Effective area (A_eff)

The effective area of a guided mode quantifies the transverse area over which optical power is concentrated, weighted by intensity:

$$A_{\text{eff}}=\frac{{[\int \mathrm{\mid }E(x,y){\mathrm{\mid }}^{2}dA]}^2}{\int \mathrm{\mid }E(x,y){\mathrm{\mid }}^{4}dA}\mathrm{.}$$

For a Gaussian mode of $1\mathrm{/}{e}^2$ radius $w_0$:

$$A_{\text{eff}}=\pi w_0^2\mathrm{.}$$

In a step-index fiber the mode is not exactly Gaussian ($J_0$ in the core and a $K_0$ tail in the cladding), so the effective area differs from $\pi w_0^2$ computed from the mode field diameter. For standard single-mode fiber the difference is a few percent: an exact step-index calculation for a V number near 2 gives $A_{\text{eff}}=0.97$ to $0.98\pi w_0^2$, about 83 μm² for a 10.4 μm mode field diameter ($\pi \times {5.2}^2=85$ μm²), close to the 80 μm² usually quoted. Fibers whose mode departs further from a Gaussian, such as dispersion-shifted designs, differ more.

Typical values at 1550 nm:

Nonlinear coefficient. $A_{\text{eff}}$ enters the fiber nonlinear coefficient $\gamma$:

$$\gamma =\frac{2\pi n_2}{\lambda A_{\text{eff}}},$$

where $n_2$ is the Kerr coefficient. Higher $\gamma$ means stronger nonlinear effects (self-phase modulation, four-wave mixing, soliton formation) per unit power. Conversely, larger $A_{\text{eff}}$ reduces nonlinear effects, which matters for high-power CW transmission.

For SMF-28 at 1550 nm: $\gamma \approx 1.3$ /(W·km). For HNLF at 1550 nm: $\gamma \approx 10$ /(W·km). For an SOI strip (500 × 220 nm): $\gamma \sim {10}^5$ /(W·km), of order 100 – 300 /(W·m).

Engineering tradeoffs. Large $A_{\text{eff}}$ favors:

• High-power transmission (less nonlinearity, less SPM)
• Long-haul telecom transmission with high per-channel power

Small $A_{\text{eff}}$ favors:

• Nonlinear devices (wavelength converters, parametric amplifiers)
• Strong tight bending in PICs
• Short-distance, high-density integration

Modern dispersion-shifted fibers (Corning LEAF, OFS True-Wave) achieve $A_{\text{eff}}\sim 70$ μm² to suppress nonlinear penalty while maintaining the low-dispersion characteristics needed for DWDM.

Distinguish from mode field diameter (MFD): MFD is a length scale (typically 10.4 μm for SMF-28 at 1550 nm), while $A_{\text{eff}}$ is an area (80 μm² for the same fiber). They are related but not identical. MFD is defined by the Petermann II second moment integral, while $A_{\text{eff}}$ is the intensity-weighted integral above.