Leaky mode

A leaky mode is a field pattern that propagates along a waveguide with a nearly fixed transverse shape while continuously shedding power into the surroundings. It differs from a guided mode, which is evanescent outside the core and lossless in an ideal structure, and from the continuum of radiation modes, which carry no confined power at all. A leaky mode has a discrete, complex propagation constant $\beta = \beta_r - i\alpha$, and its power decays as $e^{-2\alpha z}$.

Complex effective index and loss

Mode solvers report a complex effective index, $n_\text{eff} = \beta/k_0$, with $k_0 = 2\pi/\lambda$. The power attenuation coefficient is

$$\alpha_p = 2k_0\,|\operatorname{Im}(n_\text{eff})|,$$

and the loss in dB per unit length is $4.343\,\alpha_p$. At 1550 nm, $\operatorname{Im}(n_\text{eff}) = 10^{-5}$ gives $\alpha_p = 81$ /m, or 3.5 dB/cm; $10^{-6}$ gives 0.35 dB/cm. Because the leaky field keeps carrying power outward, a mode solver needs absorbing boundaries (a perfectly matched layer) to compute it, and hard walls turn the leakage into spurious standing waves.

Outside the guide the field of a leaky mode grows exponentially with transverse distance, because the light now far from the axis left the core earlier, when the mode was stronger. The mode is therefore not normalizable over an infinite cross-section and is a useful approximation only over a finite region and length.

Where leaky modes occur

• Substrate-leaky guides. When a high-index substrate lies beneath a thin lower cladding, the mode's evanescent tail couples to waves propagating into the substrate. This is substrate leakage in silicon on insulator with a thin buried oxide, and the same mechanism limits rib waveguides whose effective index falls below that of the adjacent slab mode.
• Bends. A bent guide radiates from the outer side of the bend; its modes are leaky, and their imaginary part gives the bend loss.
• Fibers near cutoff. Modes just past cutoff, and skew-ray modes in multimode fiber, are leaky with low loss and can persist for meters, which is why short multimode samples carry more high-order light than long ones.
• Antiresonant guides. In an antiresonant reflecting optical waveguide (ARROW) or an antiresonant hollow-core fiber, the core has a lower index than its surroundings, so every core mode is leaky by construction. Thin high-index layers or membranes held at antiresonance reflect strongly back toward the core, keeping the leakage low. For a silica membrane of thickness $t$ the high-loss resonances fall at $\lambda_m = 2t\sqrt{n^2-1}/m$: with $t = 0.5$ µm and $n = 1.45$, at 1050 nm and 525 nm, with low-loss windows between them.

Pitfalls

Comparing leaky-mode losses between simulations requires the same boundary treatment and window size, since a perfectly matched layer placed too close to the core absorbs part of the field and raises the apparent loss. In measurements, a cutback over too short a length can include power still carried by leaky modes and underestimate the loss of the fundamental mode; a mode filter or longer length removes them.

Common questions

What is the difference between a leaky mode and a radiation mode?

A radiation mode belongs to a continuous set with real propagation constants and fills the whole cladding. A leaky mode is a single discrete solution with a complex propagation constant, which approximates a resonant superposition of radiation modes that stays concentrated near the core for some distance.

How is the loss of a leaky mode calculated?

A mode solver with a perfectly matched layer returns the complex effective index; the power loss is $2k_0|\operatorname{Im}(n_\text{eff})|$, multiplied by 4.343 for dB per unit length.

References: A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983); D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic Press, 1991); M. A. Duguay, Y. Kokubun, T. L. Koch and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13 (1986).