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Prof. David R. Jackson Dept. of ECE

ECE 6340 Intermediate EM Waves. Fall 2013. Prof. David R. Jackson Dept. of ECE. Notes 19. Critical Angle. q i. n 1. y. q t. n 2. z. Assume lossless materials. Snell’s law:. Critical Angle. q c. n 1. y. q t. n 2. z.  t = 90 o. At the critical angle (  i =  c ) :. so.

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Prof. David R. Jackson Dept. of ECE

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  1. ECE 6340 Intermediate EM Waves Fall 2013 Prof. David R. Jackson Dept. of ECE Notes 19

  2. Critical Angle qi n1 y qt n2 z Assume lossless materials Snell’s law:

  3. Critical Angle qc n1 y qt n2 z t= 90o At the critical angle (i =c): so

  4. Critical Angle Notes: • The critical angle applies to any polarization. • The critical angle is only defined for lossless materials. • A critical angle exists only when going from a higher to a lower density medium (n1 > n2 ).

  5. Beyond Critical Angle Lets' examine the transmitted angle: Assume so

  6. Beyond Critical Angle (cont.) Power flow y Region 2 z Note: The power flow in the upper region is horizontal, and is independent of z. No power crosses the boundary Region 1 Note: The power flow in the lower region is horizontal, and decays with z.

  7. Beyond Critical Angle (cont.) All of the incident power is reflected

  8. Beyond Critical Angle (cont.) Determine the transmitted angle t : Let (real) Use or must use +sign Not possible

  9. Beyond Critical Angle (cont.) Hence

  10. Beyond Critical Angle (cont.) Choose + sign to obtain correct value for kz2: The + sign is chosen to obtain a decaying wave.

  11. Beyond Critical Angle (cont.) Hence Practical note: When dealing with inhomogeneous plane waves (complex angles), it is usually easier to avoid working with angles and use the separation equation instead. Requires complex angle Does not requires complex angle

  12. Brewster Angle No reflection Ei n1 y qtb n2 z TMz Lossless materials

  13. Brewster Angle (cont.) Perfect match:

  14. Brewster Angle (cont.) Assume m1=m2 = m:

  15. Brewster Angle (cont.) q b Hence

  16. Brewster Angle (cont.) Notes: • For non-magnetic media, only the TMz polarization has a Brewster angle. • A Brewster angle exists for any material contrast ratio.

  17. Brewster Angle (cont.) qtb q b From Snell's law: This means that tbis the angle shown in this figure. Hence

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