ReflectionReflection is the abrupt change in the direction of propagation of a wave that strikes the boundary between two different media. At least some part of the incoming wave remains in the same medium. Assume the incoming light ray makes an angle θ_{i} with the normal of a plane tangent to the boundary. Then the reflected ray makes an angle θ_{r} with this normal and lies in the same plane as the incident ray and the normal. Law of reflection: θ_{i} = θ_{r} 

Specular reflection
occurs at smooth, plane boundaries. Then the plane tangent to the
boundary is the boundary itself. Reflection at rough, irregular
boundaries is diffuse reflection.
The smooth surface of a mirror reflects light specularly, while the
rough surface of a wall reflects light diffusely. The
reflectivity or reflectance of a
surface material is the fraction of energy of the oncoming wave that is
reflected by it. The reflectivity of a mirror is close to 1.


RefractionRefraction is the change in direction of propagation of a wave when the wave passes from one medium into another, and changes its speed. Light waves are refracted when crossing the boundary from one transparent medium into another because the speed of light is different in different media. Assume that light waves encounter the plane surface of a piece of glass after traveling initially through air as shown in the figure to the right. What happens to the waves as they pass into the glass and continue to travel through the glass? The speed of light in glass or water is less than the speed of light in a vacuum or air. The speed of light in a given substance is v = c/n, where n is the index of refraction of the substance. Typical values for the index of refraction of glass are between 1.5 and 1.6, so the speed of light in glass is approximately twothirds the speed of light in air. The distance between wave fronts will therefore be shorter in the glass than in air, since the waves travel a smaller distance per period T. If f is the frequency of the wave and T = 1/f is the period, i.e. the time interval between successive crests passing a fixed point in space, then λ_{1 }= v_{1}T = cT/n_{1} and λ_{2 } = v_{2}T = cT/n_{2}, or λ_{1}/λ_{2 }= n_{2}/n_{1}. 


Now consider wave fronts and their corresponding light
rays approaching the surface at an angle.
We can see that the rays will bend as the wave passes from air to glass. The bending occurs because the wave fronts do not travel as far in one cycle in the glass as they do in air. As the diagram shows, the wave front halfway into the glass travels a smaller distance in glass than it does in air, causing it to bend in the middle. Thus, the ray, which is perpendicular to the wave front, also bends. The situation is like a marching band marching onto a muddy field at an angle to the edge of the field. The rows bend as the speed of the marchers is reduced by the mud. The amount of bending depends on the angle of incidence and on the indices of refraction of glass and air, which determine the change in speed. From the figure we can see that λ_{1}/λ_{2} = sinθ_{1}/sinθ_{2}. But λ_{1}/λ_{2 }= n_{2}/n_{1}. Therefore n_{2}/n_{1 }= sinθ_{1}/sinθ_{2}, or n_{1}sinθ_{1 }= n_{2}sinθ_{2}.



When light passes from one transparent medium to
another, the rays are bent toward the surface normal if the speed of
light is smaller in the second medium than in the first. The rays
are bent away from this normal if the speed of light in the second
medium is greater than in the first. The picture on the right
shows a light wave incident on a slab of glass.
One part of the wave is reflected, and another part is refracted as it passes into the glass. The rays are bent towards the normal. At the second interface from glass into air the light passing into the air is refracted again. The rays are now bent away from the normal. 


Reflectance and transmittance At a boundary between two transparent media, light is partially reflected and partially refracted. The ratio of the reflected intensity to the incident intensity is called the reflectance R and the ratio of the transmitted intensity to the incident intensity is called the transmittance T. Energy conservation requires that R + T = 1 (if there is no absorption). 

R and T depend on the indices of refraction of the two media n_{1} and n_{2}, the angle of incidence θ_{1}, and the polarization of the incident light. We distinguish between ppolarization and spolarization. Let the plane of incidence contain the normal to the boundary and the incident wave vector k_{1}. The electric field vector E_{1} is perpendicular to k_{1}. If we choose our coordinate system as shown on the right, then plane of incidence is the xzplane and E_{1} may be written as E_{1}= E_{p}+ E_{s}. E_{p} lies in the xzplane and E_{s} is perpendicular to the xzplane, i.e. it points in the ±ydirection. The electric field of the incident light is a linear superposition of p and spolarized fields. For ppolarized light we have R = r_{12p}^{2}, where r_{12p} is the Fresnel reflection coefficient for ppolarization. We have r_{12p }= tan(θ_{1 } θ_{2})/tan(θ_{1 }+ θ_{2}). For spolarized light we have R = r_{12s}^{2}, where r_{12s} is the Fresnel reflection coefficient for spolarization. We have r_{12s }= sin(θ_{1 } θ_{2})/sin(θ_{1 }+ θ_{2}). For a graph of the reflectance R for s and ppolarized light as a function of n_{1}, n_{2}, and θ_{1}, download this Excel spreadsheet. If θ_{1 }+ θ_{2} = π/2,then tan(θ_{1 }+ θ_{2}) = infinite and r_{12p} = 0. If light is reflected, it will have spolarization. The incident angle at which this happens is called the Brewster angle θ_{B}. We then have n_{1}sinθ_{B }= n_{2}sin((π/2)  θ_{B}) = n_{2}cosθ_{B}. tanθ_{B} = n_{2}/n_{1}. Polarized light can thus be obtained via reflection. 