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EM_scattering_homework_1_solution

# EM_scattering_homework_1_solution - EM Scattering Homework...

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Problem 1: An incident wave travels in free space (dielectric constant 0 , permeability 0 ± , wavenumber 0 0 k ² ± ³ ) along the z-direction and impinges upon a cube of dielectric material with a constant permittivity of . The faces of the dielectric cube are parallel to the x-y, y-z, z-x planes respectively. The length of each side of the cube is d . The polarization of the incident electric field is along the x-axis. Use the Born approximation to calculate the scattered electric and magnetic fields in the far-field zone along the ˆ ˆ s ³ ´ k z , ˆ ˆ s ³ ´ k x , and ˆ ˆ s ³ ´ k y directions. Solution In Born approximation we have for the far-zone scattering field: µ ¶ µ ¶ exp ˆ ˆ ( ) , f s s i jkr r · ¸ ³ E r Q k k (1.1) µ ¶ µ ¶ µ ¶ 0 ˆ ˆ ˆ ˆ ˆ ˆ , exp ( ) s i s i p i i s s V E j dV ¹ · º » ¼ ½ ³ ¸ ¾ ¸ ¾ ¼ ½ ¿ À Á Â Á Â Ã Ä Å Q k k k k r r e e k k (1.2) d 0 0 ˆ i E ³ E x x y z ˆ ˆ i ³ k z 0 EM Scattering Homework assignment 1

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Where ˆ i k k z , ˆ ˆ i e x , 0 ( ) p ±² ² ³ ´ r . Note that within this approximation no wave is scattered along the ˆ ˆ s µ k x , i.e., along a direction parallel to the polarization of the incident plane wave. Along ˆ ˆ s µ k z , ˆ ˆ s µ k y , the scattered wave is polarized along ˆ ˆ i x e as ˆ ˆ 0 i s e k . The integral in (1.2) yields for ˆ ˆ s µ k y · ¸ · ¸ · ¸ · ¸ · ¸ 0 0 2 0 2 ˆ ˆ exp exp 4 sin 2 V V j k k dV jky jkz dV a ka k ´ µ ´ ´ µ ´ ¹ º » ¼ ´ ½ ¾ ¿ À Á Â Ã Ã y z r (1.3) In the case ˆ ˆ s µ k z : · ¸ · ¸ · ¸ · ¸ · ¸ 3 0 2 0 0 ˆ ˆ , ˆ ˆ exp ˆ ˆ sin , s V s a j k k dV a ka k Ä ´ Å Æ ´ µ ´ ¹ º Ç » ¼ ´ ´ Æ È Ã k z z z r k z (1.4) Problem 2: Repeat this calculation for a dielectric cylinder with the radius a , height h , and dielectric constant . Now find the scattered field in any direction ˆ s k .
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EM_scattering_homework_1_solution - EM Scattering Homework...

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