Electromagnetic Beam Bending

Fundamentals
Title	Electromagnetic Beam Bending
Author(s)	J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level	Basic
Year of Publication	2011

As it is known, the density of the atmospheric gas and plasma is not homogeneous. This causes spatial and temporal variations in the refraction index. Due to the refractive index gradients, the rays suffer an excess of geometric path, regarding to the free space path, when crossing the atmosphere. The Snell's law states that if an electromagnetic wave travels from an environment with refractive index [math]\displaystyle{ n_1 }[/math] to a second one with refractive index [math]\displaystyle{ n_2 }[/math] and crosses the separation between them with an angle [math]\displaystyle{ \varphi_1 }[/math], it will be deviated with an angle [math]\displaystyle{ \varphi_2 }[/math]. The relation between these four values is summarised by the expression: [math]\displaystyle{ n_1 \cdot \sin \varphi_1 = n_2 \cdot \sin \varphi_2 }[/math].

Figure 1: Snell Law and consequence.

Figure 1(a) illustrates this basic postulate: [math]\displaystyle{ \varphi_2 \lt \varphi_1 }[/math] because [math]\displaystyle{ n_2 \gt n_1 }[/math]. The consequence of this law can be seen in Figure 1(b), the successive trespasses of different layers, each one with its own refractive index, causes a curvature in the ray path that causes a delay as discussed above.

For the Troposphere (see [Parkinson et al., 1996]), the effect caused by neglecting path curvature is less than [math]\displaystyle{ 3 }[/math] millimetres for elevations greater than [math]\displaystyle{ 20 }[/math] degrees, [math]\displaystyle{ 2 }[/math] centimetres for elevations close to [math]\displaystyle{ 10 }[/math] degrees and increases to [math]\displaystyle{ 17 }[/math] centimetres at elevations close to [math]\displaystyle{ 5 }[/math] degrees (mainly due to dry gases).

For the Ionosphere, these errors can reach up to few centimetres for L1 and L2 GPS signals [Hoque and Jakowsky, 2008], at low elevations and under high solar activity conditions. Nevertheless, they are mostly negligible in the ionosphere-free combination, because the excess path ray is basically proportional to the inverse of the squared frequency [Aragon-Angel et al., 2009].