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Waveguides and Optical Fibres

Solve the guided modes of a symmetric dielectric slab (exact TE and TM) and a weakly guiding step-index fibre (LP approximation). See confinement, evanescent tails, effective index, cut-off, numerical aperture, the V-number and modal dispersion, and link each mode to its zig-zag total-internal-reflection ray.

Sliders have number boxes: type a value and press Enter • Click the b–V or sweep plots to set V, λ₀ or size
V — V-number
# — Guided modes
n — n_eff
Γ — Power in core
NA — Numerical aperture
λc — Single-mode cut-off

Guided field Re{ψ(x) ei(βz − ωt)} exact slab TE

Snapshot of the transverse field in the x–z plane (diverging colours, ± peak). Grey lines are the core boundaries x = ±d/2. The cladding shows the evanescent tail e−γ|x|. Arrows show the two plane-wave (ray) directions ±θ that add up to the mode.

Ray picture ↔ computed mode β = n₁k₀ cos θ

Zig-zag ray at the angle θ = arccos(neff/n₁) taken from the solved mode. The z axis is compressed (see its tick labels) so that the bounces are visible. The dashed ray is the steepest guided ray (θmax = arccos(n₂/n₁), the critical angle at the wall). The cone on the left is the acceptance angle in air, arcsin NA. Right: the field profile of the same mode.

V = k₀(d/2)NA
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NA = √(n₁² − n₂²)
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Acceptance (air)
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Mode
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n_eff
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β
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b = (n_eff² − n₂²)/NA²
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u, w
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Γ (power in core)
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Tail 1/e depth
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Ray angle θ
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θ_max = arccos(n₂/n₁)
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Round-trip phase check
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Group index n_g
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Waveguide D_w
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Mode cut-off V
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Transverse profile peak-normalised

Field (solid) and intensity (dashed) across the guide; markers at the core edges. Move the pointer to read values.

All guided modes

Every guided mode of the chosen polarization, offset vertically; the mode with m nodes has m zero crossings.

Universal b–V diagram slab

Normalised propagation constant b against V. Dots are the current modes and the vertical line is the current V. Click to set V (changes d).

Effective index vs λ₀

neff of each mode, which lies between n₂ and n₁ (grey lines) while guided and ends where the mode is cut off. Click to set the swept variable.

Guided power fraction Γ

Fraction of the mode power travelling in the core (TE: ∫|Ey|²; TM: ∫|Hy|²/n²; LP: ∫|ψ|²). Γ → 0 at cut-off for the slab's higher modes and for the LP0m and LP1m fibre modes; the slab's TE0/TM0 have no cut-off and Γ → 0 only as V → 0. LPlm modes with l ≥ 2 keep a finite fraction Γ → (l − 1)/l in the core at cut-off.

Effective and group index vs λ₀ numerical d/dλ

neff (solid) and ng = neff − λ₀ dneff/dλ₀ (dashed) for the displayed mode, from central differences of the solved neff(λ₀).

Evanescent coupling between two identical guides exact supermodes + CMT

Power launched into guide 1 moves to guide 2 and back. P₂(z) = sin²(πz / 2Lc) for phase-matched lossless guides.

β_even − β_odd (exact)
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L_c exact = π/Δβ
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L_c coupled-mode
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CMT / exact
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Modal dispersion over 1 km non-dispersive materials

Ray estimate Δτ = L(n₁/c)(n₁/n₂ − 1)
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Mode-based spread of n_g
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Single-mode intermodal delay
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Rough bit-rate limit ≈ 1/(2Δτ)
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The ray estimate compares the axial ray with the steepest guided ray. The mode-based value is the spread of group delays of the guided modes computed here. A single-mode guide has no intermodal delay; what remains is chromatic dispersion (Dw above, plus material dispersion, which this tool leaves out).

Guided modes
💡 How to use

Learn with this tool

Learning objectives

  • Derive and solve the TE/TM eigenvalue equations of a symmetric slab. Explain why guided modes have n₂k₀ < β < n₁k₀ and evanescent cladding tails.
  • Link a guided mode to a zig-zag TIR ray through β = n₁k₀ cos θ and the transverse-resonance condition, including the Fresnel TIR phase.
  • Use V, b and cut-off conditions to count modes, and predict when a step-index fibre is single-mode (V < 2.405) and why that number does not apply to a slab.

Prerequisites

Model

Fields are E(x, z, t) = Re{ψ(x) ei(βz − ωt)} in a lossless, isotropic, non-magnetic structure with the core |x| < a = d/2 of index n₁ inside a cladding of index n₂ < n₁. A guided mode has an oscillating core field (transverse wavenumber kx) and a decaying cladding field (decay constant γ). The dimensionless numbers u = a kx and w = a γ obey u² + w² = V², with V = k₀ a √(n₁² − n₂²).

TE (Ey): even w = u tan u, odd w = −u cot u

TM (Hy): even w = (n₂/n₁)² u tan u, odd w = −(n₂/n₁)² u cot u

LPlm fibre: u Jl−1(u)/Jl(u) = −w Kl−1(w)/Kl(w)

b = w²/V² = (neff² − n₂²)/(n₁² − n₂²), β = k₀ neff = n₁ k₀ cos θ

Solver. Each equation is rewritten without poles, for example r u sin u − w cos u = 0 for even slab modes, and u Jl−1(u) + [w Kl−1(w)/Kl(w)] Jl(u) = 0 for LP modes. It is then solved with Brent's method on brackets whose ends are known analytically: slab mode m lies in u ∈ [mπ/2, (m+1)π/2); LPlm lies between its cut-off and jl,m. Therefore no mode can be skipped or counted twice. The group index is a central difference of neff(λ₀). The coupler uses the exact five-layer supermodes (slab) and coupled-mode theory (slab TE and fibre LP01).

n₁, n₂
core and cladding refractive indices (constant in λ₀ here)
d, a
full slab thickness or core diameter; half-width or core radius a = d/2
λ₀, k₀
vacuum wavelength; k₀ = 2π/λ₀
NA
numerical aperture √(n₁² − n₂²) = sin of the acceptance half-angle in air
V
normalised frequency k₀ a NA (half-width convention for the slab too)
β, n_eff
propagation constant and effective index β/k₀
u, w, b
a kx, a γ, and the normalised propagation constant w²/V²
Γ
confinement factor: fraction of the guided power flowing in the core
θ
angle of the equivalent ray to the guide axis
n_g, D_w
group index neff − λ₀ dneff/dλ₀; waveguide dispersion −(λ₀/c) d²neff/dλ₀²

Cut-offs. Slab TEm/TMm: V = mπ/2, so the slab has floor(2V/π) + 1 modes per polarization and TE0/TM0 never cut off. Fibre LP0m: V = j1,m−1 (LP01 never cuts off). LPlm, l ≥ 1: V = jl−1,m. LP11 cuts off at j0,1 = 2.405, which is the single-mode condition of the ideal, weakly guiding, step-index circular fibre only. It is not a universal waveguide constant; the slab's second mode appears at V = π/2 in this convention.

Derivation: slab TE eigenvalue equation and its ray meaning

For TE, E = ŷ Ey(x) ei(βz−ωt) satisfies Ey'' + (n²k₀² − β²)Ey = 0. In the core n₁²k₀² − β² = kx² > 0 gives cos kxx or sin kxx. In the cladding β² − n₂²k₀² = γ² > 0 gives e−γ|x|, which is the only choice that stays finite. Ey and Hz ∝ dEy/dx are tangential, so both are continuous at x = ±a. For the even solution, dividing the two conditions gives kx tan(kxa) = γ, that is w = u tan u. The odd solution gives w = −u cot u. For TM the continuous quantities are Hy and Ez ∝ (1/n²) dHy/dx, which introduces the factor (n₂/n₁)².

Write cos kxx = ½(eikxx + e−ikxx). The mode is then two plane waves travelling at ±θ with kx = n₁k₀ sin θ and β = n₁k₀ cos θ. Such a wave is guided only if it is totally internally reflected, which requires θ < arccos(n₂/n₁), exactly the condition β > n₂k₀. At each wall the TIR reflection adds the Fresnel phase φr = −2 arctan(γ/kx) (TE). The wave must reproduce itself after a round trip: 2kxd + 2φr = 2πm. Rearranged, this is the same eigenvalue equation. The readout "round-trip phase check" evaluates it for the solved mode.

Power in the core. With Sz ∝ β|Ey|² (TE), Γ = ∫core|Ey|² / ∫|Ey|². For the even mode this is [1 + sin 2u/2u] / [1 + sin 2u/2u + cos²u/w]. For non-dispersive media an energy argument gives ng neff = n₁²Γ + n₂²(1 − Γ). The tests check the numerical derivative against this identity.

Derivation: LP modes of a weakly guiding fibre

When Δ = (n₁² − n₂²)/2n₁² ≪ 1, the transverse field is almost linearly polarized and satisfies the scalar Helmholtz equation ∇t²ψ + (n²k₀² − β²)ψ = 0. Separating variables gives ψ = Jl(ur/a) cos lφ in the core and Kl(wr/a) cos lφ in the cladding. Continuity of ψ and ∂ψ/∂r at r = a, together with the Bessel recurrences, gives u Jl−1/Jl = −w Kl−1/Kl. At cut-off w → 0, and the right-hand side goes to 0 (for l ≥ 1) or K₁/K₀·w → 0 (for l = 0). Hence Jl−1(V) = 0. For LP11 this is J₀(V) = 0, so Vc = 2.405. Each LP mode groups nearly degenerate vector modes; LP11, for example, groups TE01, TM01 and HE21. These are 2 (l = 0) or 4 (l ≥ 1) degenerate field patterns, and that is where the total-mode estimate ≈ V²/2 comes from.

Exercise 1: when does a fibre become single-mode?

  1. Load SMF at 1550 nm (n₁ = 1.4492, n₂ = 1.4440, 2a = 8.2 µm). Predict the cut-off wavelength λc below which LP11 is guided.
  2. Lower λ₀ with the slider or number box until the mode count jumps from 1 to 2.
  3. Note the λ₀ where LP11 appears in the table and where the LP11 curve starts on the sweep plot.
  4. Explain why the value comes from J₀(V) = 0 rather than from V = π/2.
Show answer

λc = 2πa NA / 2.405 = 2π(4.1 µm)(0.1227)/2.405 ≈ 1314 nm. Below it V > 2.405 and LP11 (TE01, TM01, HE21) is guided. The LP11 field has one azimuthal node (cos φ), and its eigenvalue equation at w → 0 reduces to J₀(V) = 0. The value π/2 belongs to the slab's odd TE1/TM1 mode in the half-thickness convention.

Exercise 2: limiting case, a vanishingly thin slab

  1. Load Thin slab (d = 0.1 µm, V ≈ 0.05). Is TE0 still guided? What happens to b, Γ and the tail depth as d → 0?
  2. Reduce d further with the number box, for example to 0.04 µm.
  3. Record b and Γ and compare b with V².
  4. Explain why the symmetric slab's fundamental mode has no cut-off. What would change for an asymmetric slab (different cover and substrate)?
Show answer

TE0 stays guided for every V > 0 because u tan u = w always has a root in (0, π/2). For small V, u ≈ V and w ≈ u² ≈ V², so b ≈ V² ≈ 0.0024 at d = 0.1 µm. Γ ≈ 0.005, and the 1/e tail depth 1/γ ≈ 20 µm is far larger than the core, so the mode is almost entirely in the cladding. An asymmetric slab does have a TE0 cut-off. The 2.405 rule does not apply to either.

Exercise 3: rays and modes agree

  1. Load Slab, two modes. Which mode has the steeper ray, and which spends more time in the cladding?
  2. Switch the displayed mode between TE0 and TE1 and watch the ray canvas and the Γ readout.
  3. Read θ for both modes and check the round-trip phase residual 2kxd + 2φr − 2πm.
  4. Why does a steeper ray mean a smaller neff and a longer evanescent tail?
Show answer

TE0: θ ≈ 3.7°, neff ≈ 1.4969, Γ ≈ 0.96. TE1: θ ≈ 7.2°, neff ≈ 1.4883, Γ ≈ 0.80. The residual is at the 10⁻¹⁵ level, so the TIR ray picture and the wave solution are the same statement. A steeper ray has a smaller axial wavevector β = n₁k₀ cos θ and meets the wall closer to the critical angle. There γ = k₀√(neff² − n₂²) is smaller, so the tail reaches further.

Exercise 4: modal dispersion and the coupler

  1. Load Multimode 50 µm. Estimate the intermodal delay over 1 km from rays. Then load Directional coupler and predict how Lc changes when the gap grows by 1 µm.
  2. Read the modal dispersion panel. In the coupler, change s from 2 µm to 3 µm.
  3. Compare the ray and mode-based Δτ. Measure the ratio Lc(3 µm)/Lc(2 µm) and compare it with eγ·1 µm.
  4. Why is the ray estimate an upper bound? Why is the growth of Lc exponential?
Show answer

Δτray = (1.48/c)(1.48/1.46 − 1) × 1 km ≈ 68 ns. The computed modes give ≈ 63 ns because the highest modes are not at the critical angle. That limits the rate to roughly 1/(2Δτ) ≈ 7–8 Mbit/s over 1 km. In the coupler, κ ∝ e−γs because the two fields overlap only through their evanescent tails. So Lc grows by eγΔs, and the exact supermodes and CMT agree to better than 1 %.

Worked example: a step-index single-mode fibre at 1550 nm

Take n₁ = 1.4492, n₂ = 1.4440, core diameter 8.2 µm, λ₀ = 1550 nm.

  1. NA = √(1.4492² − 1.4440²) = 0.1227. The acceptance half-angle in air is arcsin 0.1227 = 7.05°.
  2. V = (2π/λ₀) a NA = (2π/1.55 µm)(4.1 µm)(0.1227) = 2.039 < 2.405, so only LP01 is guided (two polarizations).
  3. Solving u J₁(u)/J₀(u) = w K₁(w)/K₀(w) with u² + w² = V² gives u = 1.541 and w = 1.334, so b = w²/V² = 0.4285. The Rudolph–Neumann fit (1.1428 − 0.996/V)² gives 0.4280.
  4. neff = √(n₂² + b NA²) = 1.44623 and β = 5.862 rad/µm. The equivalent ray travels at θ = arccos(neff/n₁) = 3.67°, below θmax = 4.86°.
  5. Γ = 0.751, so a quarter of the power travels in the cladding. The field 1/e depth a/w is 3.07 µm, which is why the cladding must be much thicker than the core.
  6. Cut-off wavelength: λc = 2πa NA/2.405 = 1314 nm. The numerical derivative gives ng = 1.4496 and Dw ≈ −4.8 ps/(nm·km). Silica's material dispersion (about +22 ps/(nm·km) at 1550 nm, not modelled here) dominates the total.

Load SMF at 1550 nm to check each number in the readouts.

When the model fails

  • LP approximation: it needs Δ ≪ 1. For large contrast (for example silicon in silica) the true fibre modes are hybrid HE/EH/TE/TM modes with split neff. Use a full vector solver. The tool warns when Δ > 5 %.
  • No material dispersion or loss: n₁ and n₂ are constant, so ng and Dw are waveguide contributions only. There is no absorption, scattering, bend loss or leaky/cladding modes. Very near cut-off a real fibre's mode is lost to bends and finite cladding long before b = 0.
  • Geometry: the slab is symmetric and infinite in y; rectangular channel guides need 2D numerics (Marcatili and effective-index methods are approximations). The fibre is an ideal step-index circle; graded-index fibres have much lower modal dispersion than the step-index ray estimate.
  • Coupled-mode theory: it is accurate only for weak coupling (gap ≳ a tail length). At small gaps compare it with the exact slab supermodes. The fibre CMT formula has no exact counterpart here, and real couplers also have polarization dependence and fabrication asymmetry.
  • Mode counts: floor(2V/π) + 1 (slab, per polarization) and 2.405 (fibre) are exact only for these ideal structures. Asymmetric, graded or anisotropic guides have different cut-offs.

References

  • A. W. Snyder and J. D. Love, Optical Waveguide Theory, Chapman & Hall (1983): slab and fibre eigenvalue equations, Γ, coupled fibres.
  • K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed., Academic Press (2006), Ch. 2–3.
  • D. Gloge, "Weakly guiding fibers," Appl. Opt. 10, 2252 (1971): LP modes and cut-offs.
  • A. Yariv and P. Yeh, Photonics, 6th ed., Oxford (2007), Ch. 3 and 13: TIR ray picture and directional-coupler CMT.
  • B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019), Ch. 9–10.