Last modified June 2 1998

E-mail: demerson@nrao.edu

Elliptical Polarization in the Ionosphere.

Summary: This article calculates the degree to which a 100% linearly polarized wave will become elliptically polarized in its passage through the ionosphere. Although valid over a wider range of frequencies, sample calculations concentrate on the VHF range of frequencies. There is one dominant mode of generating ellipticity in the polarization (Case 1 below), although a secondary mechanism (Case 2) is also considered and shown to be of less significance.

Introduction

In general, when a linearly polarised wave propagates through the ionosphere, apart from possible attenuation of the wave, two major effects take place:

• The plane of rotation is rotated. This is Faraday rotation
• The linear polarization becomes elliptical polarization. It is possible for the wave to become 100% circularly polarized.

How important these effects are depend on the parameters of the ionosphere and the frequency. To a good approximation, above 100 MHz Faraday rotation occurs only for propagation along the magnetic field, (i.e. longitudinal propagation) while ellipticity is only introduced mainly by propagation perpendicular (i.e. transverse) to the magnetic field (Case 1 below). Some ellipticity can also be introduced into a linearly polarized wave propagating along the magnetic field lines(Case 2 below) , under conditions of high ionospheric absorption.

The two cases are considered separately:

1. Propagation perpendicular to the magnetic field lines.

   Background: the worst case of ellipticity is introduced by propagation at right angles to the magnetic field, with the E-field vector of the incident plane polarized wave at 45 degrees to the magnetic field. This wave can be considered as the superposition of two linearly polarized waves, one with the E-field parallel to the magnetic field and the other with the E-field at right angles to the magnetic field. These waves are essentially the ordinary, and the extraordinary, components for propagation transverse to the magnetic field.

   For the ordinary wave, the E-field accelerates electrons parallel to the magnetic field, which means that the magnetic field has no influence - a magnetic field only imposes a force on charged particles moving perpendicular to the field.

   For the extraordinary wave, the E-field of the incident radiation accelerates the free electrons normal to the magnetic field, which then exerts a force on the electrons and so modifies the electronic motion. This causes the ionospheric refractive index for the extraordinary wave to be different from that of the ordinary wave, and also to vary according to the magnetic field.

   The different refractive indices of the two component waves, meaning different propagation velocities, causes a progressive phase shift between the two components. If this phase shift becomes 90 degrees, then the initial 100% linearly polarized wave have been turned into a 100% circularly polarized wave. For smaller - or larger - differential phase shift, the wave in general becomes elliptically polarized.

   Details of calculations on the magnitude of the effect, assuming typical ionospheric parameters, can be found in an HTML version of a Mathcad worksheet here;   it is in effect a screen dump from Mathcad, and includes the relevant formulae, constants and calculations.

   Summary of results: A perfectly linearly polarized wave has an infinite axial ratio. A 100% circularly polarized wave has an axial ratio of 0 dB. Below is a table showing the one-way distance through a uniform ionosphere that is required to change the axial ratio of an incident linearly polarized wave to 0 dB, 6 dB, 10 dB and 20 dB. Extreme assumptions (i.e. exactly transverse propagation, and initial polarization vector 45 degrees to the magnetic field) have been made, so the results are likely to be upper limits of what's encountered in practice. Parameters for a "normal" undisturbed ionosphere have been assumed.

   Summary of results

   One-way distance through a uniform ionosphere, propagation normal to the magnetic field, in which a linearly polarised wave, E-field 45 degrees to the magnetic field lines, will become elliptically polarised to the indicated degree.
   AR is the Axial Ratio of the polarization.
   	50 MHz	144 MHz	432 MHz
   Circular: (AR=0 dB)	100	2800	76000	(km)
   AR=6 dB	70	1700	45000
   AR=10 dB	49	1000	30000
   AR=20 dB	14	400	10000

   These are fairly extreme values; Propagation along the field lines introduces no ellipticity, unless there is significant ionospheric absorption - see the next section. If the E-field of the wave propagating across the magnetic field is polarised parallel to or perpendicular to the magnetic field, NO ellipticity is introduced. If the propagation is not quite perpendicular to the magnetic field, then Faraday rotation will cause the major axis of the ellipse to rotate, in the normal way. If the Faraday rotation is an exact multiple of half-turns, then the ellipticity will cancel out exactly. This factor alone will limit how elliptical a wave can become, especially if Faraday rotation of several turns is involved. You never get more ellipticity than can be built up in a quarter-turn of Faraday rotation.

   So, the general conclusion is that at 432 MHz there will never be significant ellipticity introduced by the ionosphere. At 144 MHz 20 dB axial ratios may be common, but 6 dB axial ratio will almost never happen (although maybe it could approach that at very low elevation angles exactly perpendicular to the magnetic field?) At 50 MHz, high ellipticity, even perfectly circular polarization, should be common.

   This doesn't take account of other extreme conditions, where major ionospheric disturbances may temporarily give extreme electron densities. I believe the numbers should be representative of a "normal" ionosphere.

   I thank Kurt Weiler , K7BLT, and John Regnault , G4SWX, for comments on earlier drafts of this summary.

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2. Longitudinal propagation with attenuation

   Background:For a linearly polarized wave propagating along the magnetic field lines in the ionosphere, it is convenient to consider the wave as the sum of two circularly polarized waves, one LHCP and the other RHCP. The combination equal amplitude LHCP and RHCP waves is a 100% linearly polarized wave. The relative phase difference between the RHCP and LHCP components determines the position angle of the E-field of the linearly polarized wave after the summation. A difference in refractive index, or velocity of propagation, for the RHCP and LHCP components will lead to a gradual rotation of the linearly polarized E-field vector. This is just normal Faraday Rotation

   In normal conditions of the Earth's ionosphere, above about 100 MHz there is very little attenuation. However, at lower frequencies, or during abnormal ionospheric events, ths absorption can become significant. If one of the circularly polarized component terms of a 100% linearly polarized wave is attenuated more than the other, then the sum of the RHCP and LHCP terms will no longer give pure 100% linear polarization. In other words, differential absorption between the LHCP and RHCP waves will introduce ellipticity into the polarization of the wave. The limiting case is of course where one of the circular components is completely absorbed, leaving a wave which is completely circularly polarized.

   It is much more difficult to predict the attenuation of a wave passing through the ionosphere than to predict, say, the amount of Faraday rotation. The absorption is a function of the effective collision frequency between the atoms or molecules of the ionosphere. This effective collision frequency is very variable, ranging from about a kHz up to about 100 MHz.

   Fortunately, if the total attenuation through the ionosphere at a given frequency is known, then the differential absorption between LHCP and RHCP components propagating longitudinally along magnetic field lines is relatively simple to calculate. Details of the calculation, copied from a Mathcad worksheet, are given here. The results are summarised in the table below.

   Axial ratio degradation of an initially 100% linearly polarized wave, as a result of differential absorption of the LHCP and RHCP components.
   Ordinary ray attenuatiion	Frequency (MHz):	30	50	100	144	432
   0.1 dB	Extraordinary ray attenuation (dB)=	0.121	0.112	0.106	0.104	0.101
   	Axial Ratio(dB)=	58	63	70	73	83
   1.0 dB	Extraordinary ray attenuation (dB)=	1.21	1.12	1.06	1.04	1.01
   	Axial ratio(dB)=	38	43	50	53	63
   10.0 dB	Extraordinary ray attenuation (dB)=	12.1	11.2	10.6	10.4	10.1
   	Axial ratio(dB)=	18	23	30	33	42

   Discussion:  From various texts, at 100 MHz the night-time absorption is typically 0.01 to 0.005 dB, while the day-time absorption is about ten times greater. (See Attenuation at VHF in propagation through the Ionosphere). Under extreme conditions at 100 MHz day-time absorptions up to about 10 dB have been reported. Absorption decreases as the square of frequency. Even for 10 dB attenuation of the ordinary ray at 30 MHz, the differential absorption in longitudinal propagation along the magnetic field lines causes the axial ratio of a linearly polarized wave to be degraded to no worse than 18 dB. Ellipticity introduced by the transverse propagation differential phase shift (case [1] above] is much more important.