Module 2: Using the Petrographic Microscope

2.3 Light and Optics Part 2: Refractive Index, Polarized Light, Birefringence

Elizabeth Johnson and Juhong Christie Liu

Refractive Index

Electromagnetic energy travels at the speed of light, c, in a vacuum.  However, when electromagnetic waves travel through matter, they slow down.  The speed at which they travel through a material is specific to the density and composition of the material.

The refractive index of a material characterizes the relationship between the speed of light in a vacuum and the speed of light in that material.

The refractive index of a material is defined as:

Equation 2.3.1. Refractive Index.

where n is the refractive index, c is the speed of light in a vacuum, and is the observed speed of light in the material. Since the speed of light in matter is always less than c, the index of refraction is always greater than or equal to one.

This table lists some examples of refractive indices of different materials:

Refractive Indices of Materials
Material n
Air (0 Celsius, 1 atm) 1.000293
Water, fresh 1.333
Glass, crown 1.52
Quartz 1.544
Zircon 1.923
Diamond 2.419

Guided Inquiry

 

The amount that a light ray changes its direction depends both on the incident angle and the amount that the speed changes. For a ray at a given incident angle, a large change in speed causes a large change in direction and thus a large change in angle. The exact mathematical relationship is the law of refraction, or Snell’s law, after the Dutch mathematician Willebrord Snell (1591–1626), who discovered it in 1621. The law of refraction is stated in equation form as:

 

Equation 2.3.2 Snell's Law

Here n1 and n2 are the indices of refraction for media 1 and 2, and θ1 and θ2 are the angles between the rays and the perpendicular in media 1 and 2. The incoming ray is called the incident ray, the outgoing ray is called the refracted ray, and the associated angles are the incident angle and the refracted angle, respectively.

Figure 2.3.9. The change in direction of a light ray depends on how the speed of light changes when it crosses from one medium to another. The speed of light is greater in medium 1 than in medium 2 in the situations shown here. (a) A ray of light moves closer to the perpendicular when it slows down. This is analogous to what happens when a lawn mower goes from a footpath to grass. (b) A ray of light moves away from the perpendicular when it speeds up. This is analogous to what happens when a lawn mower goes from grass to footpath. The paths are exactly reversible.
Figure 2.3.9. The change in direction of a light ray depends on how the speed of light changes when it crosses from one medium to another. The speed of light is greater in medium 1 than in medium 2 in the situations shown here. (a) A ray of light moves closer to the perpendicular when it slows down. This is analogous to what happens when a lawn mower goes from a footpath to grass. (b) A ray of light moves away from the perpendicular when it speeds up. This is analogous to what happens when a lawn mower goes from grass to footpath. The paths are exactly reversible.

Figure 2.3.9 shows how a ray of light changes direction when it passes from one medium to another. The angles are measured relative to a perpendicular to the surface at the point where the light ray crosses it. (Some of the incident light will be reflected from the surface, but for now we will concentrate on the light that is transmitted.) The change in direction of the light ray depends on how the speed of light changes. The change in the speed of light is related to the indices of refraction of the media involved. In the situations shown in Figure 2.3.9 medium 2 has a greater index of refraction than medium 1. This means that the speed of light is less in medium 2 than in medium 1.

Note that as shown in Figure 2.3.9(a), the direction of the ray moves closer to the perpendicular when it slows down. Conversely, as shown in Figure 2.3.9(b), the direction of the ray moves away from the perpendicular when it speeds up. The path is exactly reversible.

In both cases, you can imagine what happens by thinking about pushing a lawn mower from a footpath onto grass, and vice versa. Going from the footpath to grass, the front wheels are slowed and pulled to the side as shown. This is the same change in direction as for light when it goes from a fast medium to a slow one. When going from the grass to the footpath, the front wheels can move faster and the mower changes direction as shown. This, too, is the same change in direction as for light going from slow to fast.

 

Bending Light
Figure 2.3.10. Click on the image to access “Bending Light,” an interactive simulation of light refraction.

 

 

 

Guided Inquiry

 

Polarized Light

Polaroid sunglasses are familiar to most of us. They have a special ability to cut the glare of light reflected from water or glass. Polaroids have this ability because of a wave characteristic of light called polarization. What is polarization? How is it produced? What are some of its uses? The answers to these questions are related to the wave character of light.

Watch the first 6 minutes of the video below to see a practical overview of plane polarized light, using crossed polarizers, and how a third polarizer (which is how many minerals act) can be used to increase light output from crossed polarizers.

Electromagnetic waves are transverse waves consisting of varying electric and magnetic fields that oscillate perpendicular to the direction of propagation and perpendicular to each other.

The Sun and many other light sources produce waves in which E (and B, though it is not shown) are not preferentially oriented – they exist in every direction perpendicular to the direction of propagation (see Figure 2.3.11). Such light is said to be unpolarized because it is composed of many waves with all possible directions of polarization.

Figure 2.3.11. The slender arrow represents a ray of unpolarized light. The bold arrows represent the direction of polarization of the individual waves composing the ray. Since the light is unpolarized, the arrows point in all directions.
Figure 2.3.11. The slender arrow represents a ray of unpolarized light. The bold arrows represent the direction of polarization of the individual waves composing the ray. Since the light is unpolarized, the arrows point in all directions.

In contrast, light that is plane polarized (also called linearly polarized) has E oriented in one specific direction in space (Figure 2.3.12).  The polarization direction is defined by the orientation of E (as opposed to B).

Figure 2.3.12. The polarization direction of plane polarized light is defined as the vibration direction of the electric vector E.
Figure 2.3.12. The polarization direction of plane polarized light is defined as the vibration direction of the electric vector E.

Polarizers are composed of long molecules aligned in one direction. Thinking of the molecules as many slits, analogous to those for the oscillating ropes, we can understand why only light with a specific polarization can get through. The axis of a polarizing filter is the direction along which the filter passes the electric field of an EM wave (see Figure 2.3.13).

Figure 2.3.13. he transverse oscillations in one rope are in a vertical plane, and those in the other rope are in a horizontal plane. The first is said to be vertically polarized, and the other is said to be horizontally polarized. Vertical slits pass vertically polarized waves and block horizontally polarized waves.
Figure 2.3.13. The transverse oscillations in one rope are in a vertical plane, and those in the other rope are in a horizontal plane. The first is said to be vertically polarized, and the other is said to be horizontally polarized. Vertical slits pass vertically polarized waves and block horizontally polarized waves.

To examine this further, consider the transverse waves in the ropes shown in Figure 2.3.13. The oscillations in one rope are in a vertical plane and are said to be vertically polarized. Those in the other rope are in a horizontal plane and are horizontally polarized. If a vertical slit is placed on the first rope, the waves pass through. However, a vertical slit blocks the horizontally polarized waves. For EM waves, the direction of the electric field vector E is analogous to the disturbances on the ropes (Figure 2.3.14).

Figure 2.3.14. A polarizing filter has a polarization axis that acts as a slit passing through electric fields parallel to its direction. The direction of polarization of an EM wave is defined to be the direction of its electric field.
Figure 2.3.14. A polarizing filter has a polarization axis that acts as a slit passing through electric fields parallel to its direction. The direction of polarization of an EM wave is defined to be the direction of its electric field.

 

ATOMIC EXPLANATION OF POLARIZING FILTERS

Polarizing filters have a polarization axis that acts as a slit. This slit passes electromagnetic waves (often visible light) that have an electric field parallel to the axis. This is accomplished with long molecules aligned perpendicular to the axis as shown in Figure 2.3.15.

Figure 2.3.15. Long molecules are aligned perpendicular to the axis of a polarizing filter. The component of the electric field in an EM wave perpendicular to these molecules passes through the filter, while the component parallel to the molecules is absorbed.
Figure 2.3.15. Long molecules are aligned perpendicular to the axis of a polarizing filter. The component of the electric field in an EM wave perpendicular to these molecules passes through the filter, while the component parallel to the molecules is absorbed.

Figure 2.3.16 illustrates how the component of the electric field parallel to the long molecules is absorbed. An electromagnetic wave is composed of oscillating electric and magnetic fields. The electric field is strong compared with the magnetic field and is more effective in exerting force on charges in the molecules. The most affected charged particles are the electrons in the molecules, since electron masses are small. If the electron is forced to oscillate, it can absorb energy from the EM wave. This reduces the fields in the wave and, hence, reduces its intensity. In long molecules, electrons can more easily oscillate parallel to the molecule than in the perpendicular direction. The electrons are bound to the molecule and are more restricted in their movement perpendicular to the molecule. Thus, the electrons can absorb EM waves that have a component of their electric field parallel to the molecule. The electrons are much less responsive to electric fields perpendicular to the molecule and will allow those fields to pass. Thus the axis of the polarizing filter is perpendicular to the length of the molecule.

Figure 2.3.16. Artist’s conception of an electron in a long molecule oscillating parallel to the molecule. The oscillation of the electron absorbs energy and reduces the intensity of the component of the EM wave that is parallel to the molecule.
Figure 2.3.16. Artist’s conception of an electron in a long molecule oscillating parallel to the molecule. The oscillation of the electron absorbs energy and reduces the intensity of the component of the EM wave that is parallel to the molecule.

 

 

Figure 2.3.17 shows the effect of two polarizing filters on originally unpolarized light. The first filter polarizes the light along its axis. When the axes of the first and second filters are aligned (parallel), then all of the polarized light passed by the first filter is also passed by the second. If the second polarizing filter is rotated, only the component of the light parallel to the second filter’s axis is passed. When the axes are perpendicular, no light is passed by the second.

Figure 2.3.17. The effect of rotating two polarizing filters, where the first polarizes the light. (a) All of the polarized light is passed by the second polarizing filter, because its axis is parallel to the first. (b) As the second is rotated, only part of the light is passed. (c) When the second is perpendicular to the first, no light is passed. (d) In this photograph, a polarizing filter is placed above two others. Its axis is perpendicular to the filter on the right (dark area) and parallel to the filter on the left (lighter area). (credit: P.P. Urone)
Figure 2.3.17. The effect of rotating two polarizing filters, where the first polarizes the light. (a) All of the polarized light is passed by the second polarizing filter, because its axis is parallel to the first. (b) As the second is rotated, only part of the light is passed. (c) When the second is perpendicular to the first, no light is passed. (d) In this photograph, a polarizing filter is placed above two others. Its axis is perpendicular to the filter on the right (dark area) and parallel to the filter on the left (lighter area). (credit: P.P. Urone)

Only the component of the EM wave parallel to the axis of a filter is passed. Let us call the angle between the direction of polarization and the axis of a filter θ. If the electric field has an amplitude E, then the transmitted part of the wave has an amplitude E cos θ (see Figure 2.3.18). Since the intensity of a wave is proportional to its amplitude squared, the intensity I of the transmitted wave is related to the incident wave by I = I0 cos2 θ, where I0 is the intensity of the polarized wave before passing through the filter.

Figure 2.3.18. A polarizing filter transmits only the component of the wave parallel to its axis, Ecosθ , reducing the intensity of any light not polarized parallel to its axis.
Figure 2.3.18. A polarizing filter transmits only the component of the wave parallel to its axis, Ecosθ , reducing the intensity of any light not polarized parallel to its axis.

Liquid Crystal DISPLAYS and Optically Active Materials

While you are undoubtedly aware of liquid crystal displays (LCDs) found in watches, calculators, computer screens, cellphones, flat screen televisions, and other myriad places, you may not be aware that they are based on polarization. Liquid crystals are so named because their molecules can be aligned even though they are in a liquid. Liquid crystals have the property that they can rotate the polarization of light passing through them by 90 degrees. Furthermore, this property can be turned off by the application of a voltage, as illustrated in Figure 2.3.19. It is possible to manipulate this characteristic quickly and in small well-defined regions to create the contrast patterns we see in so many LCD devices.

Figure 2.3.19. (a) Polarized light is rotated 90º by a liquid crystal and then passed by a polarizing filter that has its axis perpendicular to the original polarization direction. (b) When a voltage is applied to the liquid crystal, the polarized light is not rotated and is blocked by the filter, making the region dark in comparison with its surroundings. (c) LCDs can be made color specific, small, and fast enough to use in laptop computers and TVs. (credit: Jon Sullivan)
Figure 2.3.19. (a) Polarized light is rotated 90º by a liquid crystal and then passed by a polarizing filter that has its axis perpendicular to the original polarization direction. (b) When a voltage is applied to the liquid crystal, the polarized light is not rotated and is blocked by the filter, making the region dark in comparison with its surroundings. (c) LCDs can be made color specific, small, and fast enough to use in laptop computers and TVs. (credit: Jon Sullivan)

In flat screen LCD televisions, there is a large light at the back of the TV. The light travels to the front screen through millions of tiny units called pixels (picture elements). One of these is shown in Figure 2.3.19 (a) and (b). Each unit has three cells, with red, blue, or green filters, each controlled independently. When the voltage across a liquid crystal is switched off, the liquid crystal passes the light through the particular filter. One can vary the picture contrast by varying the strength of the voltage applied to the liquid crystal.

Figure 2.3.20. Optical activity is the ability of some substances to rotate the plane of polarization of light passing through them. The rotation is detected with a polarizing filter or analyzer.
Figure 2.3.20. Optical activity is the ability of some substances to rotate the plane of polarization of light passing through them. The rotation is detected with a polarizing filter or analyzer.

Many crystals and solutions rotate the plane of polarization of light passing through them. Such substances are said to be optically active. Examples include sugar water, insulin, and collagen (see Figure 2.3.20). In addition to depending on the type of substance, the amount and direction of rotation depends on a number of factors. Among these is the concentration of the substance, the distance the light travels through it, and the wavelength of light. Optical activity is due to the asymmetric shape of molecules in the substance, such as being helical. Measurements of the rotation of polarized light passing through substances can thus be used to measure concentrations, a standard technique for sugars. It can also give information on the shapes of molecules, such as proteins, and factors that affect their shapes, such as temperature and pH.

Figure 2.3.21. Optical stress analysis of a plastic lens placed between crossed polarizers. (credit: Infopro, Wikimedia Commons)
Figure 2.3.21. Optical stress analysis of a plastic lens placed between crossed polarizers. (credit: Infopro, Wikimedia Commons)

Glass and plastic become optically active when stressed; the greater the stress, the greater the effect. Optical stress analysis on complicated shapes can be performed by making plastic models of them and observing them through crossed filters, as seen in Figure 2.3.21. It is apparent that the effect depends on wavelength as well as stress. The wavelength dependence is sometimes also used for artistic purposes.

Birefrigence

Another interesting phenomenon associated with polarized light is the ability of some minerals and other crystals to split an unpolarized beam of light into two polarized beams (Figure 2.3.22). Such crystals are said to be birefringent.

Each of the separated rays has a specific polarization. One behaves normally and is called the ordinary ray (o or ω), whereas the other does not obey Snell’s law and is called the extraordinary ray (e or ε). Birefringent crystals can be used to produce polarized beams from unpolarized light. Some birefringent materials preferentially absorb one of the polarizations. These materials are called dichroic and can produce polarization by this preferential absorption. This is fundamentally how polarizing filters and other polarizers work. We will use the property of birefringence to help us identify and distinguish minerals in thin section!

 

Figure 2.3.22. Birefringent materials, such as the common mineral calcite, split unpolarized beams of light into two. The ordinary ray behaves as expected, but the extraordinary ray does not obey Snell’s law.
Figure 2.3.22. Birefringent materials, such as the common mineral calcite, split unpolarized beams of light into two. The ordinary ray behaves as expected, but the extraordinary ray does not obey Snell’s law.

Key Terms

  • Refractive Index
  • Snell’s Law
  • Transverse wave
  • Unpolarized
  • Plane (linear) polarization
  • Birefringence
  • Ordinary Ray
  • Extraordinary Ray

References

Lumen Learning. Physics. CC-BY. https://courses.lumenlearning.com/physics/

National Aeronautics and Space Administration, Science Mission Directorate. (2010). Anatomy of an Electromagnetic Wave. Retrieved March 9, 2019, from NASA Science website: http://science.nasa.gov/ems/02_anatomy

National Aeronautics and Space Administration, Science Mission Directorate. (2010). Introduction to the Electromagnetic Spectrum. Retrieved March 9, 2019, from NASA Science website: http://science.nasa.gov/ems/01_intro

National Aeronautics and Space Administration, Science Mission Directorate. (2010). Wave Behaviors. Retrieved March 9, 2019 from NASA Science website: http://science.nasa.gov/ems/03_behaviors

OpenStax, College Physics. Creative Commons Attribution License v4.0. https://openstax.org/details/books/college-physics. 

OpenStax, University Physics Volume 3. Creative Commons Attribution License 4.0 license.  https://openstax.org/books/university-physics-volume-3/pages/1-1-the-propagation-of-light

PhET Interactive Simulations, University of Colorado Boulder. Bending Light. CC_BY. https://phet.colorado.edu/en/simulation/bending-light

 

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