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Lecture 6 Incidence at Interfaces with Arbitrary Angle

Lecture 6 Incidence at Interfaces with Arbitrary Angle. 6.013. ELECTROMAGNETICS AND APPLICATIONS. Luca Daniel. Today’s Outline. Review of Fundamental Electromagnetic Laws Electromagnetic Waves in Media and Interfaces The EM waves in homogenous Media Electromagnetic Power and Energy

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Lecture 6 Incidence at Interfaces with Arbitrary Angle

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  1. Lecture 6Incidence at Interfaceswith Arbitrary Angle 6.013 ELECTROMAGNETICS AND APPLICATIONS Luca Daniel

  2. Today’s Outline • Review of Fundamental Electromagnetic Laws • Electromagnetic Waves in Media and Interfaces • The EM waves in homogenous Media • Electromagnetic Power and Energy • EM Fields at Interfaces between Different Media • EM Waves Incident “Normally” to a Different Medium • EM Waves Incident at General Angle • UPW in arbitrary direction • TE wave at planar interface • Phase Matching and Snell’s Law • Critical Angle • Total Reflection and Evanescent Waves • TM wave at planar interface • Brewster Angle • Duality Today

  3. wave z  “Phase fronts” x z y UPW In Arbitrary Directions z-directed wave: Arbitrary direction: k Dispersion Relation: 

  4. UPW In Arbitrary Directions y ky x  x ky 0 kx kz

  5. TE Wave at Planar Boundary Interface x kiz i kix r i i,i z y t,t kz t Case 1: TE Wave “Transverse Electric” x z y Trial Solutions:

  6. Phase Matching and Snell’s Law nwater  1.3 at visible wavelengths < 9 at audio-radio frequencies ~ is continuous at x = 0: Phase Matching ! Therefore: = = r = i Angle of incidence equals angle of reflection Snell’s Law: nvacuum = 1 nglass 1.45 – 1.66

  7. Critical Angle Case kt > ki: Case kt < ki: x x kr ki i r r kr i ki , Glass kiz krz Air z o,o z kz Air kz ktz kt • Glass, • t 2o kt t t kt < ki  t > i kt > ki  t < i Critical angle qc(for nt < ni): Therefore: t  90o  sin t  1 as i  c

  8. Wave Front Shapes at Boundaries (Case kt<ki) kt” kt’ kz’ Standard refraction: i < c “Phase Matching” at boundary i Phase fronts glass Glass z oz Air Lines of constant phase t o Beyond the critical angle, i > c: Total reflection & evanescence x i i > c glass Glass z t = 90° o = oz

  9. Total Reflection and Evanescent Waves When qi > qc,ktz > kt and: x e.g., glass Since: Therefore: ki ki i r z kiz>kt e.g., air t kt where: Fields when q > qc:

  10. Total Reflection and Evanescent Waves Standard refraction: i < c “Phase Matching” at boundary i Phase fronts glass Glass z oz Air Lines of constant phase t o Beyond the critical angle, i > c: Total reflection & evanescence x i glass i > c z ex Lines of constant amplitude t = 90° evanescent region o = oz

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