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Exercise:
Derive the Fresnel reflection and transmission formulas for s-polarized electric field E perpicular to the plane plane waves incident on a surface.

Solution:
To compute the Fresnel reflection formulas we first have to consider the specific boundary conditions for this problem. itemize item The tangential electric field E_parallel is continuous Rightarrow component of E that lies in the xz plane is continuous as one moves across the erface item The tangential magnetic field B_perp is continuous Rightarrow total B field in the plane of the erface is continuous. itemize This rewrites o: E_ixz+E_rxz E_txz -B_ixzcostheta_i + B_rxzcostheta_r -B_txzcostheta_t Now considering that BfracnEc and theta_itheta_r we can rewrite Eq. as n_iE_r-E_icos theta_i -n_tE_tcos theta_t now inserting Eq. n_iE_r-E_icos theta_i -n_tE_r+E_icos theta_t. Rearranging this we have: E_rn_icostheta_i+n_tcostheta_t E_in_icostheta_i-n_tcostheta_t Solving for fracE_rE_i yields the reflection coefficient: r_perp fracE_rE_i fracn_icostheta_i-n_tcostheta_tn_icostheta_i+n_tcostheta_t analogously the transmission coefficient is t_perp fracE_tE_i fracn_icostheta_in_icostheta_i+n_tcostheta_t
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Exercise:
Derive the Fresnel reflection and transmission formulas for s-polarized electric field E perpicular to the plane plane waves incident on a surface.

Solution:
To compute the Fresnel reflection formulas we first have to consider the specific boundary conditions for this problem. itemize item The tangential electric field E_parallel is continuous Rightarrow component of E that lies in the xz plane is continuous as one moves across the erface item The tangential magnetic field B_perp is continuous Rightarrow total B field in the plane of the erface is continuous. itemize This rewrites o: E_ixz+E_rxz E_txz -B_ixzcostheta_i + B_rxzcostheta_r -B_txzcostheta_t Now considering that BfracnEc and theta_itheta_r we can rewrite Eq. as n_iE_r-E_icos theta_i -n_tE_tcos theta_t now inserting Eq. n_iE_r-E_icos theta_i -n_tE_r+E_icos theta_t. Rearranging this we have: E_rn_icostheta_i+n_tcostheta_t E_in_icostheta_i-n_tcostheta_t Solving for fracE_rE_i yields the reflection coefficient: r_perp fracE_rE_i fracn_icostheta_i-n_tcostheta_tn_icostheta_i+n_tcostheta_t analogously the transmission coefficient is t_perp fracE_tE_i fracn_icostheta_in_icostheta_i+n_tcostheta_t
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Attributes & Decorations
Tags
electric field, fresnel, magnetic field, optics, polarisation
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Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Creator rk
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