EM Wave Basics

A cosmic source typically emits radio waves over a wide range of frequencies, but the radio telescope is sensitive to only a narrow band of emission centered on the RF. We can hence, to zeroth order, approximate this narrow band emission as a monochromatic wave. (More realistic approximations are discussed in Chapter 15). The waves leaving the cosmic source have spherical wavefronts which propagate away from the source at the speed of light. Since most sources of interest are very far away from the Earth, the radio telescope only sees a very small part of this spherical wave front, which can be well approximated by a plane wave front. Electro-magnetic waves are vector waves, i.e. the electric field has a direction as well as an amplitude. In free space, the electric field of a plane wave is constrained to be perpendicular to its direction of propagation and the power carried by the wave is proportional to the square of the amplitude of the electric field.

Consider a plane EM wave of frequency $\nu$ propagating along the Z axis (Figure 3.6). The electric field then can have only two components, one along the X axis, and one along the Y axis. Since the wave is varying with a frequency $\nu$, each of these components also varies with a frequency $\nu$, and at any one point in space the electric field vector will also vary with a frequency $\nu$. The polarization of the wave characterizes how the direction of the electric field vector varies at a given point in space as a function of time.

Figure 3.6: Electric field of a plane wave

The most general expression for each of the components of the electric field of a plane monochromatic wave^3.2 is:

$$E_X = A_X\cos(2 \pi \nu t + \delta_X)$$

$$E_Y = A_Y\cos(2 \pi \nu t + \delta_Y)$$

where $A_X,~A_Y,~\delta_X,~\delta_Y$ are constants. If $A_Y = 0$, then the field only has one component along the X axis, which increases in amplitude from $-A_X$ to $+A_X$ and back to $-A_X$ over one period. Such a wave is said to be linearly polarized along the X axis. Similarly if $A_X$ is zero then the wave is linearly polarized along the Y axis. Waves which are generated by dipole antennas are linearly polarized along the length of the dipole. Now consider a wave for which $A_X=A_Y, \delta_X=0$, and $\delta_Y=-\pi/2$. If we start at a time at which the X component is a maximum, then the Y component is zero and the total field points along the +X axis. A quarter period later, the X component will be zero and the Y component will be at maximum, the total field points along the +Y direction. Another quarter of a period later, the Y component is again zero, and the X component is at minimum, the total field points along the -X direction. Thus over one period, the tip of the electric field vector describes a circle in the XY plane. Such a wave is called circularly polarized. If $\delta_Y$ were $= \pi/2$ then the electric field vector would still describe a circle in the XY plane, but it would rotate in the opposite direction. The former is called Right Circular Polarization (RCP) and the latter Left Circular Polarization (LCP).^3.3 Waves generated by Helical antennas are circularly polarized. In the general case when all the constants have arbitrary values, the tip of the electric wave describes an ellipse in the XY plane, and the wave is said to be elliptically polarized.

Any monochromatic wave can be decomposed into the sum of two orthogonal polarizations. What we did above was to decompose a circularly polarized wave into the sum of two linearly polarized components. One could also decompose a linearly polarized wave into the sum of LCP and RCP waves, with the same amplitude and $\pi$ radians out of phase. Any antenna is sensitive to only one polarization (for example a dipole antenna only absorbs waves with electric field along the axis of the dipole, while a helical antenna will accept only one sense of circular polarization). Note that the reflecting surface of a telescope could well ^3.4 work for both polarizations, but the feed antenna will respond to only one polarization. To detect both polarizations one need to put two feeds (which could possibly be combined into one mechanical structure) at the focus. Each feed will require its own set of electronics like amplifiers and mixers etc.

EM waves are usually described by writing explicitly how the electric field strength varies in space and time. For example, a plane wave of frequency $\nu$ and wave number $k$ ( $k = 2\pi/\lambda,\ \lambda = c/\nu$) propagating along the Z axis and linearly polarized along the X axis could be described as

$$E(z,t) = A \cos(2\pi\nu t - kz)$$

This could also be written as

$$E(z,t) = {\rm Real}(A e^{j(2\pi\nu t - kz)})$$

where Real$()$ implies taking the real part of $()$ and $j$ is the imaginary square root of $-1$. Since all the time variation is at the same frequency $\nu$, one could suppress writing it out explicitly and introduce it only when one needs to deal with physical quantities. So, one could equally well describe the wave by the complex quantity A, where ${\bf A } = ~A\ e^{ - jkz}$, and understand that the physical field is obtained by multiplying A by $e^{j 2\pi\nu t}$ and taking the real part of the product. The field A is called the phasor field^3.5. So for example the phasor field of the wave

$$E = A \cos(2 \pi \nu t - kz+ \delta)$$

is simply ${\bf A }e^{j \delta}$.

Footnotes

... wave^3.2

Monochromatic waves are necessarily 100% polarized. As discussed in Chapter 15 quasi-monochromatic waves can be partially polarized.

... Polarization (LCP).^3.3

This RCP-LCP convention is unfortunately not fixed, and the reverse convention is also occasionally used, leading to endless confusion. It turns out however, that most cosmic sources have very little circular polarization.

... well^3.4

Not all reflecting radio telescopes have surfaces that reflect both polarizations. For example, the Ooty radio telescope's (Figure 3.16) reflecting surface consists of a parallel set of thin stainless steel wires, which only reflect the polarization with the electric field parallel to the wires.

... field^3.5

For qasi monochromatic waves, (see Chapter 1), one has the related concept of the complex analytical signal