Polarization Part #1: Scattering

You will have probably noticed with time that I do not spend much time talking about the electromagnetic nature of light on this blog but rather focus on experimental optics. For the following few posts, I’m going to break this habit a bit because we need just a bit of theory to understand what is going on in our experiments. That being said, this post, and the following ones, will mostly focus on hands-on experiments that you can perform at home or in a classroom to illustrate the theory. Furthermore, all files required to reproduce the experiment performed here (CAD, drawings etc.) are available [∞] here and are free for all my [∞] Premium members.

First, let’s summarize what is the electromagnetic wave theory for light :) I will keep it short here using some well-established equations such as Faraday’s law, Gauss’s law and Ampère’s law. The motivation is to give you pointers on how to explain the physical observation of the experiment we will perform, not necessarily to make you solve differential equations.

Long ago, [∞] Faraday observed that any time-varying magnetic field, B, is always accompanied by a spatially variable electric field, E, according to the law

where we introduce the curl operator in the reference frame XYZ

As an illustration, if we conveniently choose a function dB/dt aligned with the reference frame axis Z, this leads to

which implies that the electrical field is constrained to the XY plane

Plotting this on a graph shows that the electric field describes a swirling, vortex-like, motion with an amplitude and direction equal to the time-variation magnitude of B. An illustration is provided in Figure 1 for our illustrative function. Please note that the resulting distribution is specific to our example and that real distributions might be more complicated, depending on the exact expression of dB/dt, although it always possible to consider that the function corresponds to Figure 1 when looking at infinitely small region of space. One key take-away from Figure 1 that I wanted to highlight is that the curl operator is inherently perpendicular to its input – an aspect that will have large importance later on.

Also, there is a reciprocal of Faraday’s law but for magnetic fields, known as [∞] Ampère’s law, stating that

with the same graphical interpretation as in Figure 1.

Taking the curl of Faraday’s law (first mathematical twist), we can express the equation in terms of the electric field only

with µ₀ the permeability of vacuum and ε₀ the permittivity of vacuum, quantities that reflects how well vacuum supports magnetic and electrical fields respectively and which just acts as scaling constants in the equations.

Using the mathematical property of the [∞] triple product (second mathematical twist),

we obtain the differential equation at the origin of the electromagnetic representation of light

where the new constant, c, has the unit of speed (m/s) and is defined as

Intuitively, you can understand c as the speed at which a spatial variation of the field E will propagate away from its origin by moving the second component of the equation on the other side of the equal sign.

Continuing with our derivations, and for the sake of simplicity, I will analyze a version of this equation where the electric field, E, depends only on the spatial coordinates, z, and the time, t,

which admits solutions of the form (you can inject the solution into the equation and check by yourself)

where

This equation describes a wave oscillating in the plane defined by XZ with a frequency ω and a speed c≈3×10⁸ m/s. Electrical fields therefore propagates in vacuum in the form of waves. You may also recognize that c is the speed of light.

Note that, by virtue of Ampère’s law, this electric wave is always accompanied by a perpendicular magnetic wave

An illustration of such an electromagnetic wave propagating along the Z axis is given in Figure 2 with electrical components along X and magnetic component along Y.

It is important to mention that all the derivations performed here are only valid in vacuum where Gauss’s law on electrical fields states that

which we used to help decomposing our triple products earlier.

You can understand Gauss’s law as the fact that all electrical fields, in a vacuum, must form closed lines (with lines potentially extending towards infinity). That is, every exiting field vector must be compensated by an equivalent incoming field vector over an infinitely small volume – there are no sources or sinks for electrical fields in vacuum. When sources are present, the derivations are a bit more complicated to perform.

Also, you should note that all these derivations only shows that vacuum supports the propagation of electric and magnetic fields in the forms of waves – they do not explain where the oscillating electric field originates from. While the maths are a bit more intricated, it is possible to show that oscillating electrical charges create such variation of the electrical field with the shape of the wave strongly dependent on the oscillation motion of the charge. Near the electrical charge, the distribution does not look much like a wave, but as we go away from the charge, a solution looking like the one of Figure 2 emerges. This is precisely what happens when divE≠0, something that I will come back on in a specific post later.

Any oscillating electrical charge will create an electromagnetic wave that will follow the equations we just derived once sufficiently far away from the source. However, we call light only those electromagnetic waves that have frequencies between 10¹⁴ rad/s and 10¹⁶ rad/s. Such electromagnetic waves are created when atoms decay from high electronic states to lower electronic states. However, these processes are transient and only last for a very short period of time, typically from 10^-8 sec to 10^-9 sec, leading to short pulses of electromagnetic waves.

Real light consists of many transitions occurring over a volume. The direction of the electric field for each of these waves depends on the orientation of the charge variation and we end up with a mixture of many different orientations at the same time – the light is said randomly polarized or natural or unpolarized, although this latter term might be misleading physically speaking. Furthermore, if the transitions result in a narrow range of oscillation frequencies, we say that the light is monochromatic. If the transitions result in a broader range of frequencies, we end up with what we would call polychromatic light or even white light as the spectrum gets broader.

Let’s come back to the electric field orientation as it is going to be the topic of this post and the following ones.

The electric field of a single electromagnetic wave can be oriented anywhere in the XY plane, but it is always possible to represent it as a combination of the two orthogonal directions X and Y by virtue of the superposition principle:

such that

A single electromagnetic wave can therefore be seen as the superposition of two orthogonal electromagnetic waves where the overall direction is given by the proportion of E_0,x and E_0,y. The tricky part is to manage to flip your mind to accommodate that, here, there is no “two waves” nor “one wave” but just different decompositions thanks to the superposition principle. Also, there is almost no restriction on how we choose X and Y, although it is convenient to choose XYZ mutually orthogonal to simplify computations. So, we can always choose any two orthogonal vectors X and Y provided that they are perpendicular to the light propagation direction, Z, as illustrated in Figure 3. This will conveniently be used to divide an incoming wave into the reference frame of a physical phenomenon later on.

Now that we have a good description of electromagnetic waves, let’s have a look at what happens when the wave passes through a scattering liquid.

Our scattering liquid is made of dipoles and a dipole is made of a positive an electric charge, separated in space. A typical example of a dipole, at the scale of light waves, are atoms that are made of a negatively charged electron cloud and a positively charged nucleus. The mean center of charge of the nucleus may super-impose on the mean center of charge of the electron cloud, but, because of the constant electron motion, temporary dipoles arise even in atoms. In molecules, electrons clouds deform depending on the relative electronegativity of each atom – leading to permanent dipoles. In most cases, it is very common for dipoles to be oriented in random directions.

Being permanent or transient, the electromagnetic field produces a [∞] Lorentz force on the electric charge, q,

This force tends to pull the dipole apart in the direction of the electric field. However, due to [∞] Coulombian attraction, the dipole contracts like a spring of constant k

The total force acting on the dipole is therefore

leading to the standard [∞] forced oscillator equation

with ω₀ the resonant natural frequency of this oscillator

These equations apply to the electron cloud as well as to the nucleus, but since the nucleus is much heavier than the electron clouds, we observe a large motion of the electron cloud for an extremely small variation of the nucleus position.

Since the electric field oscillates at a frequency ω, we expect the dipole to oscillate at the same frequency (just like in any forced oscillator). We can therefore expect a solution of the form

yielding

such that

and

where α is the polarizability of the dipole, an extremely important concept in [∞] Raman spectroscopy.

In conclusion, dipoles will oscillate in the direction of the input electric field to the extent of how well the oscillation frequency of the electromagnetic wave corresponds to the natural resonant frequency of the dipole. Note that the amplitude of this phenomenon is proportional to the squared frequency of the input electric field and we can therefore expect that the magnitude of the scattered wave will be proportional to ω²-ω₀², emphasizing scattering for some colors of light.

Now, remember that an oscillating charge itself produces an electromagnetic wave. So, the dipole starts emitting its own electromagnetic wave at the same frequency – a process known as scattering. We are however not creating light per se as energy conservation implies that the incoming electromagnetic wave has been absorbed by the dipole to make it oscillate, and this energy is freed again with the emission of a new electromagnetic wave. Because of the susceptibility of the dipole to radiation, only a very small fraction of light follows scattering in normal conditions.

Recalling how the curl operator drives the electromagnetic equations, as we have shown in Figure 1, the emitted electromagnetic wave orientation will be stronger perpendicular to the oscillating direction (so perpendicular to the incoming electric field) and null along the oscillation direction. This is represented in Figure 4 for a wave travelling along Z with an electrical field oscillating along the Y direction (left) and along the X direction (right). You immediately see that the scattered wave goes either along X or Y, depending on the initial direction of the electrical field. Dipoles will therefore scatter light differently, depending on the different polarizations states of the incoming light.

Since natural light contains many electromagnetic waves of random electric field orientation, and that we can always divide any random orientation in XY into orthogonal components X and Y according to the superposition principle, we end up with the results of Figure 5 where the light along Z is still unpolarized (mixed polarization states) but fully polarized along X and Y. The scattering process therefore acts as a polarization filter! Diagonal directions of the XY plane contains mixes of polarization states, with mix content decreasing as we approach pure X and Y axis.

This phenomenon inspired the experimental setup of Figure 6 which can be downloaded [∞] here for reproduction (STEP, drawings and BOM files). A vial containing a scattering solution is illuminated either from the side, bottom, or from forward/in-line using a LED and a detector is placed with a polarizing filter – a special optical element that let passes only light that have a specific oscillation direction. By rotating the polarizer, we can choose the direction of polarization. Concerning the scattering solution, [∞] formazine with low NTU would be ideal, but very diluted milk does the job as well. For the experiment here, it is important that the solution is almost transparent.

The concept of the experiment is to verify the predictions of Figure 5. When placing the LED at 90° from the detector, the light should be fully polarized, but, when placing the light inline with the detector, the light should be completely unpolarized. Using a photodiode is not mandatory for the experiment as you can see the extinction of light with your eye when turning the polarizer. I included a photodiode in this experiment to quantify the degree of polarization. Note that I used pulsed light at 100 Hz (sine wave) and read the resulting amplitude using an oscilloscope after amplifying the signal using my [∞] photodiode amplifier circuit to obtain precise results.

To express the degree of polarization, we now introduce the polarization ratio obtained as the contrast between the two polarizations axis of the analyzer noted here I_0° and I_90° (this represent the axis of the analyzer – not to be confused with the system placed at 90°),

A polarization ratio of 1 means fully polarized light, with the sign indicating the direction of the polarization. A polarization ratio of 0 means fully unpolarized light.

Experimentally, with the LED at 90°, I found a minimum amplitude of 32 mV and a maximum amplitude of 82 mV. When illuminating from below, I got 109 mV and 58.5 mV with the same polarizer orientation as for the previous measurement (LED at 90°). Finaly, in the inline configuration, the signal was constant at 3.13 V.

The results are summarized in Figure 7.

While the theory predicts fully polarized light when not inline, we observe only partial polarization at 90°. This is due to the fact that we don’t have a single scattering dipole in the experiment and that the light we collect has been scattered multiple times, decreasing its overall polarization – reason for which we need some scattering but not too much scattering for the experiment. This also explains why illuminating from below resulted in less polarization (more depolarization) since there is more optical path to traverse. Nonetheless, we can say that the results of Figure 7 validates the theoretical predictions of polarization by scattering.

The purpose of today’s post was to present linearly polarized light and how some physical processes can affect the degree of polarization. However, polarization by scattering has consequences in your everyday life – especially if you are a photographer! When sun light passes through the atmosphere, a tiny fraction is scattered back to you; making the sky looks blue (scattering in more intense in the low region of the spectrum due to its dependency with ω⁴). But due to the polarization by scattering effect of Figure 5, this light is partially polarized. Photographers take advantage of this process to reduce sky diffusion as illustrated in Figure 8. Note that the amount of polarization depends on position of the observer and sun, as polarization is maximum when they are 90° apart as explained previously. If you use a polarizer for your pictures to decrease sky haze, it is therefore recommended to avoid wide angle lenses and to focus on a region of the sky that forms a 90° angle with the sun and your position.

That’s all for today! In the next post, I will present another experimental setup to get your hands on linear polarization! As we progress into the series, we will go towards more advanced usage of polarization, but I want to introduce this step by step such as to offer the best experience I can on understanding polarization in optics.

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I would like to give a big thanks to Sebastian, Alex, Stephen, Lilith, James, Jon, Jesse, Kausban, Karel, Michael, Zach, Sivaraman, Samy, Shaun, Onur, Sunanda, Benjamin, Themulticaster, Tayyab, Marcel, Dennis, M, Natan and RottenSpinach who have supported this post through [∞] Patreon. I also take the occasion to invite you to donate through Patreon, even as little as $1. I cannot stress it more, you can really help me to post more content and make more experiments!