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Some Consequences of the Fresnel Equations

Module by: Paul Padley. E-mail the author

Summary: We look at some of the consequences of the Fresnel equations.

The Fresnel Equations

r ( E 0 r E 0 i ) = n i cos θ i n t cos θ t n i cos θ i + n t cos θ t r ( E 0 r E 0 i ) = n i cos θ i n t cos θ t n i cos θ i + n t cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n i cos θ i + n t cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n i cos θ i + n t cos θ t

r ( E 0 r E 0 i ) = n t cos θ i n i cos θ t n t cos θ i + n i cos θ t r ( E 0 r E 0 i ) = n t cos θ i n i cos θ t n t cos θ i + n i cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n t cos θ i + n i cos θ t t ( E 0 t E 0 i ) = 2 n i cos θ i n t cos θ i + n i cos θ t

Why Polaroid sunglasses work

Lets put these equations to work and figure out something practical. Consider light reflecting off a surface, such as the road in front of you when you are driving a car. The light hitting the road surface can have any polarization but that will be some addition of light that has E E and to the plane of incidence. From r ( E 0 r E 0 i ) = n t cos θ i n i cos θ t n t cos θ i + n i cos θ t r ( E 0 r E 0 i ) = n t cos θ i n i cos θ t n t cos θ i + n i cos θ t and r ( E 0 r E 0 i ) = n i cos θ i n t cos θ t n i cos θ i + n t cos θ t r ( E 0 r E 0 i ) = n i cos θ i n t cos θ t n i cos θ i + n t cos θ t we see there is an angle where there is no and only light, namely n t cos θ i n i cos θ t = 0 n t cos θ i n i cos θ t = 0 or n t cos θ i = n i cos θ t . n t cos θ i = n i cos θ t . Now we can use Snell's law rewritten as sin θ i n t = sin θ t n i sin θ i n t = sin θ t n i and multiply both sides to get n t cos θ i = n i cos θ t n t cos θ i = n i cos θ t n t cos θ i sin θ i n t = n i cos θ t sin θ t n i n t cos θ i sin θ i n t = n i cos θ t sin θ t n i cos θ i sin θ i = cos θ t sin θ t cos θ i sin θ i = cos θ t sin θ t which can only be true if θ t = π / 2 θ i . θ t = π / 2 θ i . In this case Snell's law can be written n t n i = sin θ i sin θ t = sin θ i sin ( π / 2 θ i ) = tan θ i n t n i = sin θ i sin θ t = sin θ i sin ( π / 2 θ i ) = tan θ i The angle that gives this effect is known as Brewster's angle θ i θ B r e w s t e r = tan 1 n t n i θ i θ B r e w s t e r = tan 1 n t n i At this angle light is completely polarized, it only has E E to the plane of incidence (or parallel to the surface). Thus the glare you get from reflected light tends to be polarized in this way. In 1929 Edwin Land invented a method for making celluoid filters to filter out light with given polarizations. He then manufactured sunglasses with these polarizers lined up to filter out the E E light and thereby reduce glare.

Other cases

Look at the Fresnel equations again and examine what happens when θ i θ i approaches 90 degrees. The reflection approaches 1 (ignore the signs) Thus at Glancing incidence you get lots of reflection. In fact X-ray telescopes use this to focus the x-rays onto their detector.

Another effect, if n i > n t n i > n t then there is an angle of incidence beyond which light is only reflected. That is the angle where θ t = 90 θ t = 90 degrees. n t sin θ t = n i sin θ i n t sin θ t = n i sin θ i θ t π / 2 θ t π / 2 sin θ t 1 sin θ t 1 sin θ i n t n i sin θ i n t n i The critical angle at which this occurs is θ c = sin 1 n t n i θ c = sin 1 n t n i

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