Two hundred years ago, Augustin-Jean Fresnel published the foundations for modern polarization optics[1]. The goal is simple: compute how much light reflects from materials such as water, or metal.
In present times the Fresnel equations are understood as a consequence of basic laws of physics. Energy cannot be created nor destroyed, light slows down in a material, and electric fields are continuous across a boundary of two materials.
Description of Light
In the simplest terms, light can be represented by a traveling wave.
There are two important terms used to characterize the light wave. The wavelength of light describes the distance between the peaks in the wave.
The second parameter is the frequency of light. This describes the number of times per second that a single point can make a round-trip oscillation from top to bottom and back to top. In the figure above, the left edge can be observed to take 4 seconds to make a complete oscillation, and so this wave has a frequency of 1/4 cycle per second.
Frequency commonly appears in optics because the energy of light is proportional to frequency. This is important because a fundamental law in physics states that energy must always be conserved. This means that as long as the light wave exists, the rate at which a point oscillates up and down can never change.
Material Properties
Now what happens when the light wave travels through a material? The simple answer is the wave slows down. Change the slider below to see how the wave transforms when the material property changes.
Different materials slow down light by different amounts. This is characterized by a parameter called the "index of refraction" and denoted by the letter "n". The value n = 1 represents perfect vacuum, water is near n = 1.3, and diamond is approximately n = 2.4.
In changing the slider above, you may notice two effects. First, the speed at which the peaks of the wave move from left-to-right slows down when n is increased. This is the main effect of the index of refraction for the material. Higher index of refraction means slower speed.
Secondly, the spacing between peaks also changes. As discussed earlier, the frequency of oscillation can never change. For every value of "n" in the plot above, each point always oscillates at 1/4 cycle per second. Having a slower speed with the same frequency means the wavelength must decrease.
Law of Refraction
The consequence of light slowing down in a material is the direction of travel must change when light transitions from one material to another. This behavior is known as the law of refraction, and is the basis for everything from camera lens design to the angle a rainbow makes in the sky.
It is worth noting the law of refraction is commonly referred to as "Snell's law" for work by Willebrørd Snellius approximately 400 years ago. However it's actual use has been traced more than 600 years earlier to work by Abu Said al-Ala Ibn Sahl[2].
The best way to understand the law of refraction is to consider a collection of light waves all traveling in the same direction. In the diagram below, peaks of the waves are represented by red lines that are at right angles to the direction of motion.
angle (θ1):
As light slows down upon entering a material with higher index of refraction, the direction of travel changes. Calculation of this new direction angle is the law of refraction and satisfies the following equation: n1 sin(θ1) = n2 sin(θ2) In this equation θ1 is the angle of incidence, and θ2 is the angle of propagation in the second material. Both angles are measured relative to the surface normal. The values n1 and n2 are the indices of refraction for the two materials.
Fresnel Equations
Developed in the years 1821-1823, the Fresnel equations[1] describe the amplitude of transmitted and reflected light at the boundary between two materials. For over 1,000 years the direction of light has been calculated by the law of refraction, but equations to describe the amplitude of reflected light are only 200 years old.
The reason it took so long for the Fresnel equations to appear is because it had to wait for another discovery; light is a transverse wave. Until the 1800s, it was believed that light could oscillate along the direction of propagation. However, experiments by Thomas Young in 1818 proved that light only oscillates orthogonal to the propagation direction[3]. In other words, light is a transverse wave.
s and p Polarization
For transverse waves at an interface between two materials, it is convenient to describe the oscillations in terms of symmetry at the interface. The input direction of propagation and a line at right angles to the surface will form a plane. If the wave oscillations are fully contained in that plane, modern terminology designates it as "p-polarization". When the light oscillation is perpendicular to that plane, it is known as "s-polarization". The letters "p" and "s" are abbreviations of the German words for parallel (parallel) and perpendicular (senkrecht).
The figure below only shows the incident light wave, but allows you to switch between s and p polarizations. The vertical dashed line represents the surface normal. Between the surface normal and the solid black line that represents the light propagation direction is a shaded green region. This region is part of a plane of symmetry that is parallel to your computer screen, and p-polarization is completely contained in that plane.
In this figure, a red dot has been added to help visualize how the light oscillates at a single point along the propagation direction. Because this figure represents a 2D view of a 3D phenomena, the s-polarization may be difficult to visualize. In this perspective, s-polarized light oscillates into and out of your computer screen.
Continuity of Electric and Magnetic Fields
The final concept on this journey to the Fresnel equations is continuity of electric and magnetic waves. In the diagram below, all three waves (incident, reflected, and transmitted) oscillate in sync at the boundary between materials. This is what it means for the tangential component of the electric field to be continuous across the boundary, and this is an important requirement for constructing the Fresnel equations.
angle (θ1):
In this figure, the opacity of each wave indicates its magnitude. When the wave disappears, no light propagates in that direction. Can you find the angle where the reflected wave disappears? This is known as Brewster's angle, and is named for David Brewster's work in 1815.[4].
Fresnel Equations
In previous figures, only the electric wave is shown because it is the most significant for optical sciences. However light also has a magnetic field which also has continuity across the boundary.
When everything is combined, the Fresnel equations can be derived[5]. Conservation of energy, law of refraction, s and p polarizations, and also the continuity of electric and magnetic fields. These are the concepts that define how bright reflected and transmitted light is.
Letters "r" and "t" represent reflection and transmission coefficients for electric field amplitude. Angles θ1 and θ2 represent the incident and transmitted angles relative to the surface normal. These angles are related by the law of refraction. Variables n1 and n2 are the indices of refraction for the incident and transmitted material.
As a final step, it is often desired to convert from electric field amplitude to measurable irradiance quantities. This involves the norm-square of the amplitude with a few additional factors required for conservation of energy.
Following a common naming convention, lower-case letters "r" and "t" represent amplitude terms while upper-case letters "R" and "T" represent irradiance terms. Irradiance is what we observe, but electric field amplitude is the fundamental property of light.
In the Fresnel irradiance equations, the ratio of indices of refraction accounts for the change in wavelength at a boundary. The ratio of cosines is necessary for conservation of energy. A transmitted beam of light has greater width than an incident beam, as can be observed in the diagram for waves of light refracting at an interface between materials.
It is interesting to note that conservation of energy requires the irradiance components always sum to 100% (Rs + Ts = 1 and Rp + Tp = 1) but the amplitude terms do not follow the same behavior. In some cases, the transmitted amplitude can be larger than the incident amplitude!
Final Comments
Fresnel equations are a fundamental building block in optics and for this reason, a Fresnel equation calculator is provided on a separate page for quick reference. Also included on that page are links to downloadable code in several programming languages commonly used in the optical sciences.
Although this discussion focused on interactions of light with transparent materials such as water, glass, diamond and clear plastic, the concepts also apply to metals and crystals. However those materials require careful attention because the direction of energy flow is not the same direction where phase evolves. Details can be found in textbooks[6].
If you have any questions or comments, please contact me.
References
- A. Fresnel, "Considérations mécaniques sur la polarisation de la lumiére", Annales de Chimie et de Physique, vol.17, 1821, pp.192-194. Retrieved online Sept 2022: https://archive.org/details/s3id13207850/
- A.Kwan, J.Dudley and E.Lantz, "Who really discovered Snell's law?", Phys. World 15 (4), 2002, pp. 64. https://doi.org/10.1088/2058-7058/15/4/44
- A. Fresnel, "Considérations mécaniques sur la polarisation de la lumiére", Annales de Chimie et de Physique, vol. 17, 1821, p.184. Retrieved online Sept 2022: https://archive.org/details/s3id13207850/
- D. Brewster, "On the laws which regulate the polarisation of light by reflexion from transparent bodies", Philosophical Transactions of the Royal Society of London, vol. 105, 1815, pp.125-159. https://doi.org/10.1098/rstl.1815.0010
- For complete derivation, see J.D. Jackson, "Classical Electrodynamics" 3rd ed., section 7.3, "Reflection and Refraction of Electromagnetic Waves at a Plane Interface Between Dielectrics", pp.302-306. (ISBN: 978-0-471-30932-1)
- The authoritative reference for optics is: M.Born and E.Wolf, "Principles of Optics", 7th ed. Cambridge University Press, 2019. (ISBN: 978-1-108-47743-7) https://doi.org/10.1017/9781108769914. In this book, an entire chapter is devoted to the optics of metals and a separate chapter to the optics of crystals.