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Understanding Optical Fiber Waveguides: TE Modes and Eigenvalues

Explore the dielectric slab guide in optical fibers and the wave equation governing it, focusing on TE modes and Eigenvalue equations. Learn about impedance continuity at interfaces, cutoff frequencies, and dispersion diagrams for TE waves in dielectric guides. Understand field profiles, confinement factors, and guided mode partitioning.

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Understanding Optical Fiber Waveguides: TE Modes and Eigenvalues

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

  2. 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:

  3. 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

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

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

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

  7. Additional comprehension for waveguide E(y) profile: n1=1.5, n2=1.495, d=10mm, l=1mm TE1 TE2 x Core 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

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

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