1. 16.711 Lecture 4 Optical fibers Last Lecture • Fiber modes • Fiber Losses • Dispersion in single-mode fibers • Dispersion induced limitations • Dispersion management • The Graded index fibers

2. 16.711 Lecture 4 Mode-coupling Today • Mode–coupling theory • Directional coupler • Coupling between guided and un-guided modes • Coupling from waveguide to free space • Mode coupling by scattering • Coupling to radiating modes at waveguide bands

3. 16.711 Lecture 4 Mode-coupling Mode–coupling theory • Why knowledge of mode coupling is important? • DFB Lasers/DBR Lasers • VCSEL structure • March-Zehnder interferometer • Direction coupler • Optimized device structures

4. 16.711 Lecture 4 Mode-coupling Mode–coupling theory • Recall mode equations TE Mode: Notice: • Mode profile E(x, y) doesn’t change with z if no index perturbation. • No coupling between different modes if no index perturbation.

5. 16.711 Lecture 4 Mode-coupling Mode–coupling theory • index perturbations • Generalized propagation equation

6. 16.711 Lecture 4 Mode-coupling • perturbation theory, TE mode Mode–coupling equation

7. 16.711 Lecture 4 Mode-coupling • discussion of the mode-coupling equation • if no index perturbation, A(z) is an constant, goes back to the normal equation. • Cn measures how much the index perturbation changes the propagation constant n. • If index perturbation doesn’t vary with z, coupling between the mode depends on the .

8. 16.711 Lecture 4 Mode-coupling • directional coupler

9. 16.711 Lecture 4 Mode-coupling • directional coupler --- solution of the differential equation set (1) E2(z=0) =0: • Output field power is half of the exciting waves. • the field of the unexcited waveguide is delayed by /2 with respect to the exciting wave.

10. 16.711 Lecture 4 Mode-coupling • directional coupler --- Applications (1) Signal switch (2) WDM coupler

11. 16.711 Lecture 4 Mode-coupling (2) if • Application: Y modulator

12. 16.711 Lecture 4 Mode-coupling • Phase-mismatched directional coupler • General solution E10 =1, E20=0:

13. 16.711 Lecture 4 Mode-coupling • Distributed Bragg Reflective (DBR) Structure Mode–coupling equation

14. 16.711 Lecture 4 Mode-coupling • Distributed Bragg Reflective (DBR) Structure Small variations Phase matching condition:

15. 16.711 Lecture 4 Mode-coupling • General solution Phase matching condition:

16. 16.711 Lecture 4 Mode-coupling • Example General Phase matching condition: Grating assistant coupling:

17. 16.711 Lecture 4 Mode-coupling • Phase mismatched Bragg Mirror Phase mismatching is related to wavelength, wavelength sensitive mirror.

18. 16.711 Lecture 4 Mode-coupling • Coupling between guided and unguided modes (a) prism coupling (b) grating assisted coupling (c) end-fired coupling

19. 16.711 Lecture 4 Mode-coupling • end-fired coupling • Coupling efficiency: • fm(x) is the outgoing guided mode profile, and fg(x) is the incoming waveguide excited mode profile.

20. 16.711 Lecture 4 Mode-coupling • Coupling form waveguide to Free-space • Model coupling by scattering:

21. 16.711 Lecture 4 Mode-coupling • Coupling to radiating modes at waveguide bends • Example: