Guided Waves in Layered Media

1. Guided Waves in Layered Media

2. Guided Waves in Layered Media Layered media can support confined electromagnetic propagation. These modes of propagation are the so-called guided waves, and the structures that support guided waves are called waveguides.

3. Symmetric Slab Waveguides • Dielectric slabs are the simplest optical waveguides. • It consists of a thin dielectric layer sandwiched between two semi-infinite bounding media. • The index of refraction of the guiding layer must be greater than those of the surround media.

4. Symmetric Slab Waveguides • The following equation describes the index profile of a symmetric dielectric waveguide:

5. Symmetric Slab Waveguides • The propagation of monochromatic radiation along the z axis. Maxwell’s equation can be written in the form

6. Symmetric Slab Waveguides • For layered dielectric structures that consist of homogeneous and isotropic materials, the wave equation is • The subscript m is the mode number

7. Symmetric Slab Waveguides • For confined modes, the field amplitude must fall off exponentially outside the waveguide. • Consequently, the quantity (nω/c)^2-β^2 must be negative for |x| > d/2.

8. Symmetric Slab Waveguides • For confined modes, the field amplitude must fall off sinusoidally inside the waveguide. • Consequently, the quantity (nω/c)^2-β^2 must be positive for |x| < d/2.

9. Guided TE Modes • The electric field amplitude of the guided TE modes can be written in the form

10. Guided TE Modes • The mode function is taken as

11. Guided TE Modes • The solutions of TE modes may be divided into two classes: for the first class and for the second class • The solution in the first class have symmetric wavefunctions, whereas those of the second class have antisymmetricwavefunctions.

12. Guided TE Modes • The propagation constants of the TE modes can be found from a numerical or graphical solution.

13. Guided TE Modes

14. Guided TM Modes • The field amplitudes are written

15. Guided TM Modes • The wavefunction H(x) is

16. Guided TM Modes • The continuity of Hy and Ez at the two (x=±(1/2)d) interface leads to the eigenvalue equation

17. Asymmetric Slab Waveguides • The index profile of a asymmetric slab waveguides is as follows • n2 is greater than n1 and n3, assuming n1<n3<n2

18. Asymmetric Slab Waveguides Typical field distributions corresponding to different values of β

19. Guided TE Modes • The field component Ey of the TE mode can be written as • The function Em(x) assumes the following forms in each of the three regions

20. Guided TE Modes • By imposing the continuity requirements , we get • or

21. Guided TE Modes • The normalization condition is given by • Or equivalently,

22. Guided TM Modes • The field components are

23. Guided TM Modes • The wavefunction is

24. Guided TM Modes

25. Guided TM Modes • We define the parameter • At long wavelengths, such that No confined mode exists in the waveguide.

26. Guided TM Modes • As the wavelength decreases such that One solution exists to the mode condition

27. Guided TM Modes • As the wavelength decreases further such that Two solutions exist to the mode condition

28. Guided TM Modes • The mthsatifies

29. Surface Plasmons • Confined propagation of electromagnetic radiation can also exist at the interface between two semi-infinite dielectric homogeneous media. • Such electromagnetic surface waves can exist at the interface between two media, provided the dielectric constants of the media are opposite in sign. • Only a single TM mode exist at a given frequency.

30. Surface Plasmons • A typical example is the interface between air and silver where n1^2 = 1 and n3^2 = -16.40-i0.54 at λ = 6328(艾).

31. Surface Plasmons • For TE modes, by putting t=0 in Eq.(11.2-5), we obtain the mode condition for the TE surface waves, p + q =0 Where p and q are the exponential attenuation constants in media 3 and 1. • It can never be satisfied since a confined mode requires p>0 and q>0.

32. Surface Plasmons • For TM waves, the mode funcionHy(x) can be written as • The mode condition can be obtained by insisting on the continuity of Ez at the interface x = 0 of from Eq.(11.2-11)

33. Surface Plasmons • The propagation constant β is given by • A confined propagation mode must have a real propagation constant, since <0,

34. Surface Plasmons • The attenuation constants p and q are given

35. Surface Plasmons • The electric field components are given

36. Surface Plasmons • Surface wave propagation at the interface between a metal and a dielectric medium suffers ohmic losses. The propagation therefore attenuates in the z direction. This corresponding to a complex propagation constants β • Where α is the power attenuation coefficient.

37. Surface Plasmons • In the case of a dielectric-metal interface, is a real positive number and is a complex number (n – iκ)^2, and the propagation constant of the surface wave is given

38. Surface Plasmons • In terms of the dielectric constants • The propagation and attenuation constants can be written as

39. Surface Plasmons • The attenuation coefficient can also be obtained from the ohmic loss calculation and can be written as • σ is the conductivity