Propagation of electromagnetic (EM) waves through the air is difficult to predict, due to atmospheric effects. Raindrops, for example, can act as scatterers that attenuate and affect the polarization of the propagating EM waves. Scattering theory has treated effects on the far-field regions of receive and transmit antennas in communications links, but not near-field effects. In this first of a two-part article, this analysis examines the nearfield effects of distributed scatterers, such as rainfall and what influence they can have on the EM propagation of a communications system, such as attenuation and depolarization.

In some communications platforms, such as microwave satellite communications systems, the near field may contain a significant portion of the rain path for the communications link. At frequencies of 10 GHz and above^{l,2}, precipitation can cause attenuation and depolarization of the transmitted EM radiation.^{3,4} Both effects can limit system performance, especially for systems employing orthogonal polarization schemes to handle higher data rates.^{5} Early research on the atmospheric effects^{6} on radar echoes found that the main atmospheric cause of microwave attenuation is rainfall. Spherical raindrops were assumed in the study.

Another study examined EM wave propagation in a medium containing independent particles that scatter and absorb the incident energy.^{7} The effects of the particles were expressed in terms of an effective complex index of refraction that predicts both attenuation and phase shifts caused by the particles. Following this, a further study repeated and extended the earlier work on radar echoes comparing predictions with measurements and finding a lack of close agreement, with predictions falling short of measured attenuation values.^{8} The lack of agreement was felt to be due possibly to omission of the effects of multiple scatterers along the transmission path in the predictions, as well as the possibility that the rain structure was more complex than initially assumed and was not adequately represented by theory. Additional studies showed that even with no variations in rainfall rate or rain drop size, theory and measurements failed to agree.^{9} A study on the near-field effects of rain on antennas revealed that modifications to earlier theory were needed.^{10} Measurements of rainfall in the near field of antennas showed significant polarization effects for both phase and amplitude. But work in 1978^{11} found flaws in the antenna correction factors developed previously and noted the importance of understanding the effects of rainfall in the near field of an antenna, especially for microwave satellite communications systems.

In a terrestrial microwave link, the near-field regions of both antennas may only be a small part of the link's total path. But in satellite communications systems, the near-field region of a large ground-station antenna could represent a significant fraction of the rain path. For that reason, it is important to know the effects of scatterers, such as rain, on the near-field regions of antennas in a microwave communications system. Analysis will be performed on an example link in four parts: in the absence of precipitation, with the introduction of a raindrop, in a uniform distribution of raindrops, and for the attenuation and phase shifts due to a thick precipitation layer. For EM propagation in the absence of precipitation, consider a case where two aperture antennas are pointed at each other (* Fig. 1*).

For the antennas in * Fig. 1*, EM field

**E**

_{1}

**H**

_{1}is the field from the transmitter (antenna 1) and field

**E**

_{2}

**H**

_{2}is the field of the receiver (antenna 2) when it acts as a transmitter. To understand the behavior of the fields, consider a volume, V, formed by the sides of the antenna, S

_{1}S

_{2}, S

_{3}, S

_{4}, and S

_{5}. Within this volume there are no impressed currents and both fields (

**E**

_{1}

**H**

_{1}and

**E**

_{2}

**H**

_{2}) satisfy Maxwell's equations in free space, i.e.,

Equation 2 can be used in Eq. 1 to solve for Eq. 2 in Eq. A. Use of Gauss' theorem gives Eq. 3, where n^ is the outward normal on the respective surfaces. Equation 3 is the Lorentzian form of the Reciprocity Theorem for free space.

The next step in this analysis is to consider each surface separately. Surface S_{1} is the aperture plane of antenna 1. For the case of this analysis, these two antennas are large compared to the transmit/receive wavelength. Assuming matched polarizations, the fields over Surface S_{1} can be described by

For this surface, n^ = k^, with Eq. 4 being true.

Surface S_{2} is located in the transmission line connected to antenna 2. For simplicity, the line is assumed to be matched and with only a normal transmission mode. As a result, on S_{2}

where Bi is the complex amplitude of the inward traveling wave resulting when antenna 1 transmits; A is the complex amplitude of the outward traveling wave when antenna 2 is transmitting; Z0 is the transmission line wave impedance; u^ is a unit vector normal to k^; and f(x,y) is the transmission- line-mode distribution function. For this surface, n^ = k^, and the conditions shown in Eqs. C and 6.

The integral over surfaces S3 and S4 vanishes. To show this, note that E2 and E2 are parallel with the normal n^ everywhere over S3 and S4, which are assumed to be perfectly conducting surfaces. As a result, **E _{1} = n^E_{1}** and

**E**can be written over S

_{2}= n^E_{2}_{3}and S

_{4}as Eq. D and 7. Surface S

_{5}can be treated as a large sphere tending to infinity, so that:

with the conditions shown in Eq. E, so that Eq. 8 is true. Using Eqs. 4, 6, 7, and 8 results in Eqs. F and 9.

* Figure 2* is similar to

*, with the addition of an arbitrarily shaped raindrop (scatterer). Let*

**Fig. 1****E**be the field due to the transmitter (antenna 1) in the presence of precipitation and

_{3}H_{3}**E**be the field of the receiver (antenna 2) when it acts as a transmitter in free space. Note that

_{2}H_{2}**E**is the same as described previously, although

_{2}H_{2}**E**is not the same as

_{3}H_{3}**E**.

_{1}H_{1} Let **E _{s1}H_{3s}S_{1}** represent the scattered field in free space due to

**E**incident on the raindrop. Both

_{3}H_{3}**E**and

_{s1}H_{s1}**E**satisfy Maxwell's free-space equations in the volume V bounded by the surfaces S

_{2}H_{2}_{2, 3}S

_{7}. As a result, following the form of the analysis already performed for EM wave propagation in the absence of any precipitation, Eq. 10 results, where n^ is the outward normal on the respective surfaces.

Integration over the surfaces S3, S4, and S5 yields zero, as shown in Eq. 11. For surface S2, Eq. 12 holds, where A, Z0, and f(x,y) are the same as defined previously and Bs is the complex amplitude of the inward traveling wave at S2 due to the scattered field EsS1Hs1, which itself is due to **E _{3}H_{3}**.

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Surface S7 can be analyzed by considering that both **E _{s3}H_{s3}** and

**E**give rise to traveling waves in the same direction across S7. This can be used to show the conditions in Eq. 13.

_{3}H_{3} Surface S_{6} is the surface of the raindrop, with 0 representing a point anywhere within the scatterer (* Fig. 2*). By incorporating the scatterer, fields.

**E _{3}H_{3}** and

**E**can be written as

_{2}H_{2}where E_{30} is the complex amplitude of E_{3} at point 0 and E_{20} is the complex amplitude of E_{2} at point 0. Equation 14 assumes that both E_{3} and E_{2} are linearly polarized in the same direction since the two antennas are illuminating a narrow common volume. Over the region occupied by the raindrop, **E _{3} H_{3}** and

**E**can be expressed as plane waves traveling in opposite directions. Taking point 0 as the origin of the z' axis (

_{2}H_{2}*),*

**Fig. 2**It can now be seen that

where * denotes the complex conjugate. Using Eq. 16 yields Eq. 17.

The scattered field **E _{S1}H_{S}S_{1}** is due to

**E**incident on the raindrop. The integral on the right-hand side of Eq. 17 is related to the forward scattering complex vector amplitude F(0) of a scatterer12 according to Eq. 18. For any scatterer, the forward scattering complex vector amplitude can be shown in scalar form as Eq. 19:

_{3}H_{3}where parameter S(0) is the forward scattering complex scalar amplitude of a scatterer.

Using Eqs. 17, 18, and 19 and remembering that E*_{30} = i^E*_{30}, results in Eq. 20. Using Eqs. 11, 12, 13, and 20 in Eq. 10 yields Eq. G, where the conditions in Eq. 21 apply.

From Eq. 15, in the region occupied by the raindrop, Eq. 22 applies, where E_{3} and E_{2} are the complex scalar amplitudes of vectors E_{3} and E_{2}, respectively. Over this region,

Equation 23 shows that the product E3E2 can be assumed constant over the region occupied by the raindrop. It can also be assumed that E3E2 is constant over an elemental volume dv in space. Equations 23 and 21 can be used to find the EM wave detected picked up by the receiving antenna from any raindrop located within this volume, in the form of Eq. 24.

If the elemental volume dv contains similar-sized raindrops that are uniformly distributed in dv and if N is the number of raindrops per unit volume, then by applying single scattering theory it follows from Eq. 24 that the EM wave detected by the receiving antenna due to scattered radiation from the raindrops in dv is shown in Eq. 25.

Next month, this two-part article series on the effects of EM scatterers will conclude with further development of the scatterer model that includes a precipitation region with a large number of raindrops. The study will show that a sufficient number of raindrops can have measurable nearand far-field EM effects, causing both attenuation and phase shifts.