## Faraday Rotation

Linearly polarized radiowaves may be considered as equivalent to the superposition two equal amplitude circularly polarized waves, but of opposite sense. Faraday rotation occurs in the ionosphere as a result of the fact that two modes (i.e., ordinary and extraordinary waves) propagate in the magneto-ionic medium (i.e., the ionosphere) at different phase velocities. The phase difference between the two modes is proportional to the product of the magnetic field strength and the EC, and inversely proportional to the square of the radio frequency. At any position along the ray trajectory one may compute the resultant orientation of the electric vector of the resultant linearly polarized wave. The amount of (one-way) Faraday rotation throughout the ionosphere is given by the approximate formula:

where Q is in radians, HL is the magnetic field component in ampere-turns per meter, and i N ds is the slant electron content (EC) in electrons/m2. The magnetic field component is given by HL = H cos 9, where 9 is the angle between the ray path and the magnetic field vector. In its most general form Equation 4-7 retains the parameter Hj under the integral sign since it is not a constant but varies slowly along the path of integration. However we may invoke the mean value theorem to bring out a representative value of HL; in this way we can conveniently isolate the electron content. The ionospheric "mean" is given by <h> = \hN dh)/\N dh. It is also convenient to replace ds by dh sec x where dh is a height increment and x is the ray zenith angle. Using the mean value theorem, we extract the value of the so-called magnetic field parameter H cosO secx = ^ evaluated at the mean height <h>. A typical value for W = ~ 50 ampere turns/meter for a mean altitude of <h> = 400 km. Taking the EC to be 3 x 1018, this implies Q = 1.75 radians at 1.6 GHz. This can be serious if linearly polarized antennas are employed. At lower frequencies, the amount of Faraday rotation may be enormous, and may translate to periodic fading. The rate of fading will depend upon the motion of the satellite (or target) and/or the diurnal variation of the EC. Figure 4-14 illustrates Faraday rotation as a function of frequency with EC as parameter.

Manual methods for computing the amount of Faraday rotation for arbitrary path appear in an NRL report [Goodman, 1965]. It should be recognized that for a radar situation, the amount of Faraday rotation is doubled. For the radar situation, the amount of Faraday rotation becomes:

where the rotation angle Q is given in radians and, as before, MKS units are employed.

The parameter W may be computed for various ground stations and values for the ionospheric height. Figure 4-15 shows the elevation and azimuthal dependence of for a site near Washington DC.

Frequency

Figure 4-14: Faraday rotation angle as a function of frequency with the electron content as a parameter. Remember this is the "slant" electron content and not the Total Electron Content (TEC), which is defined along the vertical. From ITU-R [1996],

### Frequency

Figure 4-14: Faraday rotation angle as a function of frequency with the electron content as a parameter. Remember this is the "slant" electron content and not the Total Electron Content (TEC), which is defined along the vertical. From ITU-R [1996],

Figure 4-15: The WHcosQ sec/ values for an ionospheric height of400 km and a site near Washington, D.C. For the Northern Hemisphere, it is seen that Faraday rotation peaks toward the south and reaches a minimum toward the north (from Goodman [1965]).

Figure 4-15: The WHcosQ sec/ values for an ionospheric height of400 km and a site near Washington, D.C. For the Northern Hemisphere, it is seen that Faraday rotation peaks toward the south and reaches a minimum toward the north (from Goodman [1965]).

For system design purposes where only order of magnitude values are needed, the following expression suffices for estimation of the amount of Faraday for a superionospheric radar target, an arbitrary radar path, and midlatitude location:

Q(radians) ~ Order of: {Iff* (foF2/f)2 ; fÂ»foF2 (4.9)

This is only a representative number, not a maximum value. In Equation 4.9, we have assumed that the effective slab thickness of the ionosphere is 360 km, and W = 40 ampere turns/meter. (Note that the TEC is given by the product of the F2 maximum electron density and the effective slab thickness.). Thus, if the value of foF2, obtained from a sounder, is 10 MHz, and the transmission frequency is 100 MHz, the value of Q ~ 100 radians. If the value of f = 1000 MHz, then Q ~ 1 radian. Since this is for a two-way radar path, the one-way radio path would be '/2 of these values.

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