Polarization has many different meaning in various context. So it would be hard to find the simple and well described described (try to google 'Polarization' and see if it helps). To me, I think the following one sentence from Wikipedia sounds simple and well desribed.

The polarization of an antenna refers to the orientation of the electric field (E-plane) of the radio wave with respect to the Earth's surface

Especially the green part is the statement that may apply to almost everything in this page. 'with respect to the Earth's surface' would better be changed to 'in 3D cartesian coordinate' to make it better fit for the description in this page.

The Wikipedia shows a couple of very intuitive animation. However, in many cases especially at a very early stage of learning curve, the animation would cheat you to think that you understand it (but in reality it is not) mainly because the animiation does not give your brain enough time to think of the concept.  However, the animation would be helpful to trigger various questions in your mind at early stage of learning, and helpful to confirm on your understanding after you studied enough.

So my personal approach is

i) If available, search and look into some short video clips and animations.. have some of your own questions.

ii) search for various of written descriptions with static image or figures / plots, and spend enough time for you to understand or at least get familiar with

iii) go back to the animation at step i). The animation would look very differently

One of the problems of written descriptions at step ii) would be that those material tend to give you only a few image, graphs/figures but does not give such a diverse cases of image / graphs. My note is also only one example for step ii). This page would not be enough to give you a complete understanding. But I hope my note here would give you at least some portions of understanding and help you to better understand other materials that you may search further.

To help you further and to help myself as well, I wrote two Python script to draw all the plots in this page and animate these plots. I strongly recommand you to try chaning the parameter values in the script on your own. Have Fun !!!

Sorry for too long prelude. Now getting down to the topic.

Here is the plan. We start from two dipoles and add their fields. Then we write the same sum as an equation, and change one parameter at a time to get the three kinds of polarization. The last section asks the practical question: how much signal is lost when the transmit and receive antennas do not have the same polarization?

• How do two dipoles make one polarized wave ?
• Mathematical Representation of Polarization
• Linear Polarization
• Circular Polarization
• Elliptic Polarization
• How much signal is lost when the polarizations do not match ?
• Reference :

How do two dipoles make one polarized wave ?

Every polarization on this page comes from one simple setup: two dipoles at right angles, fed with the same frequency. The receiver far away cannot see the two dipoles separately. It sees only the sum of their fields, so the question is what shape that sum traces over time.

Let's assume that you have two dipole antenna labelled as V (Blue) and H (Green). The electric field of V (Blue) is oscillating as in Blue sinusoidal plot. The direction of oscillation is aligned to the vertical axis of the coordinate system, we would say 'Blue wave is vertically polarized'.  The electric field of H (Green) is oscillating as in Green sinusoidal plot. The direction of oscillation is aligned to the horizontal axis of the coordinate system, we would say 'Green wave is Horizontally polarized'.

However, if you are standing far away from these two antenna and feels for the radiating field, you would feel the summed energy of the two polarized and you will feel a wave polarized in a new pattern as in Red sinusoidal plot.

If you observe the summed field in Red projected onto V and H plane, you will see a trajectory as shown in the plot on the right side.

The main purpose this page is the pattern (trajectory) of this page is to see how the summed wave(Red plot on the left) and the pattern on the right depending on various parameters of V and H wave.

The red wave is the vector sum of the blue V wave and the green H wave. Its projection on the V and H plane is the polarization.

• On the left, the black line is the direction of travel. The blue V wave oscillates along the V axis and the green H wave along the H axis.
• The red curve, labelled (H+V) wave and Polarized Wave, is the sum of the two. At the red dot, the dashed blue and green lines show its V and H components.
• On the right, the plot labelled Polarization Status looks at the same wave along the direction of travel. The red dot is the tip of the (H+V) vector at one moment, and the black line is the path of that tip over time.
• Here the V and H waves have the same amplitude and the same phase, so the tip moves on a straight line at 45 deg between the H and V axes.

This is why the plot on the right matters more than the 3D plot on the left. The 3D plot shows how the wave travels. The plot on the right shows what an antenna at the receiver sees, and it removes the direction of travel. From here on, we read each polarization from the shape of that path.

• Polarization is the direction of the electric field : it is read in the plane at right angles to the direction of travel.
• Any polarization is a sum of V and H : two orthogonal components with their own amplitude and phase describe every case on this page.
• The path of the field tip names the polarization : a line, a circle or an ellipse.

Mathematical Representation of Polarization

The drawing gives the picture, and the equation tells us which parameters we can change. There are only four: two amplitudes and two phases. The rest of the page changes them one at a time.

The illustration shown above can be represented in mathematical equation as shown below. The Red curve is represented by Ep and the Green curve is represented by the first term on the right hand side. The blue curve is represented by the second term on the right hand side.

The polarized wave is the sum of a horizontal and a vertical wave. The amplitudes and the phase difference decide the polarization.

• u_h and u_v are unit vectors along the horizontal and the vertical axis.
• |E_h| and |E_v| are the amplitudes of the two waves, and φ_h and φ_v are their phase shifts.
• The term ωt - kz is the travel of the wave in time and in space. It is the same in both terms, because both waves have the same frequency and travel the same way.
• The notes at the bottom list the polarization for each phase difference: none gives linear, nπ/2 gives circular, and any other value gives elliptical.

The notes at the bottom of the equation need two corrections. First, a circle needs equal amplitudes, |E_h| = |E_v|, as well as the right phase difference. With unequal amplitudes, a phase difference of π/2 gives an ellipse whose axes lie along H and V. Second, only the odd values of n give a circle. With n = 2 the phase difference is π, and the two waves are in exact opposition. The tip then moves on a straight line again, at -45 deg instead of +45 deg. So the complete rule is this. A phase difference of 0 or π gives linear polarization for any amplitudes. A phase difference of π/2 or 3π/2 with equal amplitudes gives circular polarization. Every other case is elliptical.

• Four parameters decide the polarization : the two amplitudes and the two phases, and only the difference of the phases matters.
• A phase difference of 0 or π gives a line : whatever the amplitudes are.
• A circle needs two conditions : a phase difference of an odd multiple of π/2, and equal amplitudes.

Linear Polarization

Linear polarization is the case most antennas produce, and a single dipole is the simplest example. The examples below show that neither a common phase shift nor unequal amplitudes can turn the line into anything else.

In this section, I will show you several examples of the V, H propagation and the summed wave. Each of  these example will is plotted by applying different set of parameter values. But you will notice that the trajectory shown on the right (Polarization pattern)all of these example are all straight line (i.e, linear). That is, the summed wave in all the example in this section has linear polarization.

Equal amplitudes and equal phases. The common phase moves the red dot along the line, but the line itself never changes.

• The panels above use the same amplitude of 1 for H and V. The phase of both waves is 0, π/5, 2π/5, π/2 and π in turn.
• The difference between φ_H and φ_V is 0 in every panel, and every panel shows the same line at 45 deg.
• Only the red dot moves. With a phase of π/2 it sits at the origin, and with a phase of π it sits at the lower left end of the line.

Now let's suppose a case where the Amplitue of H and V wave is different. You still see that the summed wave shows linear polarization since the phase of V and H are still same. But the angle of polarization changes.

Unequal amplitudes tilt the line toward the stronger component.

The tilt is easy to compute. The line makes an angle of atan(A_V/A_H) with the H axis. With A_H = 0.5 and A_V = 1, this is atan(2), about 63.4 deg, which matches the steeper line in the plot above.

As an extreme case, let's set the amplitude of H wave to 0 (i.e, turnning off H wave). Then you see the summed wave superimpose (overlap) onto V wave.

As another extreme case, let's set the amplitude of V wave to 0 (i.e, turnning off V wave). Then you see the summed wave superimpose (overlap) onto H wave.

With one component switched off, the sum is simply the other component: pure vertical or pure horizontal polarization.

These two extreme cases are the everyday ones. A vertical whip or a vertical dipole produces the case with A_H = 0. A dual-polarized antenna simply contains both dipoles and feeds them separately. Base station panels for LTE and NR usually place the two dipoles at +45 deg and -45 deg, so one panel gives two independent receive or transmit branches.

• A common phase does not change the polarization : it only moves the field tip along the same line.
• The amplitude ratio sets the angle of the line : the angle to the H axis is atan(A_V/A_H).
• Vertical and horizontal are special cases : one of the two components is zero.

Circular Polarization

Circular polarization keeps the field strength constant and turns its direction instead. That property makes the receive level independent of how the receive antenna is rotated around the direction of travel.

Now let's try to set the phase difference between H and V to be pi/2 (90 degree) and see how the summed wave(Red) and polarization pattern changes. The result is as shown below. As you see on the trajectory on the right side is perfect circle. This type of polarization is called Circular Polarization.

A phase difference of π/2 with equal amplitudes. The field tip moves around a circle, and its length never changes.

• The labels at the top give φ_H = π/2, φ_V = 0, and an amplitude of 1 for both waves.
• On the left, the red sum wave is a spiral around the direction of travel rather than a flat wave.
• On the right, the red vector has the same length at every moment, and its tip runs around the black circle.

The direction of rotation is set by the sign of the phase difference. If H leads V by π/2, the tip turns one way. If H lags V by π/2, it turns the other way. These are the two senses of circular polarization, right-hand and left-hand. A receive antenna must use the same sense as the transmitter. Satellite links use circular polarization because the rotation of the satellite or of the receiver then does not matter. For example, GPS transmits right-hand circular polarization.

• Circular polarization has a constant field strength : only the direction of the field rotates.
• The sign of the phase difference sets the sense : the two senses are right-hand and left-hand circular polarization.
• Rotation of the antenna does not matter : this is why satellite systems such as GPS use it.

Elliptic Polarization

Elliptic polarization is the general case, and linear and circular are its two limits. A real circularly polarized antenna is always slightly elliptical, so we need a number that says how close to a circle it is.

Now let's try to set the phase difference between H and V to be different by some value other than pi/2 and see how the summed wave(Red) and polarization pattern changes. The result is as shown below. As you see on the trajectory on the right side is elliptic. This type of polarization is called Elliptic Polarization.

A phase difference of π/4 with equal amplitudes. The path lies between the line and the circle, as an ellipse tilted at 45 deg.

That number is the axial ratio, the ratio of the major axis to the minor axis of the ellipse. A line has an infinite axial ratio, and a circle has an axial ratio of 1, or 0 dB. For equal amplitudes and a phase difference Δφ, the axial ratio is cot(Δφ/2). In the plot above, Δφ = π/4, so the axial ratio is cot(π/8) = 2.41, which is 7.7 dB. With Δφ = π/3 it falls to 1.73, or 4.8 dB, and at π/2 it reaches 0 dB. Antenna datasheets for circular polarization quote the axial ratio in dB, and a lower value means a better circle.

• Elliptic polarization is the general case : linear and circular polarization are its two limits.
• The axial ratio measures the shape : 0 dB is a circle, and a line has an infinite axial ratio.
• For equal amplitudes the axial ratio is cot(Δφ/2) : π/4 gives 2.41, about 7.7 dB.

How much signal is lost when the polarizations do not match ?

So far we have looked at one wave on its own. In a link there are two antennas, and the receive antenna picks up only the part of the field that lies along its own polarization. Let's put a number on the part it misses.

For two linear antennas the answer is the polarization loss factor, cos²ψ, where ψ is the angle between the two polarizations. The table below lists some values. At 45 deg half of the power is lost, which is 3 dB. At 90 deg the antennas are cross-polarized, and in theory nothing is received. In practice reflections rotate part of the field, so a real link keeps some signal. But that signal is much weaker than in the matched case.

A mix of linear and circular polarization gives a fixed loss instead. A circular wave has equal H and V parts, so a linear antenna at any angle picks up half of the power. The loss is 3 dB, and it does not change when the antenna rotates. Two circular antennas of opposite sense are the circular version of cross-polarization, and again in theory nothing is received.

The same fact is also an opportunity. Two orthogonal polarizations can carry two different signals on the same frequency at the same time. That is how the +45 deg and -45 deg dipoles of a base station panel give two MIMO branches.

• Linear to linear loses cos²ψ : 3 dB at 45 deg, and in theory everything at 90 deg.
• Linear to circular always loses 3 dB : the linear antenna sees only one of the two equal components.
• Orthogonal polarizations are two channels : a dual-polarized antenna uses them for two MIMO branches.

Reference :

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