Confinement factor (Γ)

The optical confinement factor is the fraction of the guided mode's power that lies within a specified region of the cross-section, usually the active region of a laser or the absorber of a detector:

$$\mathrm{\Gamma }=\frac{{\displaystyle {\int }_{active}\mathrm{\mid }E(x,y){\mathrm{\mid }}^{2}dxdy}}{{\displaystyle {\int }_{all}\mathrm{\mid }E(x,y){\mathrm{\mid }}^{2}dxdy}}$$

where $E(x,y)$ is the transverse mode field. A material gain $g$ in the active region, computed from the carrier density and the band structure, becomes a modal gain $\mathrm{\Gamma }g$ for the guided mode, and a material absorption $\alpha$ becomes a modal loss $\mathrm{\Gamma }\alpha$; the modal loss of a whole structure is the confinement-weighted sum over its layers. The definition above is the power confinement in the weak-guidance sense. In high-index-contrast waveguides the rigorous quantity, the ratio of the electromagnetic energy stored in the region to the total energy weighted by the group velocity, can differ from the power fraction and can exceed unity in slot and slow-light structures, which is a property of the energy density rather than a paradox.

The symmetric slab

For a symmetric slab of thickness $d$, core index $n_1$ and cladding index $n_2$, the TE₀ confinement factor has a closed form. With the transverse wavenumbers $\kappa$ in the core and $\gamma$ in the cladding, related by the eigenvalue equation $\tan (\kappa d\mathrm{/}2)=\gamma \mathrm{/}\kappa$ and ${\kappa }^2+{\gamma }^2=k_0^2(n_1^2-n_2^2)$,

$$\mathrm{\Gamma }=\frac{d\mathrm{/}2+\sin (\kappa d)\mathrm{/}2\kappa }{d\mathrm{/}2+\sin (\kappa d)\mathrm{/}2\kappa +{\cos }^2(\kappa d\mathrm{/}2)\mathrm{/}\gamma }$$

and the approximation $\mathrm{\Gamma }\approx D^2\mathrm{/}(2+D^2)$ with $D=k_{0}d\sqrt{n_1^2-n_2^2}$ reproduces it within 2 % for any thickness. Evaluated at 1550 nm this gives 0.81 for a 220 nm silicon slab between oxide claddings, 0.66 for a 150 nm slab, 0.72 for a 400 nm silicon nitride slab and 0.92 for an 800 nm one, and 0.38 for a 200 nm InGaAsP core between InP claddings: the high-contrast silicon platform reaches the same confinement in a film less than half as thick, since about 86 nm of silicon gives the 0.38 that takes 200 nm of InGaAsP. The same formula shows why a single quantum well cannot guide: an 8 nm layer with a 0.15 index step has $D=0.03$ and would confine 0.05 % of a mode on its own. The mode is instead shaped by the separate-confinement heterostructure around the wells, and each well then takes the slice of that mode that passes through it.

Typical values

Wells, threshold and speed

More wells give a larger $\mathrm{\Gamma }$ and a larger modal gain at a given carrier density, but the same current is then shared among more wells, so each well reaches transparency later. The threshold condition of a Fabry-Pérot laser,

$$\mathrm{\Gamma }{g}_{th}={\alpha }_i+{\alpha }_m,{\alpha }_m=\frac{1}{L}\ln \frac{1}{\sqrt{R_1{R}_2}}$$

sets the material gain the wells must supply, and because gain saturates logarithmically with carrier density, a small $\mathrm{\Gamma }$ pushes the operating point onto the flat part of the gain curve where the differential gain is low and the threshold current and its temperature sensitivity are high. The usual compromises follow: 5 to 8 wells for Fabry-Pérot edge emitters, 5 to 10 for DFB lasers, whose grating adds outcoupling loss, and 1 to 3 wells for VCSELs, placed at the antinode of the standing wave so that each well sees twice the average intensity. In the VCSEL the relevant quantity is the longitudinal confinement, the overlap of a few nanometers of active material with the standing-wave pattern along a cavity a wavelength long, and the enhancement factor of 2 at the antinode is part of the design.

Waveguides and detectors

In passive waveguides the confinement factor of the core decides how much the mode feels the boundaries. The 20 to 25 % of a silicon wire mode that travels in the cladding is what scatters from sidewall roughness and what couples across the gap of a directional coupler, and the rib waveguide trades confinement for a lower field at the etched wall. A ring resonator sensor or an evanescent-field absorption cell wants the opposite, the largest possible fraction of the mode outside the core, and thin cores or slot waveguides push the cladding fraction past 50 %. In a germanium-on-silicon photodiode the absorption length is $1\mathrm{/}(\mathrm{\Gamma }{\alpha }_{Ge})$, so with a confinement of 0.6 in a material absorbing a few thousand per centimeter at 1550 nm, 90 % absorption takes 10 to 20 µm, and the device is made short enough for bandwidth by choosing the geometry that maximizes $\mathrm{\Gamma }$ rather than by thickening the germanium.

Computing it

Beyond the slab, $\mathrm{\Gamma }$ requires the transverse mode of the full structure from a finite-difference or finite-element mode solver, followed by the integral over the region of interest; every commercial photonic and laser design package reports it as a standard output, and Marcatili's separable approximation gives rectangular-core results within 10 to 20 % of the numerical answer. The Waveguide Mode Explorer solves the slab case exactly and reports the confinement in the core along with the effective index and the evanescent tails.

References: L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), appendix 5 and chapter 4. D. Botez, "Analytical approximation of the radiation confinement factor for the TE₀ mode of a double heterojunction laser," IEEE Journal of Quantum Electronics 14, 230 (1978). J. T. Robinson, K. Preston, O. Painter, and M. Lipson, "First-principle derivation of gain in high-index-contrast waveguides," Optics Express 16, 16659 (2008). T. Visser, H. Blok, B. Demeulenaere, and D. Lenstra, "Confinement factors and gain in optical amplifiers," IEEE Journal of Quantum Electronics 33, 1763 (1997).