# Effective Mode Area

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a quantitative measure of the area which a waveguide or fiber mode effectively covers in the transverse dimensions

Categories: fiber optics and waveguides, nonlinear optics

Units: m^{2}

Formula symbol: <$A_\textrm{eff}$>

The effective mode area is a frequently encountered concept in the context of fiber optics and other waveguides.

Modes of fibers or other waveguides have smooth transverse profiles with no well-defined outer boundary. The definition of a mode area is thus not straightforward, particularly for complicated mode shapes where e.g. some <$1/e^2$> intensity criterion as for Gaussian beams is not sensible. A common and useful definition for the effective mode area is

$${A_{{\rm{eff}}}} = \frac{{{{\left( {\int {{{\left| E \right|}^2}\;{\rm{d}}A} } \right)}^2}}}{{\int {{{\left| E \right|}^4}\;{\rm{d}}A} }} = \frac{{{{\left( {\int {I\;{\rm{d}}A} } \right)}^2}}}{{\int {{I^2}\;{\rm{d}}A} }}$$where <$E$> is the electric field amplitude and <$I$> is the optical intensity. The integration is done not only over the core area of a fiber, but over the whole plane. For a Gaussian beam with radius <$w$>, the effective area is <$\pi w^2$>, and the same equation is valid for relating the mode area to the effective mode radius. The equation would also hold for a rectangular (flat-top) intensity profile with radius <$w$>.

Note that the article on mode radius contains an equation that can be used for calculating the mode radius (and thus the effective mode area) for a step-index fiber.

Figure 1 shows a Gaussian mode and a rectangular profile with the same effective mode area. It may look as if the Gaussian profile had a much larger area, since it has a larger area under the curve. However, this is misleading: one needs to do a two-dimensional integration.

For the same optical power, the Gaussian mode has twice the peak intensity.

For a higher-order mode of a fiber (Figure 2), the intensity peaks are substantially higher than for the rectangular profile:

From the effective mode area and the nonlinear index <$n_2$>, one can calculate the nonlinear phase shift resulting from the Kerr effect:

$${\varphi _{{\rm{nl}}}} = \frac{{2\pi }}{\lambda }{n_2}\;\frac{P}{{{A_{{\rm{eff}}}}}}L$$where <$P$> is the optical power and <$L$> is the length of the medium. It is assumed that the nonlinear index is constant over the whole area, which is not accurate for all fibers, as the index-raising dopants can also influence the nonlinearity. Note that the nonlinear phase shift applies to the whole mode area, rather than e.g. only to the point with highest intensity, as the waveguide effect prevents the build-up of any significant transverse variations of the optical phase (e.g. a curvature of the wavefronts).

Note that depending on the shape of the mode profile, the effective area may deviate substantially from <$\pi w^2$> where the radius <$w$> is calculated either as an <$1/e^2$> intensity value or with the D4σ method. In most practical cases concerning single-mode fibers, however, these deviations are not very large.

Typical single-mode fibers, as used e.g. for optical fiber communications, have effective mode areas of the order of 100 μm^{2}. Large mode area fibers have several times higher mode areas, sometimes even above 1000 μm^{2}.
On the other hand, there are highly nonlinear fibers with particularly small mode areas. Some photonic crystal fibers can have mode areas below 10 μm^{2}.

An important consequence of a small mode area is that the optical intensities for a given power level are high, so that nonlinearities become important. Also, small mode areas are usually the consequence of strong guiding, where bend losses and other effects of external disturbances are weak.

If two fibers with different effective mode areas are spliced together, this will lead to some optical power loss. The article on fiber joints contains a formula for estimating that coupling loss. Note that although large mode areas make the splicing process less sensitive to transverse offsets, they increase the angular sensitivity.

For efficiently coupling fibers with substantially different mode areas, certain *mode field converters* (or *mode area adapters*) are sometimes used. Often, these are made from tapered fibers, adiabatically expanding or compressing the fiber modes.

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