lecture19 - Lecture 19 Non-Normal Incidence of Waves at...

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1 ECE 303 – Fall 2005 – Farhan Rana – Cornell University Lecture 19 Non-Normal Incidence of Waves at Interfaces In this lecture you will learn: • What happens when waves strike an interface between two different media coming at an angle • Reflection and transmission of waves at interfaces • Application of E-field and H-field boundary conditions • Total internal reflection • Brewster’s angle ECE 303 – Fall 2005 – Farhan Rana – Cornell University Waves at Interfaces – TE and TM Waves z =0 z i k r E i H i o i µ ε o t x Transverse Electric (TE) wave Plane of Incidence: The plane containing the incident wavevector and a vector that is normal to the interface is called the plane of incidence (in the figure above the x-z plane is the plane of incidence) i k r TE Wave: If the E-field of the wave is perpendicular to the plane of incidence then the wave is called a TE-wave TM Wave: If the H-field of the wave is perpendicular to the plane of incidence then the wave is called a TM-wave i k r E i H i Transverse Magnetic (TM) wave

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2 ECE 303 – Fall 2005 – Farhan Rana – Cornell University TE Wave - Wavevectors z =0 z o i µ ε o t x Transverse Electric (TE) wave r k r E r H r t k r E t H t i k r E i H i i θ r t ( ) ( ) [] z x k z k x k k i i i iz ix i ˆ cos ˆ sin ˆ ˆ + = + = r c n k i i o i ω = = ( ) ( ) z x k z k x k k r r r rz rx r ˆ cos ˆ sin ˆ ˆ = + = r c n k k i i o i r = = = ( ) ( ) z x k z k x k k t t t tz tx t ˆ cos ˆ sin ˆ ˆ + = + = r c n k t t o t = = ECE 303 – Fall 2005 – Farhan Rana – Cornell University z =0 z o t x t k r E t H t t o i Transverse Electric (TE) wave r k r E r H r i k r E i H i i r TE Wave – First Boundary Condition () r k j r r k j i z r i e E y e E y r E r r r r r r . . 0 ˆ ˆ < + = r k j t z t e E y r E r r r r . 0 ˆ > = ( )( ) [ ] z x k k i i i i ˆ cos ˆ sin + = r ( ) [ ] z x k k r r r r ˆ cos ˆ sin = r ( ) [ ] z x k k t t t t ˆ cos ˆ sin + = r Use boundary conditions: (1) At z = 0 the E-field parallel to the interface must be continuous across the interface for all x This gives: ( ) ( ) x k j t x k j r x k j i t t r r i i e E e E e E sin sin sin = +
3 ECE 303 – Fall 2005 – Farhan Rana – Cornell University TE Wave – Phase Matching Condition z =0 z o t µ ε x t k r E t H t t θ o i Transverse Electric (TE) wave r k r E r H r i k r E i H i i r () ( ) ( ) x k j t x k j r x k j i t t r r i i e E e E e E sin sin sin = + ( ) ( ) ( ) t t r r i i k k k sin sin sin = = The first equality gives ( using k i = k r ) : ( ) ( ) r i r i = = sin sin angle of incidence equals the angle of reflection tx rx ix k k k = = The only way the above boundary condition can be satisfied for all x is if all the x - dependent phase factors are the same (this is called “phase matching”) ECE 303 – Fall 2005 – Farhan Rana – Cornell University TE Wave – Snell’s Law z =0 z o t x t k r E t H t t o i Transverse Electric (TE) wave r k r E r H r i k r E i H i i r ( ) ( ) ( ) t t r r i i k k k sin sin sin = = The second equality gives: ( ) ( ) () () t t i i t t i i t t i i n n c n c n k k ω sin sin sin sin sin sin = = = Snell’s Law ( ) ( ) t r i x k j t x k j r x k j i E E E e E e E e E t t r r i i = + = + sin sin

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lecture19 - Lecture 19 Non-Normal Incidence of Waves at...

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