 Note transmission, reflection at slab boundaries 5 Verify that H ( t = 0 ) = 0 ⇒ right & left pulses 6 Investigate resonator modes for plane waves with nodes at 7 boundaries 9 / 10

Title Prob Maxwell Algor Implementation Assess Extension: Circularly Polarized Waves CircPolartzn.py 10 / 10