Optical Equations 4 Solution Techniques for Optical Model

1. Optical Equations 4 Solution Techniques for Optical Model

2. Solving the Eigen-Value Problem to Find the Mode • The above eigen-value problem can generally be solved numerically in a 2D domain once the space refractive index distribution in the cross-section (i.e., in the x-y plane) is given – the boundary condition is already defined by Maxwell’s equations. • Therefore, once a waveguide is deigned with its geometric dimensions and material refractive indices specified, the set of guided modes of this waveguide can be found.

3. Solution Techniques for Solving the Eigen-Value Problem • Numerical Methods • Finite difference method (easy to implement in mesh and formulation, hard to deal with boundary condition except for a few special cases) • Finite element (finite volume, boundary element) method (hard to implement in mesh and formulation, stable and accurate) • Collocation method (easy mesh, hard formulation, the easiness on boundary treatment in between) • Global wave expansion method (suitable for dealing with waveguides with continuous refractive index distribution)

4. Solution Techniques for Solving the Eigen-Value Problem • Analytical Methods • Direct method for treating triple layer slab waveguide • Transfer matrix method for treating slab waveguide with arbitrary layers • Effective index method • Variational method (when a reference waveguide with a known solution exist)

5. Solving the Traveling-Wave Equation to Find the Field envelope • The above coupled traveling wave equations can be solved numerically along the cavity (i.e., in z-direction) once the material gain, refractive index change, and optical loss (non-inter-band absorption) are given under specified spontaneous emissions. • Therefore, once a cavity is deigned with its geometric dimensions and material properties (i.e., the susceptibility or the gain and refractive index change) specified, the field envelops can be found.

6. Solution Techniques for Solving the Traveling-Wave Equation • Numerical Methods • Time domain method (traveling wave method) • Frequency domain method (standing wave method) • Mixed domain method (to deal with gain/index dispersion in broadband devices)

7. Solution Techniques for Solving the Traveling-Wave Equation • Analytical Method for Oscillation (Lasing) Condition Unified solution form with Matching the facet condition to obtain: Zero facet reflections, for study of pure grating effects General case when there are arbitrary facet reflections ?

8. Purely Index Coupled Grating Couple mode equation leads to an eigen equation >0 <0 Spectral symmetry causes dual mode operation. Threshold gain and lasing wavelength n1 If  is a root, -, * are all roots. Therefore, if (, ) is a solution set, (, -) is a solution set too. Solution at =0 is forbidden. n2 Fully symmetrical structure

9. Complex (Gain/Loss-Index) Coupled Grating Spectral symmetry removed by complex coupled grating Couple mode equation leads to an eigen equation >0 <0 Threshold gain and lasing wavelength n1 g1 If  are roots for , * are roots for *. Once  is complex, *. Hence, if (, ) is a solution set, (, -) is not a solution set. =0 can also be a solution when  is purely imaginary. n2 g2 Fully symmetrical structure with g1 > g2

10. Phase Shifted Grating Uniform grating forms a BRF. To obtain a BPF, we can cascade two BRFs, with one’s right edge aligned to the other’s left edge. It is also equivalent to fold the cavity in the middle to form a uniform grating with unsymmetrical facets. Hence the structural symmetry is broken and single mode operation is possible inside the stop-band (e.g., at =0). Degeneracy removed by phase-shifted grating =0 n1 n2 /4 phase-shift in centre

11. Other Structures/Devices • FP cavity (can be viewed as special case of DFB) • Broadband device such as SOA and SLED • Gain and index dispersion must be considered • Traveling wave model has to be used since there is no standing wave at all • Absorbing devices such as EAM and waveguide photo-detector • - Similar to SOA and SLED • - Only absorption (negative gain) needs special treatment • High power device with horn waveguide