Planar Boundaries, Total Internal Reflection, Beamsplitters

Refraction is an important effect in ray optics. The refractive index of a material influences how rays react when entering or leaving a boundary. For instance, if the ray is exiting a medium of smaller refractive index and entering a medium with a higher refractive index, the angle will tend towards being perpendicular to the boundary line. The angle of refraction is also greater than the angle of incidence. This case is called external refraction (n1 < n2) and (θ1 > θ2). If the ray is exiting a medium of higher refractive index into a medium with a lower refractive index, the rays will tend towards being closer to parallel with the medium boundary. This case is referred to as internal refraction (n1 > n2) and (θ2 > θ1). Both of these situations are governed by Snell’s Law:

n1*sin(θ1) = n2*sin(θ2)

When the rays are paraxial, the relation between θ1 and θ2 is linear (n1*θ1 = n2*θ2).


The critical angle occurs when n1*sin(θ1) = n2*sin(pi/2) = n2. θ1 in this case is then equal to the critical angle. If θ1 is greater than the critical angle θC, refraction cannot occur and the situation is characterized by a phenomenon known as total internal reflection (TIR). Total internal reflection is the basis for many optical systems and devices. Systems with total internal reflection are understood to be highly efficient even under more rigorous approaches to optics such as electromagnetic optics.



Prisms are common applications of refraction. A prism of apex angle α and refractive index n deflects a ray incident at an angle of θ:


This is taken by using Snell’s law twice along two planar boundaries.



A beamsplitter is an optical component that divides a ray into a reflected and refracted (or transmitted) ray. The proportions of reflected to refractive light is a problem dealt with in electromagnetic optics. Beamsplitters are also used to combine two rays.


Beam directors apply Snell’s law and the rules governing refraction to direct rays in different directions. Three methods of directing waves are the biprism, the Fresnel biprism and the axicon.



4 thoughts on “Planar Boundaries, Total Internal Reflection, Beamsplitters

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