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# 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 n1 to a second one with refractive index n2 and crosses the separation between them with an angle $\varphi_1$, it will be deviated with an angle $\varphi_2$. The relation between these four values is summarised by the expression: $n_1 \cdot \sin \varphi_1 = n_2 \cdot \sin \varphi_2$.

Figure 1: Snell Law and consequence.

Figure 1(a) illustrates this basic postulate: $\varphi_2 < \varphi_1$ because n2 > n1. 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, cause a curvature in the ray path that causes a delay as discussed above.

For the Troposphere (see [Parkinson et al., 1996][1]), the effect caused by neglecting path curvature is less than 3 millimetres for elevations greater than 20 degrees, 2 centimetres for elevations close to 10 degrees and increases to 17 centimetres at elevations close to 5 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][2], at low elevations and under high solar activity conditions. On the other hand the bending angle is proportional to the inverse of the squared frequency and mostly cancels in the ionosphere-free combination [3] (see also the tutorial [4]).

## References

1. ^ [Parkinson et al., 1996] Parkinson, B., Spilker, J. and Enge, P., 1996. Global Positioning System, Vol. I and II. Theory and Applications. American Institute of Aeronautics, USA.
2. ^ [Hoque and Jakowsky, 2008] Hoque, M. and Jakowsky, N., 2008. Estimate of higher order ionospheric errors in GNSS positioning. Radio Science. 43,pp. 1-15.
3. ^ Hajj, G., Kursinsky, E., Romans, L., Bertier, W. and Leroy, S., 2002. A technical description of atmospheric sounding by GPS occultation. Journal of Atmospheric and Solar-Terrestrial Physics 64, pp. 451–469
4. ^ Sanz, J., Juan, J. and Hernandez-Pajares, M., 2012. Analysis of propagation effects from GNSS observables based on laboratory exercises. Course on Propagation effects, channel models and related error sources on GNSS, ESA/ESAC., Madrid, Spain. http://gage.upc.edu/tutorials
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