Effective area (A_eff)

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

$$A_\text{eff} \;=\; \frac{\left[\int |E(x,y)|^2 \, dA \right]^2}{\int |E(x,y)|^4 \, dA}.$$

For a Gaussian mode of $1/e^2$ radius $w_0$:

$$A_\text{eff} \;=\; \pi w_0^2 / 2 \cdot 2 \;=\; \pi w_0^2.$$

Wait — the correct relation for a Gaussian mode is $A_\text{eff} = \pi w_0^2$. For a step-index fiber, the effective area differs from $\pi w_0^2$ by 10–20% because the actual mode profile is closer to a Bessel function $J_0$ than a Gaussian.

Typical values at 1550 nm:

Fiber	$A_\text{eff}$
SMF-28	80 μm²
LEAF (NZDSF)	70 μm²
Ultra-large effective area fiber	110 – 150 μm²
Dispersion-compensating fiber	15 – 25 μm² (much smaller)
Highly nonlinear fiber (HNLF)	12 – 20 μm²
Solid-core photonic crystal fiber	1 – 100 μm² (designed)
Hollow-core PCF	50 – 200 μm² (or much larger)
SOI strip waveguide	0.05 – 0.1 μm² (much smaller)

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 — important 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 SOI strip: $\gamma$ can exceed $10^4$ /(W·km) in nm-thick waveguides.

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.