1. Wave Incidence at Oblique angles Sandra Cruz-Pol ECE Dept. UPRM

2. Quiz VIERNES  Uno de los problemas asignados en el prontuario  Enviar contestaciones a preguntas en Word por email

3. Wave Incidence I.Normal incidence Wave arrives at 90 o from the surface Standing waves II.Oblique incidence (lossless) Wave arrives at an angle Snell’s Law and Critical angle Two types: Parallel or Perpendicular Brewster angle

4. Oblique incidence First, we need to define:  Expression for uniform plane wave traveling in any direction  The Normal, a n  Plane of incidence  Angle of incidence z y z=0 Medium 1 :  1,  1 Medium 2 :  2,  2 rr ii tt x

5. Oblique incidence  Uniform plane wave in general form the radius or position vector the wave number or propagation vector Where:  For lossless unbounded media, k = 

6. Oblique Incidence x z y z=0 Medium 1 :  1,  1 Medium 2 :  2,  2 rr ii tt krkr kiki ktkt

7. Look at the direction of travel x z y z=0 Medium 1 :  1,  1 Medium 2 :  2,  2 rr ii tt krkr kiki ktkt k ix k iz

8. Expression for Waves kiki k ix k iz ii

9. Tangential E must be Continuous From this we know that frequency is a property of the wave. So is color. So 700nm is not always red!! Give example of bathing suit.

10. Snell Law  Equating, we get or where, the index of refraction of a medium, n i, is defined as the ratio of the phase velocity in free space ( c ) to the phase velocity in the medium.

11. Critical angle,  c …All is reflected  When  t =90 o, the refracted wave flows along the surface and no energy is transmitted into medium 2.  The value of the angle of incidence corresponding to this is called critical angle,  c.  If  i >  c, the incident wave is totally reflected.

12. Fiber optics  Light can be guided with total reflections through thin dielectric rods made of glass or transparent plastic, known as optical fibers.  The only power lost is due to reflections at the input and output ends and absorption by the fiber material (not perfect dielectric).

13. Optical fibers have cylindrical fiber core with index of refraction n f, surrounded by another cylinder of lower, n c < n f, called a cladding.  For total reflection: Acceptance angle [Figure from Ulaby, 1999] Use Snell and critical angle to derive:

14. Exercise: optical fiber An optical fiber (in air) is made of fiber core with index of refraction of 1.52 and a cladding with an index of refraction of 1.49.  Find the acceptance angle: Answer: ≤17.5 o

15. Parallel polarization  It’s defined as E is || to incidence plane z y z=0 Medium 1 :  1,  1 Medium 2 :  2,  2 rr ii tt krkr kiki ktkt ErEr EiEi EtEt x z

16. Equating for continuity, the tangent fields Which components are tangent to the interface between two surfaces?  y and x At z = 0 (interface):

17. Reflection and Transmission Coefficients: Parallel Incidence  Reflection

18. Reflection and Transmission Coefficients: Perpendicular Incidence

19. Java Animation  http://www.amanogawa.com/archive /Oblique/Oblique-2.html http://www.amanogawa.com/archive /Oblique/Oblique-2.html

20. Exercise A plane wave in air with Is incident upon planar surface of nonmagnetic dielectric material with  r =4, on z >0, Find  The incidence polarization of this wave  The angle of incidence  The time-domain expressions for the reflected electric and magnetic fields.  The average power density carried by the wave in the dielectric medium.

21. Brewster angle,  B  Is defined as the incidence angle at which the reflection coefficient is 0 (all transmission).  The Brewster angle does not exist for perpendicular polarization for nonmagnetic materials.  B *  B is known as the polarizing angle

22. Exercise A plane wave in free space with Is incident upon planar surface of nonmagnetic lossless dielectric material with  r =4, on z >0, Find  The polarization of the incident wave  The angle of incidence and transmission  The reflection and transmission coefficients.  The E field that is reflected and in dielectric  Brewster angle

23. Reflection vs. Incidence angle. Reflection vs. incidence angle for different types of soil and parallel or perpendicular polarization.

24. Exercise (Brewster angle) A wave in air is incident upon the flat boundary of a nonmagnetic soil medium with  r =4, at  i =50 o.  Find reflection and transmission coefficients for both incident polarizations, and the Brewster angle.  Answers: EMAG

25. Summary: wave incidence PropertyNormal Incidence PerpendicularParallel Reflection coefficient Transmission coefficient Relation Power Reflectivity Power Transmissivity Snell’s Law: