| Snell's Law |
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OVERVIEW 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 ''n''1 (on the left) and ''n''2 (on the right) meet at a surface or interface (vertical line). ''n''2 > ''n''1, 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. Snell's law gives the relation between the angles θ1 and θ2: : 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 ''n''1 and ''n''2. In other words, 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. A handy way to remember this is to visualize 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. Snell's law is only generally true for isotropic media (such as Glass ). In Anisotropic media such as some Crystal s, 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. TOTAL INTERNAL REFLECTION When moving from a dense to a less dense medium (i.e. ''n''1 > ''n''2), it is easily verified that the above equation has no solution when θ1 exceeds a value known as the '' Critical Angle '': : When θ1 > θcrit, no refracted ray appears, and the incident ray undergoes Total Internal Reflection from the interface. 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) : : : : 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. 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. 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. HISTORY Snell's law was first discovered and described by Ibn Sahl in a manuscript written c. 984 , who used it to work out the shapes of Anaclastic Lens es (lenses that focus light with no geometric aberrations). It was discovered again by Thomas Harriot in 1602 , 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 in terms of sines in his 1637 treatise '' Discourse On Method '', and used it to solve a range of optical problems. In French , Snell's Law is called "la loi de Descartes" or "loi de Snell-Descartes". SEE ALSO REFERENCES | ||
| Author | Rashed, Roshdi |
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| Title | A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses |
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| Journal | ISIS |
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| Year | 1990 volume= 81 pages= 464–491 |
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