Optical Waveguide Modes
Symmetric slab waveguide: V-number, number of TE and TM modes, exact effective indices, core power fraction, b–V diagram and TE mode fields.
V-number
7.784
(πd/λ)√(n₁² − n₂²); NA = 0.3841
Guided modes
5 TE + 5 TM
⌊2V/π⌋ + 1; single-mode for d < λ/(2NA) = 2.018 µm
TE₀ effective index
1.498429
b = 0.96806; TM₀: 1.498406
TE₀ power in the core
99.63 %
θ_c = 75.16°, acceptance ±22.59° in air
A symmetric slab of thickness d and index n₁ between two claddings of index n₂. With u = V√(1 − b) and w = V√b, TE mode m satisfies u = mπ/2 + arctan(w/u), and TM modes have w multiplied by n₁²/n₂². The page solves these exactly by bisection. Mode m is cut off at V = mπ/2, so the fundamental TE₀ and TM₀ modes are always guided, and the normalized index b = (n_eff² − n₂²)/(n₁² − n₂²) runs from 0 at cutoff towards 1. Here V uses the half-thickness, so it is half the V of Kogelnik and Ramaswamy.
Normalized index b vs V (TE modes)
- TE0
- TE1
- TE2
- TE3
- TE4
- TE5
- This guide
TE mode fields across the slab
- TE0
- TE1
- TE2
- TE3
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Exact eigenvalue equations of a symmetric step-index slab (TE and TM), solved by bisection. Asymmetric slabs, graded profiles and channel (2D) waveguides are not covered.
- Kogelnik H., Ramaswamy V. (1974). Scaling rules for thin-film optical waveguides. Appl. Opt. 13, 1857. doi:10.1364/AO.13.001857