Snell's law

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In Optics and physical optics, Snell's law (also known as Decartes' Law or the law of refraction) is a formula named after one of its discoverers, Dutch mathematician Willebrord Snellius, used along with Fermat's principle to determine the extent of refraction caused when a light passes beyond a boundary into a medium of a different density. The theory is most often re-arranged to find the angle of incidence or refraction in a given situation, and is also used to calculate the refractive index of a given material, which is a measure of the extent that light will deflect when passing through the medium.

Despite the fact that the majority of the iterations of snell's law use angles to determine the refraction caused by a medium, the formula has been expanded to allow the use of other variables such as speed and wavelength to determine the change. Both of these expansions of the law use the average speed and/or wavelength of a photon.

1 History
2 Explanation
- 2.1 Total internal reflection
- 2.2 Derivation
3 Uses
- 3.1 Calculating refractive indices
- 3.2 Vector form
4 See also
5 References
6 External links

[edit] History

Snell's law was first discovered and described by Ibn Sahl in a manuscript written c.984 ^[1], who used it to work out the shapes of lenses that focus light with no geometric aberrations, known as anaclastic lenses . It was discovered again by Thomas Harriot in 1602 ^[2], who did not publish his work.

In 1621, it was discovered yet again by Willebrord Snell, in a mathematically equivalent form, but unpublished during his lifetime. René Descartes independently derived the law using heuristic momentum conservation arguments in terms of sines in his 1637 treatise Discourse on Method (though detractors such as Fermat accused Descartes of working toward the already known answer with sophistic reasoning), and used it to solve a range of optical problems. Rejecting Descartes' solution, Pierre de Fermat arrived at the same solution based solely on his principal of least time.

In French, Snell's Law is called "la loi de Descartes" or "loi de Snell-Descartes".

[edit] Explanation

The law itself is used, or re-arranged in the majority to ascertain the nature of refraction within, or through a refractive media with varying indices of refraction. In common parlance, the indices of refraction are labelled $n 1, n 2$ and so on, which are used to represent the factor by which light has "slowed down" within a refractive medium, such as glass or water.

As light passes or "impinges" on the border between 2 media, it passes through into the second medium; depending upon the refractive index of the medium, the light will either be refracted to a lesser angle, or a greater one. The extent of diffraction is measured with relevance to the normal line, which is drawn 90^o perpendicular to the boundary. In the case of light travelling from air into water, light would be deflected towards the normal line, due to the fact the light has a more densely packed medium to travel through - light travelling from water to air would refract away from the normal line.

Diffraction between 2 surfaces is also referred to as reversible due to the fact that if all conditions were identical, the deflection would be the same when performed in the opposite manor.

Snell's law is only generally true for isotropic media (such as glass). In anisotropic media such as some crystals, birefringence may split the refracted ray into two rays, the ordinary or o-ray which follows Snell's law, and the other extraordinary or e-ray which may not be co-planar with the incident ray.

Although in several areas of physics, angles are used, the law also works for wavelengths and speeds which are relative to those of a photon travelling at the speed of light. The full equation taking wavelength into account is usually shown as:

$\frac {\lambda_o}{\lambda_1} = \frac{v_o}{v_m} = \frac{c / n_1}{c / n_2} = \frac{n_2}{n_1}$

Which are also expressed as:
$λ 1 sinθ 1 = λ 2 sinθ 2$ for wavelength-based calculations
$n 1 sinθ 1 = n 2 sinθ 2$ for angle-based calculations

Where:

$λ o$ denotes the wavelength of a photon propogated at 3x10⁸ms^-1	$λ 1$ denotes the wavelength of the light within the material
$v o$ represents 3x10⁸ms^-1	$v m$ represents the speed of the ray within the material
$c / n 1$ and $c / n 2$ represent the shorthand calculation for calculating phase velocity or refractive index.	$n 1$ and $n 2$ represent the overall refraction by division of refractive indecies.

[edit] Total internal reflection

An example of the angles involved within total internal reflection.

When light moves from a dense to a less dense medium, such as the process of moving from water into air -- the law formulae cannot be used to calculate the refractive angle due to the fact that the resolved sine value is higher than 1.

This is when the light is known to refract at the critical angle, which is described as an angle of refraction that is actually refracting light backwards onto itself, without there to be a change in phases, like in other forms of optical phenomena.

A common example to illustrate this point, is used when a ray of light is incident at $50 o$ towards a water-air boundary. If the angle is calculated using snell's law, then the resulting sine value will not invert, and thus, the refracted angle cannot be calculated:

$\theta_2 = sin^{-1} (\frac{n_1}{n_2}\sin\theta_1) = sin^{-1} (\frac{1.333}{1.000}0.766) = sin^{-1} 1.021$

The usage of snell's law to calculate such angles does not work due to the fact that there is no refracted outgoing ray which is relative to the normal. In order to calculate the refractive angle, the assumption that $n 2 < n 1$ and the refracted angle is 90 degrees. From this, the law can be expanded to
$n 1 sinθ c = n 2 sin90 o = n 2,$ or $(\sin \theta_c = \frac{n_1}{n_2})$ or as $\theta_{\mathrm{crit}} = \arcsin \left( \frac{n_2}{n_1} \right)$

When θ₁ > θ_crit, no refracted ray appears, and the incident ray undergoes total internal reflection from the interface medium.

[edit] Derivation

Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time. By taking the derivative of the optical path length, the stationary point is found giving the path taken by the light (though it should be noted that the result does not show light taking the least time path, but rather one that is stationary with respect to small variations as there are cases where light actually takes the greatest time path, as in a spherical mirror). Alternatively, it can be derived using interference of all possible paths of light wave from source to observer - it results in destructive interference everywhere except extrema of phase (where interference is constructive) - which become actual paths. In a classic analogy by Feynman, the area of lower refractive index is replaced by a beach, the area of higher refractive index by the sea, and the fastest way for a rescuer on the beach to get to a drowning person in the sea is to run along a path that follows Snell's law.

[edit] Uses

[edit] Calculating refractive indices

$Refraction of light at the interface between two media of different refractive indices, with n2 > n1.$

Refraction of light at the interface between two media of different refractive indices, with n₂ > n₁.

In the diagram on the right, two media of refractive indices n₁ (on the left) and n₂ (on the right) meet at a surface or interface (vertical line). n₂ > n₁, and light has a slower phase velocity within the second medium.

A light ray PO in the leftmost medium strikes the interface at the point O. From point O, we project a straight line at right angles to the line of the interface; this is known as the normal to the surface (horizontal line). The angle between the normal and the light ray PO is known as the angle of incidence, θ₁.

The ray continues through the interface into the medium on the right; this is shown as the ray OQ. The angle it makes to the normal is known as the angle of refraction, θ₂.

n 1 sinθ 1 = n 2 sinθ 2

or $\sum_{k=x,y}^N x=n_x\sin\theta_x,n_y \sin\theta_y$

$n_1 = \frac{n_2 \sin\theta_2}{\sin\theta_1}$

$n_2 = \frac{n_1 \sin\theta_1}{\sin\theta_2}$

Note that, for the case of θ₁ = 0° (i.e., a ray perpendicular to the interface) the solution is θ₂ = 0° regardless of the values of n₁ and n₂-- a ray entering a medium perpendicular to the surface is never bent.

The above is also valid for light going from a dense to a less dense medium; the symmetry of Snell's law shows that the same ray paths are applicable in opposite direction.

A qualitative rule for determining the direction of refraction is that the ray in the denser medium is always closer to the normal. An analogy often used to remember this is done by visualizing the ray as a car crossing the boundary between asphalt (the less dense medium) and mud (the denser medium). Depending on the angle, either the left wheel or the right wheel of the car will cross into the new medium first, causing the car to swerve.

[edit] Vector form

Given a normalized ray vector v and a normalized plane normal vector p, one can work out the normalized reflected and refracted rays: (note that the actual angles θ₁ and θ₂ are not worked out)

$\cos\theta_1=\mathbf{v}\cdot\mathbf{p}$

$\cos\theta_2=\sqrt{1-\left(\frac{n_1}{n_2}\right)^2\left(1-\left(\cos\theta_1\right)^2\right)}$

$\mathbf{v}_{\mathrm{reflect}}=\mathbf{v}-\left(2\cos\theta_1\right)\mathbf{p}$

$\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{v} + \left(\cos\theta_2 - \frac{n_1}{n_2}\cos\theta_1\right)\mathbf{p}$

Note: $\mathbf{v}\cdot\mathbf{p}$ must be negative.

The cosines may be recycled and used in the Fresnel equations for working out the intensity of the resulting rays. During total internal reflection an evanescent wave is produced, which rapidly decays from the surface into the second medium. Conservation of energy is maintained by the circulation of energy across the boundary, averaging to zero net energy transmission.

[edit] See also

[edit] References

^ Rashed, Roshdi (1990). "A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses". ISIS 81: 464–491.
^ Kwan, A., Dudley, J., and Lantz, E. (2002). "Who really discovered Snell's law?". Physics World 15 (4): 64.