By default, far field projections assume that the material at the monitor location extends to infinity. In the following figure, this implies the substrate material extends to infinity. Obviously this is not always true.
This page describes how to calculate the far field distribution assuming the refractive index in the far field is different from the index in the near field, which makes it possible to include effects such as the Substrate-Air interface shown above.
Note: The descriptions and examples of the far field projection calculation on the following pages are primarily intended for users of FDTD. For users interested in calculating far field projections with MODE, these descriptions are basically still correct, although some subtle differences do exist.
Two methods are available: Directly setting the far field refractive index in the far field projection functions, and applying the Fresnel equations to the far field data in an additional post processing step.
Setting the far field refractive index in the far field projection functions
The far field projection functions have an optional argument to set the far field refractive index:
out = farfield3d("mname",f, na, nb, illumination,
periodsa, periodsb, index, direction);
To understand this option, it is important to remember that the far field projection calculation is basically a plane wave expansion of the near field data. To calculate the expansion, it is necessary to know the background refractive index, since the ratio between E and H fields in a plane wave depends on the refractive index. By default, the refractive index at the monitor location is used for the expansion, but the 'index' property of the projection function allows a different index to be specified.
Expanding the fields using a different refractive index can provide useful information, but some aspects of the data (particularly the field amplitude and power measurements) can be slightly challenging to interpret. To illustrate the basic issue, consider a single plane wave in air, propagating in the forward direction. If this plane wave has an electric field amplitude of 1 V/m, the magnetic field amplitude will be sqrt(eps0/mu0) = 0.0026.
Now imagine expanding this field profile, using plane waves that exist in a medium with a refractive index = 2. The key point is that the ratio between E and H will be different in this medium by a factor of the the refractive index: If E=1, then H=2*sqrt(eps0/mu0) in this medium. To represent the original field profile using these plane waves, it is necessary to combine a forward propagating wave with a electric field amplitude of 0.75 propagating in the forward direction and a backwards wave propagating with an amplitude of 0.25. Combining these two plane waves allows us to reconstruct the original field profile. Notice that the sign of H is reversed for the backward propagating wave.
E field: 0.75 + 0.25 = 1
H field: 0.75*2 - 0.25*2 = 1
This method can be used to quickly see how refraction from a far field interface will affect the angular distribution of radiation. It is also used in various types of advanced data analysis. However, it is generally not the best option for calculating the effect of a far field interface, especially when the results will be compared to experimental measurements (i.e. power or field intensity measurements). Instead, the Fresnel correction method described below should be used.
Example
The associated simulation file has a gaussian beam propagating at a 10 degree angle of incidence in a medium with a refractive index of 2. The analysis script will plot the far field for a refractive index of 2 and 1.
Efar = farfield3d("T",1,res,res);
Efar_air = farfield3d("T",1,res,res,1,1,1,n_air);
Notice how the angle of the beam changes. In the FDTD simulation (with a refractive index of 2), the gaussian beam propagates at an angle of 10 degrees. The standard far field projection in the substrate shows the beam continues to propagate at a 10 degree angle. When the far field refractive index is changed to 1, the angle of the beam shifts to about 20 degrees. Snell's law can be used to confirm this is the expected shift:
\begin{aligned} \theta _2 &= asin \left( \frac{n_1 sin(\theta_1)}{n_2} \right) \\ &= asin \left( \frac{2 sin(10)}{1} \right) \\ &= 20.3 \deg \end{aligned}
Fresnel correction
Alternatively, it is possible to calculate the far field data using the 'near' field refractive index, then use Snell's law and the Fresnel equations to account for the far field interface. Snells' law is used to calculate the change in propagation direction that occurs when the fields pass from one material to the other. The Fresnel equations are used to account for the fraction of power that is lost due to reflection from the interface. The details of this calculation can be found in the associated example files. In particular, see the analysis script of the 'far_field_change_index' analysis group.
This approach can be slightly more complicated that the first method, but it is recommended when it is necessary to get correctly normalized field amplitudes and power measurements. The example script will calculate the |E|^2 in the far field using the default projection (assuming the substrate with an index of 2 extends to infinity), using the plane wave expansion method with a far field index of 1, and using the Fresnel correction using a far field index of 1.
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Note: It is important to understand that multiple reflections (between the interface and the device in the FDTD simulation) effects are not taken into account by this technique. Fortunately, such reflections are often small, making this approximation valid in many situations. |