Refractive Index

Imagine you are going for a run along a smooth dirt path, and you approach a waist-deep river. You need to cross the river and don't want to slow your run, so you decide to press forward through it. As you enter the water, you try to maintain the same speed as before, but quickly realize that the water is slowing you down. Finally, making it to the other side of the river, you pick up the same speed as before and continue with your run. In the same way that the speed of your run decreased as you ran through the water, optics tells us that the propagation speed of light decreases as it travels through different materials. Every material has a refractive index that gives the ratio between the speed of light in the vacuum and the speed of light in the material. The refractive index allows us to determine the path a light beam will take as it travels through the material. Let's learn more about the refractive index in optics!

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    Refraction Index A man running through the water StudySmarterFig. 1 - Water slows down a runner like different materials slow down the propagation speed of light.

    Definition of Refractive Index

    When light is traveling through a vacuum, or empty space, the light's speed of propagation is simply the speed of light, \(3.00\times10^8\mathrm{\frac{m}{s}}.\) Light travels slower when it goes through a medium such as air, glass, or water. A light beam passing from one medium to another at an incident angle will experience reflection and refraction. Some of the incident light will be reflected off of the surface of the medium at the same angle as the incident angle with respect to the surface normal, while the rest will be transmitted at a refracted angle. The normal is an imaginary line perpendicular to the boundary between both media. In the image below, a light ray experiencing reflection and refraction as it passes from medium \(1\) to medium \(2,\) appears in light green. The thick blue line depicts the boundary between both media whereas the skinny blue line perpendicular to the surface represents the normal.

    Refraction Index Reflected and refracted light as it passes from one material to another StudySmarterFig. 2 - A light beam is reflected and refracted as it passes from one medium to another.

    Every material has an index of refraction that gives the ratio between the speed of light in a vacuum and the speed of light in the material. This helps us to determine the refracted angle.

    The refractive index of a material is the ratio between the speed of light in a vacuum and the speed of light in the material.

    A light beam traveling at an angle from a material that has a lower refractive index to one with a higher refractive index will have a refraction angle that bends toward the normal. The refraction angle bends away from the normal when it travels from a higher refractive index to a lower one.

    Formula for Refractive Index

    The refractive index, \(n,\) is dimensionless since it's a ratio. It has the formula \[n=\frac{c}{v},\] where \(c\) is the speed of light in vacuum and \(v\) is the speed of light in the medium. Both quantities have units of meters per second, \(\mathrm{\frac{m}{s}}.\) In a vacuum, the refractive index is unity, and all other media have a refractive index that is greater than one. The index of refraction for air is \(n_\mathrm{air}=1.0003,\) so we generally round to a few significant figures and take it to be \(n_{\mathrm{air}}\approx 1.000.\) The table below shows the refractive index for various media to four significant figures.

    MediumRefractive Index
    Air1.000
    Ice1.309
    Water1.333
    Crown Glass1.517
    Zircon1.923
    Diamond2.417

    The ratio of the refractive indices of two different media is inversely proportional to the ratio of the light's propagation speed in each one:

    \[\begin{align*}\frac{n_2}{n_1}&=\frac{\frac{c}{v_2}}{\frac{c}{v_1}}\\[8pt]\frac{n_2}{n_1}&=\frac{\frac{\bcancel{c}}{v_2}}{\frac{\bcancel{c}}{v_1}}\\[8pt]\frac{n_2}{n_1}&=\frac{v_1}{v_2}.\end{align*}\]

    The law of refraction, Snell's law, uses the refractive index to determine the refracted angle. Snell's law has the formula

    \[n_1\sin\theta_1=n_2\sin\theta_2,\]

    where \(n_1\) and \(n_2\) are the indices of refraction for two media, \(\theta_1\) is the incident angle, and \(\theta_2\) is the refracted angle.

    Critical Angle of the Index of Refraction

    For light traveling from a medium of a higher index of refraction to a lower one, there is a critical angle of incidence. At the critical angle, the refracted light beam skims the surface of the medium, making the refracted angle a right angle with respect to the normal. When the incident light hits the second medium at any angle greater than the critical angle, the light is totally internally reflected, so that there is no transmitted (refracted) light.

    The critical angle is the angle at which the refracted light beam skims the surface of the medium, making a right angle with respect to the normal.

    We calculate the critical angle using the law of refraction. As mentioned above, at the critical angle the refracted beam is tangent to the surface of the second medium so that the refraction angle is \(90^\circ.\) Thus, \(\sin\theta_1=\sin\theta_\mathrm{crit}\) and \(\sin\theta_2=\sin(90^\circ)=1\) at the critical angle. Substituting these into the law of refraction gives us:

    \[\begin{align*}n_1\sin\theta_1&=n_2\sin\theta_2\\[8pt]\frac{n_2}{n_1}&=\frac{\sin\theta_1}{\sin\theta_2}\\[8pt]\frac{n_2}{n_1}&=\frac{\sin\theta_\mathrm{crit}}{1}\\[8pt]\sin\theta_\mathrm{crit}&=\frac{n_2}{n_1}.\end{align*}\]

    Since \(\sin\theta_\mathrm{crit}\) is equal to or less than one, this shows that the refractive index of the first medium must be greater than that of the second for total internal reflection to occur.

    Measurements of Refractive Index

    A common device that measures the refractive index of a material is a refractometer. A refractometer works by measuring the refraction angle and using it to calculate the refractive index. Refractometers contain a prism upon which we place a sample of the material. As light shines through the material, the refractometer measures the refraction angle and outputs the refractive index of the material.

    A common use for refractometers is to find the concentration of a liquid. A hand-held salinity refractometer measures the amount of salt in salt water by measuring the refraction angle as light passes through it. The more salt there is in the water, the greater the refraction angle is. After calibrating the refractometer, we place a few drops of salt water on the prism and cover it with a cover plate. As light shines through it, the refractometer measures the refraction index and outputs the salinity in parts per thousand (ppt). Beekeepers also use hand-held refractometers in a similar way to determine how much water is in honey.

    Refraction Index A hand-held refractometer for liquids StudySmarterFig. 3 - A hand-held refractometer uses refraction to measure the concentraion of a liquid.

    Examples of the Refractive Index

    Now let's do some practice problems for the refractive index!

    A light beam initially traveling through air hits a diamond with an incident angle of \(15^\circ.\) What is the propagation speed of the light in the diamond? What is the refracted angle?

    Solution

    We find the propagation speed by using the relation for the index of refraction, speed of light, and propagation speed given above:

    \[n=\frac{c}{v}.\]

    From the table above, we see that \(n_\text{d}=2.417.\) Solving for the propagation speed of the light in a diamond gives us:

    \[\begin{align*}v&=\frac{c}{n_\text{d}}\\[8pt]&=\frac{3.000\times10^8\,\mathrm{\frac{m}{s}}}{2.417}\\[8pt]&=1.241\times10^8\,\mathrm{\tfrac{m}{s}}.\end{align*}\]

    To calculate the refracted angle, \(\theta_2,\) we use Snell's law with the incident angle, \(\theta_1,\) and indices of refraction for air, \(n_\mathrm{air},\) and diamond, \(n_\mathrm{d}\):

    \[\begin{align*}n_\mathrm{air}\sin\theta_1&=n_\mathrm{d}\sin\theta_2\\[8pt]\sin\theta_2&=\frac{n_\mathrm{air}}{n_\mathrm{d}}\sin\theta_1\\[8pt]\theta_2&=\sin^{-1}\left(\frac{n_\mathrm{air}}{n_\mathrm{d}}\sin\theta_1\right)\\[8pt]&=\sin^{-1}\left(\frac{1.000}{2.147}\sin(15^\circ)\right)\\[8pt]&=6.924^\circ.\end{align*}\]

    Thus, the refraction angle is \(\theta_2=6.924^\circ.\)

    When using your calculator to calculate cosine and sine values for an angle given in degrees, always make sure that the calculator is set to take degrees as inputs. Otherwise, the calculator will interpret the input as given in radians, which would result in an incorrect output.

    Find the critical angle for a light beam traveling through crown glass to water.

    Solution

    According to the table in the section above, the refractive index of crown glass is higher than that of water, so any incident light coming from the crown glass that hits the glass-water interface at an angle greater than the critical angle will be totally internally reflected into the glass. The refractive indices of crown glass and water are \(n_\mathrm{g}=1.517\) and \(n_\mathrm{w}=1.333,\) respectively. So, the critical angle is:

    \[\begin{align*}\sin\theta_\mathrm{crit}&=\frac{n_\mathrm{w}}{n_\mathrm{g}}\\[8pt]\sin\theta_\mathrm{crit}&=\frac{1.333}{1.517}\\[8pt]\sin\theta_\mathrm{crit}&=0.8787\\[8pt]\theta_\mathrm{crit}&=\sin^{-1}(0.8787)\\[8pt]&=61.49^{\circ}.\end{align*}\]

    Thus, the critical angle of a light beam traveling from crown glass to water is \(61.49^{\circ}.\)

    Refractive Index - Key takeaways

    • The refractive index of a material is the ratio between the speed of light in the vacuum and the speed of light in the material, \(n=\frac{c}{v},\) and is dimensionless.
    • The propagation speed of light is slower in media with a higher refractive index.
    • The law of refraction, or Snell's law, relates the angles of incidence and refraction and the indices of refraction according to the equation: \(n_1\sin\theta_1=n_2\sin\theta_2.\)
    • When light travels from a medium with a low refractive index to one with a high refractive index, the refracted beam bends toward the normal. It bends away from the normal when traveling from a medium with a high refractive index to a low one.
    • At the critical angle, light traveling from a medium of a higher refractive index to a lower one skims the surface of the medium, making a right angle with the normal to the surface. Any incident beam that hits the material at an angle greater than the critical angle is totally internally reflected.
    • A refractometer calculates the refractive index of a material and can be used to determine the concentration of a liquid.

    References

    1. Fig. 1 - Running in Water (https://pixabay.com/photos/motivation-steeplechase-running-704745/) by Gabler-Werbung (https://pixabay.com/users/gabler-werbung-12126/) licensed by Pixaby License (https://pixabay.com/service/terms/)
    2. Fig. 2 - Reflected and Refracted Light, StudySmarter Originals
    3. Fig. 3 - Hand-held Refractometer (https://en.wikipedia.org/wiki/File:2020_Refraktometr.jpg) by Jacek Halicki (https://commons.wikimedia.org/wiki/User:Jacek_Halicki) licensed by CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/deed.en)
    Frequently Asked Questions about Refractive Index

    What is a refractive index?

    The refraction index of a material is the ratio between the speed of light in vacuum and the speed of light in the material.

    What are examples of refractive indices?

    Examples of refractive indices for different materials include approximately one for air, 1.333 for water, and 1.517 for crown glass. 

    Why does refractive index increase with frequency?

    The refractive index increases with frequency in dispersion when white light is split into different wavelengths. The wavelengths of light travel at different speeds, and the refractive index for a wavelength increases with shorter wavelengths and greater frequencies. 

    How to calculate refractive index?

    The refractive index of a material is calculated by finding the ratio between the speed of light in vacuum and the speed of light in the material. A refractometer can be used to find the refraction angle of a material, and then the refractive index can be calculated.

    What is the refractive index of glass?

    The refractive index of crown glass is approximately 1.517.

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    Test your knowledge with multiple choice flashcards

    The speed of light is the same when traveling through all materials. 

    What is the refractive index in vacuum?

    How does the ratio of the refractive indices for two materials relate to the ratio of the speed of light in the materials?

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