1.2 Basic Principles

The properties of electromagnetic radiation propagation and of radio antennas can be deduced from a few basic physical principles, i.e.

1.

the notion that Electromagnetic Radiation are Waves of a certain Wavelength ($\lambda$), or Frequency ($\nu$), and Amplitude (A) and Phase ($\varphi$);

2.

from Huygens Principle which says that each element of a wavefront is the origin of a Secondary Spherical Wavelet;

3.

the notion that the Optical Instrument (like a single-dish antenna, a telescope, etc.) combined with a receiver manipulates the incident wavefront through their respective phase and amplitude transfer functions.

  

Figure 1.1: Illustration of Huygens Principle. The individual points of the plane wavefront in the aperture plane are the origin of secondary spherical wavelets, which propagate to the right, and superpose to form a plane wavefront in the image plane. The optical instrument is placed in between.

Summarized in one sentence, and proven in the following, we may say that the radio antenna transforms the radiation incident on the aperture plane ($\cal A$) to an image in the image plane ($\cal I$), also called focal plane. Following Huygens Principle illustrated in Figure 1.1, the point a(x,y) $\equiv$ a($\vec r$) of the incident wavefront in the aperture plane $\cal A$ is the origin of a spherical wavelet of which the field $\delta$E(a') at the point a'(u,v) $\equiv$ a($\vec u$) in the image plane $\cal I$ is

$${\delta}{\rm E}({\vec u}) = {\rm A}({\vec r}){\rm exp[iks]/s}$$ (1.1)

with k = 2 $\pi/\lambda$. The ensemble of spherical wavelets arriving from all points of $\cal A$ at the point a$'(\vec u)$ of the image plane $\cal I$ produces the field

$${\rm E}({\vec u}) = {\int}_{{\cal A}}{\rm A}({\vec r}){\Lambda}({\beta}) [{\rm exp(iks)/s}] dx dy$$ (1.2)

For the paraxial case, when the rays are not strongly inclined against the direction of wave propagation (i.e. the optical axis), the inclination factor $\Lambda$ can be neglected since $\Lambda$($\beta$) $\approx$cos($\beta$) $\approx$ 1. Also, s $\approx$ s₀ for paraxial rays, but exp[iks] $\neq$ exp[iks₀] since these are cosine and sine terms of s where a small change in s may produce a large change of the cosine or sine value. Thus, for the paraxial approximation we may write

$$s = [{\rm (x-u)}^2 + {\rm (y-v)}^2 + {\rm z}^2]^{1/2}\ \approx\ {\rm R + g(x,y,R) - (xu + yv)/R}$$ (1.3)

with

$${\rm R} = ({\rm x}^2 + {\rm y}^2 + {\rm z}^2)^{1/2}\ \ and\ \ {\rm g(x,y,R)} = ({\rm x}^2 + {\rm y}^2)/2{\rm R}$$ (1.4)

When using these expressions in Eq.1.2, we obtain

$${\rm E(u,v) = [exp(ikR)/s}_{0}]{\int}_{{\cal A}}{\rm A(x,y)} {\rm exp[ik(g(x,y,R) - (ux + vy)/R)] dx dy}$$ (1.5)

This equation describes the paraxial propagation of a wavefront, for instance the wavefront arriving from a very far away star. In particular, this equation says, that without disturbances or manipulations in between $\cal A$ and $\cal I$ the plane wavefront continues to propagate in straight direction as a plane wavefront.