1. Lecture 18 Chapter XI Propagation and Coupling of Modes in Optical Dielectric Waveguides – Periodic Waveguides Highlights (a) Periodic (corrugated) WG and filters; (b) Distributed feedback lasers; (c) EO mode coupling; (d) Directional couplers. Propagation of optical modes in dielectric films with thicknesses comparable to the wavelength Generate, guide, modulate, and detect light in thin film configuration ---- integrated optics Basic problem of TE and TM mode propagation in slab dielectric waveguides. A coupled-mode formalism to describe situations involving exchange of power between modes.

2. §11.1 Waveguide Modes – A General Discussion A mode of a dielectric waveguide is a solution of the wave equation The solutions are subject to the continuity of the tangential components of E and H at the dielectric interfaces. We will limit our treatment to planar waveguides … I II III

3. §11.1 Waveguide Modes – A General Discussion Region I Region II Region III Assume Let ’ s consider the physical nature of solution as a function of the propagation constant

4. §11.1 Waveguide Modes – A General Discussion (a) in all 3 regions IIIIII (b) & (c) in region II in region I & III IIIIII (b) (c) Exponential behavior Confined modes is discrete

5. §11.1 Waveguide Modes – A General Discussion (d) (e) in all 3 regions in region II & III in region I IIIIII Substrate radiation modes IIIIII Radiation modes is continuous variable in (d) & (e)

6. §11.1 Waveguide Modes – A General Discussion Confined Modes in a Symmetric Slab Waveguide From Maxwell ’ s curl equations

7. §11.1 Waveguide Modes – A General Discussion Propagate along the z direction Transverse Electric (TE) modes : Electric field is restricted to the transverse plane Transverse Magnetic (TM) modes : Magnetic field is restricted to the transverse plane

8. §11.1 Waveguide Modes – A General Discussion Let ’ s consider the TE mode only Waveguide is symmetric about the plane x=0 Even modesOdd modes Interface continuous condition at x=d

9. §11.1 Waveguide Modes – A General Discussion From wave equations

10. §11.1 Waveguide Modes – A General Discussion Odd modes Interface continuous condition at x=d

11. §11.3 A perturbation Theory of Coupled Modes in Dielectric Optical Waveguides Let ’ s consider the modes coupling, we start with wave equation Assume perturbed field is a superposition of confined modes: where

12. §11.3 A perturbation Theory of Coupled Modes in Dielectric Optical Waveguides Substitute perturbed field into wave equation: “ slow ” variation approximation

13. §11.3 A perturbation Theory of Coupled Modes in Dielectric Optical Waveguides Above equation can be used to treat a large variety of mode interactions. Each physical example involves, in general, a different perturbation polarization P.

14. §11.4 Periodic Waveguide Guiding Layer Substrate Applications: Optical filtering, distributed feedback laser, etc. Corrugation couples only TE to TE and TM to TM modes, but not TE to TM

15. §11.4 Periodic Waveguide For TE mode propagation Assume index changes as:

16. §11.4 Periodic Waveguide Both source and wave need to have the same frequency and nearly the same phase dependence.

17. Some General Properties of the Coupled Mode Equations §11.4 Periodic Waveguide If the modes A and B carry power in the same direction If the modes A and B carry power in opposite direction

18. §11.4 Periodic Waveguide Two modes contradirectional coupling

19. §11.4 Periodic Waveguide The coupling constant for square-wave corrugation waveguide where So that

20. §11.4 Periodic Waveguide For l odd Guide wavelength of the sth mode

21. §11.4 Periodic Waveguide The problem of two-wave coupling by a corrugation has thus been reduced to a pair of coupled differential equations and an expression for the coupling constant.