Polarization; and The Human Eye

8-4-99

Polarization

To talk about the polarization of an electromagnetic wave, it's easiest to look at polarized light. Just remember that whatever applies to light generally applies to other forms of electromagnetic waves, too. So, what is meant by polarized light? It's light in which there's a preferred direction for the electric and magnetic field vectors in the wave. In unpolarized light, there is no preferred direction: the waves come in with electric and magnetic field vectors in random directions. In linearly polarized light, the electric field vectors are all along one line (and so are the magnetic field vectors, because they're perpendicular to the electric field vectors). Most light sources emit unpolarized light, but there are several ways light can be polarized.

One way to polarize light is by reflection. Light reflecting off a surface will tend to be polarized, with the direction of polarization (the way the electric field vectors point) being parallel to the plane of the interface.

Another way to polarize light is by selectively absorbing light with electric field vectors pointing in a particular direction. Certain materials, known as dichroic materials, do this, absorbing light polarized one way but not absorbing light polarized perpendicular to that direction. If the material is thick enough to absorb all the light polarized in one direction, the light emerging from the material will be linearly polarized. Polarizers (such as the lenses of polarizing sunglasses) are made from this kind of material.

If unpolarized light passes through a polarizer, the intensity of the transmitted light will be 1/2 of what it was coming in. If linearly polarized light passes through a polarizer, the intensity of the light transmitted is given by Malus' law:

A third way to polarize light is by scattering. Light scattering off atoms and molecules in the atmosphere is unpolarized if the light keeps traveling in the same direction, is linearly polarized if at scatters in a direction perpendicular to the way it was traveling, and somewhere between linearly polarized and unpolarized if it scatters of at another angle.

There are plenty of materials that affect the polarization of light. Certain materials (such as calcite) exhibit a property known as birefringence. A crystal of birefringent material affects light polarized in a particular direction differently from light polarized at 90 degrees to that direction; it refracts light polarized one way at a different angle than it refracts light polarized the other way. Looking through a birefringent crystal at something, you'd see a double image.

Liquid crystal displays, such as those in digital watches and calculators, also exploit the properties of polarized light.

Polarization by reflection

One way to polarize light is by reflection. If a beam of light strikes an interface so that there is a 90 angle between the reflected and refracted beams, the reflected beam will be linearly polarized. The direction of polarization (the way the electric field vectors point)is parallel to the plane of the interface.

The special angle of incidence that satisfies this condition, where the reflected and refracted beams are perpendicular to each other, is known as the Brewster angle. The Brewster angle, the angle of incidence required to produce a linearly-polarized reflected beam, is given by:

This expression can be derived using Snell's law, and the law of reflection. The diagram below shows some of the geometry involved.

Using Snell's law:

The scattering of light in the atmosphere

The way light scatters off molecules in the atmosphere explains why the sky is blue and why the sun looks red at sunrise and sunset. In a nutshell, it's because the molecules scatter light at the blue end of the visible spectrum much more than light at the red end of the visible spectrum. This is because the scattering of light (i.e., the probability that light will interact with molecules when it passes through the atmosphere) is inversely proportional to the wavelength to the fourth power.

Violet light, with a wavelength of about 400 nm, is almost 10 times as likely to be scattered than red light, which has a wavelength of about 700 nm. At noon, when the Sun is high in the sky, light from the Sun passes through a relatively thin layer of atmosphere, so only a small fraction of the light will be scattered. The Sun looks yellow-white because all the colors are represented almost equally. At sunrise or sunset, on the other hand, light from the Sun has to pass through much more atmosphere to reach our eyes. Along the way, most of the light towards the blue end of the spectrum is scattered in other directions, but much less of the light towards the red end of the spectrum is scattered, making the Sun appear to be orange or red.

So why is the sky blue? Again, let's look at it when the Sun is high in the sky. Some of the light from the Sun traveling towards other parts of the Earth is scattered towards us by the molecules in the atmosphere. Most of this scattered light is light from the blue end of the spectrum, so the sky appears blue.

Why can't this same argument be applied to clouds? Why do they look white, and not blue? It's because of the size of the water droplets in clouds. The droplets are much larger than the molecules in the atmosphere, and they scatter light of all colors equally. This makes them look white.

Optics of the eye

The human eye is a wonderful instrument, relying on refraction and lenses to form images. There are many similarities between the human eye and a camera, including:

The way the eye focuses light is interesting, because most of the refraction that takes place is not done by the lens itself, but by the aqueous humor, a liquid on top of the lens. Light is refracted when it comes into the eye by this liquid, refracted a little more by the lens, and then a bit more by the vitreous humor, the jelly-like substance that fills the space between the lens and the retina.

The lens is critical in forming a sharp image, however; this is one of the most amazing features of the human eye, that it can adjust so quickly when focusing objects at different distances. This process of adjustment is known as accommodation.

Consider the lens equation:

With a camera, the lens has a fixed focal length. If the object distance is changed, the image distance (the distance between the lens and the film) is adjusted by moving the lens. This can't be done with the human eye: the image distance, the distance between the lens and the retina, is fixed. If the object distance is changed (i.e., the eye is trying to focus objects that are at different distances), then the focal length of the eye is adjusted to create a sharp image. This is done by changing the shape of the lens; a muscle known as the ciliary muscle does this job.

Nearsightedness

A person who is nearsighted can only create sharp images of close objects. Objects that are further away look fuzzy because the eye brings them in to focus at a point in front of the retina. To correct for this, a diverging lens is placed in front of the eye, diverging the light rays just enough so that when the rays are converged by the eye they converge on the retina, creating a focused image.

Farsightedness

A farsighted person can only create clear images of objects that are far away. Close objects are brought to a focus behind the retina, which is why they look fuzzy. To correct for this, a converging lens is placed in front of the eye, allowing images to be brought into sharp focus at the retina.

Diopters

If you go to an optometrist to get glasses or contact lenses, you will get a prescription specified in units of diopters. This is a measure of the refractive power of the lens needed, which means it's a measure of the focal length of the lens. The two are, in fact, inversely related:

refractive power in diopters = 1 / focal length in meters

A diopter has units of 1 / m .

If the lenses you get are specified as 5.0 diopters, it means they have a focal length of 0.2 m, meaning that they are converging lenses that bring parallel rays of light to a focus 0.2 m beyond the lens. Similarly, lenses of -2.0 diopters correspond to a focal length of -0.5 m; these would be diverging lenses with a focal point 0.5 m from the lens.

Lens aberrations

Lenses can distort the image of an object for a number of reasons. One kind of distortion is known as spherical aberration; this occurs because parallel rays of light are not all focused to a single point by a spherical lens. The further a ray is from the principal axis, the more it misses the focal point.

A second kind of distortion is chromatic aberration. This occurs because a lens will have a slightly different index of refraction for light of different wavelengths (i.e., light of different colors). In other words, chromatic aberration is caused by dispersion. Parallel rays of red light, therefore, would be brought to a different focal point than parallel rays of light of another color, leading to a blurry image.

To minimize chromatic aberration, many high-quality lenses are made up of two lenses made from different materials. One lens will be converging, and the other diverging; the compound lens will still be converging overall, generally. The chromatic aberration introduced by one lens is corrected for in the second lens, bringing parallel rays of any color light to about the same focal point.

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