Chapter 6 OPTICAL FIBERS AND GUIDING LAYERS ◈ The dielectric slab guide (Waveguide) ▪ Wave equation (Governing eq.): TIR ▪ Solution: ▪ Direction separation: TE & TM

Transverse Electric (TE) Modes (1/3) ▪ TE field: ▪ Wave equation (previous): ▪ We can get the Eigen-value equation: TIR Each eigenfunction has one eigenvalue associated with it, ie, eigenfunctions and eigenvalues come in pairs . ▪ Considering : ▪ For core, we select a symmetric solution:

Transverse Electric (TE) Modes (2/3) ▪ To match the boundary condition, the impedance should be continuous (at the interface): moves toward the origin and intersections are lost ▪ All higher-order modes (m>0) have a cutoff  Waves are not guided below a certain critical frequency

Transverse Electric (TE) Modes (3/3) -- Even -- Odd r ▪ Let (Normalized term), then the previous solutions are represented as: - even case: - odd case: ▪ Graphical representation - Discrete # of the TE solutions (modes) - - Mode depends on the radius of the circle m=1 m=0 m=2 ▪ [Ex]Higher mode  

Dispersion diagram for TE waves in dielectric guide Higher mode  Less β

Numerical/Graphical representation ▪ Field profile of dominant mode for three different frequencies ▪ Dominant TE mode

Additional comprehension for waveguide E(y) profile: n1=1.5, n2=1.495, d=10mm, l=1mm TE1 TE2 Core x Cladding Even function solution Odd function solution TE3 →  E or energy penetrates (leaks) at the boundary Even function solution  TIR backward and forward in x-direction: Standing wave case

Additional comprehension for waveguide -- Even -- Odd r ▪ Confinement factor: G - How much power is confined within the core - How does G change for different modes? →  Energy penetrates (leaks) at the boundary ▪ Partitioning of input field into different guided modes. - Discrete modes  Summation of the solutions + n2 n1

Supplement studying materials

TE1 TE2 (Haus Fig. 6.4)

E(y) profile: n1=1.5, n2=1.495, d=10mm, l=1mm TE1 TE2

E(y) profile: n1=1.5, n2=1.495, d=10mm, l=1mm TE1 TE2 TE3

How does neff changes for different modes? - Effective index: neff= b/k0 d n1 n2 How does neff changes for different modes? - Confinement factor: G How much power is confined within the core How does G change for different modes?

+ + Partitioning of input field into different guided modes. (Sturm-Liouville theory) Dot product between Ein(y) and Em(y) Or projection of Ein(y) into basis Em(y)

For a given waveguide: V,a  b from the diagram cladding Asymmetric waveguide? d n1 core - Numerical solution n2 - Graphical solution b-V diagram for TE mode V b For a given waveguide: V,a  b from the diagram Then, determine b (Haus 6.11)