Rough surface reflections Gregory Hart Steve Turley BYU XUV

Why do we care? Astronomy Computers Microscopy He Photolithography imaging BYU XUV

XUV 1-100 nm Low reflectance 𝑛≈1 𝜅≈1 𝑎𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑣𝑒 BYU XUV

BYU XUV

The Equations 𝐸 𝑧 𝑖𝑛𝑐𝑙 𝑥,𝑦 = 𝐾 𝑡 𝑥,𝑦 −𝑖 𝑘 1 𝜂 1 𝐽 𝑧 𝑥 ′ , 𝑦 ′ 𝑖 4 𝐻 0 1 𝑘 1 𝑟 𝑑 𝑠 ′ + 𝜕 𝜕𝑥 𝑦 ∙ 𝑡 𝑥 ′ , 𝑦 ′ 𝐾 𝑡 𝑥 ′ , 𝑦 ′ 𝑖 4 𝐻 0 1 𝑘 1 𝑟 𝑑 𝑠 ′ − 𝜕 𝜕𝑦 𝑥 ∙ 𝑡 𝑥 ′ , 𝑦 ′ 𝐾 𝑡 𝑥 ′ , 𝑦 ′ 𝑖 4 𝐻 0 1 𝑘 1 𝑟 𝑑 𝑠 ′ 0= 𝐾 𝑡 𝑥,𝑦 −𝑖 𝑘 2 𝜂 2 𝐽 𝑧 𝑥 ′ , 𝑦 ′ 𝑖 4 𝐻 0 1 𝑘 2 𝑟 𝑑 𝑠 ′ + 𝜕 𝜕𝑥 𝑦 ∙ 𝑡 𝑥 ′ , 𝑦 ′ 𝐾 𝑡 𝑥 ′ , 𝑦 ′ 𝑖 4 𝐻 0 1 𝑘 2 𝑟 𝑑 𝑠 ′ 𝑦 ∙ 𝑡 𝑥 ′ , 𝑦 ′ 𝐾 𝑡 𝑥 ′ , 𝑦 ′ 𝑖 4 𝐻 0 1 𝑘 2 𝑟 𝑑 𝑠 ′ − 𝜕 𝜕𝑦 𝑥 ∙ 𝑡 𝑥 ′ , 𝑦 ′ 𝐾 𝑡 𝑥 ′ , 𝑦 ′ 𝑖 4 𝐻 0 1 𝑘 2 𝑟 𝑑 𝑠 ′ 𝑥 ∙ 𝑡 𝑥 ′ , 𝑦 ′ 𝐾 𝑡 𝑥 ′ , 𝑦 ′ 𝑖 4 𝐻 0 1 𝑘 2 𝑟 𝑑 𝑠 ′ 𝛻∙( 𝜖 0 E)= 𝜌 𝑒 𝛻∙( 𝜇 0 𝐇)= 𝜌 𝑚 𝛻×𝐄=𝑖ω 𝜇 0 𝐇−𝑲 𝛻×𝐇=−𝑖ω ϵ 0 𝐄+𝑱 𝛻 2 + 𝑘 2 𝐄=0 𝛻 2 − 𝑘 2 𝐁=0 Assumption time dependence is harmonic, mu = mu0, epsilon is a scalar. S polarized makes it a scalar problem in z. BYU XUV

Numerically 𝐸= 1 2 𝐾 𝑡 + 𝑖𝑗 𝑀 𝑖𝑗 1 𝐽 𝑧 𝑖 + 𝑖𝑗 𝑁 𝑖𝑗 1 𝐾 𝑡 0=− 1 2 𝐾 𝑡 + 𝑖 𝑀 𝑖 2 𝐽 𝑧 𝑖 + 𝑖 𝑁 2 𝐾 𝑡 𝑀 𝑖𝑗 𝜔 = 𝑘 𝜔 𝜂 𝜔 4 𝑐 𝑖 𝑆 𝑖 𝐻 0 1 𝑘 𝜔 𝑟 𝑖𝑗 𝑁 𝑖𝑗 𝜔 = 𝑖 𝑘 𝜔 4 𝑐 𝑖 𝑆 𝑖 𝐻 1 1 ( 𝑘 𝜔 𝑟 𝑖𝑗 ) 𝑟 𝑖𝑗 [ cos 𝜃 𝑖 ′ 𝑦 𝑗 − 𝑦 𝑖 ′ − sin 𝜃 𝑖 ′ 𝑥 𝑗 − 𝑥 𝑖 ′ ] 𝐸 0 = ( 1 2 + 𝑁 1 ) 𝑀 1 (− 1 2 + 𝑁 2 ) 𝑀 2 𝐾 𝐽 BYU XUV

Test .Smooth surface incident at 20° BYU XUV

Test .Cylindical surface incident at 20° BYU XUV

Correlation BYU XUV

Discussion The program works for simple cases Found single processor limits Random roughnesses have same overall effect 2D surface Parallel processing BYU XUV

Acknowledgement Steve Turley Jed Johnson Fulton Supercomputing Lab BYU Physics and Astronomy BYU XUV