Effective area

Effective Area

The effective area A_eff of a mode is a quantity with dimensions of a surface that, roughly speaking, characterizes the cross section on which the modal fields are concentrated. The effective area appears naturally in the study of nonlinear guided wave optics, where it is defined as the square of the cross sectional spatial integral of the square of the electric field divided by the cross sectional spatial integral of the fourth power of the electric field or ((\sum\sum |E|²dA)²)/(\sum\sum |E|⁴dA). The nonlinear coefficient gamma is then proportional to the non linear index of the fibre material n₂ divided by the effective area. Alternate definitions use the integration of the Poynting vector ((\sum\sum S_zdA)^2)/(\sum\sum S_z²dA), or different fields, but int he context of nonlinearity the most accurate definition is that using the norm of the electric field.

While this definition and use in the equation for gamma are valid in the weak guidance approximation, where the non linear index n₂ can be assumed constant over the cross section of the fibre, care should be taken with photonic crystal fibes containing different materials (eg silica with air holes). For such geometries, the non-linear coefficient should be obtained directly from integrating n₂E² rather than via the use of the effective area, which does not take into account spatial variations of n₂. In fact for tightly confined modes, where modes are strongly non transverse, the integration should be done using the tensor products of the nonlinearity tensor with the mode's electric field.

In the CMU, the effective area can be calculated using winfield or using batchfield.

Calculating the Effective Area with Winfield

The effective area can be calculated in Winfield, in the "Effective Core Area" section of the "Winfield Advanced Tools" window that opens when clicking "More Options".

The effective area is calculated by numerically integrating the fields over the window in which the fields have been calculated (that is, over the range of coordinates that are displayed in the field plot, and using the resolution used in displaying, without interpolation). According to the above section, the correct way of calculating the effective area is by using the E field and using the 'norm' component - and also un-checking 'normalize to pitch'. That uses the definition of ((\sum\sum |E|²dA)²)/(\sum\sum |E|⁴dA) from Agrawal.

Note that for leaky modes, the effective area is ill defined and effectively very highly dependent on the spatial extent over which one calculates the fields. Indeed, leaky modes diverge (almost) exponentially with radial distance past the radiation caustic (Snyder and Love). If one is using an integration region that is too large, Aeff will start diverging with increasing region size, while if one uses an integration region that is too small, it will not cover the entire region where the fields are. This is only really a problem for very leaky modes.

Resolution should be high for accurate numerical integration (at least 200), local expansions should be unchecked, and any interpolation settings are ignored in the integration.

Calculating the Effective Area with Batchfield

Batchfield (bfld.exe) also can be used to calculate the effective area. Batchfield always uses the norm of E (ie the definition of ((\sum\sum |E|²dA)²)/(\sum\sum |E|⁴dA) from Agrawal). The effective area data is found in the mode_table.dat under column Aeffn. Note that this is the normalized effective area and is thus Aeff/pitch^2 where pitch is the pitch, defined as the smallest cylinder to cylinder distance found in the structure, and is also given in the mode_table.dat file in the column under the 'pitch' header. If there is a single cylinder in the structure, the pitch is set to the external radius of the cylinder.

Using external software

The effective area can also be calculated using external software, such as matlab. For this, the fields need to be exported (eg using write file in the Advanced Tools dialog), then imported into matlab and numerically integrated there. This method is recommended for calculating actual nonlinear properties when index contrasts are high, since the standard definition of the effective area is then of limited use.