Light as a ray

     When we think of light as a ray, we think of it as typically moving in a straight line until it runs into something.  So sunlight comes to the earth in a straight line and then enters the atmosphere.  Once the light encounters matter such as the atmosphere or perhaps the surface of the ocean, it can do several things.  It can be reflected or scattered, it can be absorbed, or it can be transmitted.  If it is transmitted, it typically does not keep moving in the exact same direction.  It is refracted at the interface between two media, and so slightly changes directions. 

     In the real world, light will do all of these things.  Our goal is to figure out how much of the light follows each path as it passes through one medium or encounters a boundary between two materials.  The amount of each process depends on the index of refraction of the two materials, with the index depending on wavelength.  If we deal with smooth interfaces, we need only think about reflection, as scattering into all random directions is negligible.  The fraction of light reflected at this smooth interface is determined by the difference in index of refraction of the two materials and the angle of incidence of the light ray.  At an interface between materials that have indices of refraction n₁ and n₂, for light incident perpendicular to the surface, the fraction of light reflected is

          f_R = [(n₂ – n₁)/( n₂ + n₁)]²

Typically this is pretty small.  For air (n₁ = 1.0) and water (n₂=1.33), f_R is 0.02 so only 2% is reflected.  

     The light that is transmitted through the air - water interface will be refracted. This is a result of the slowing down of the light in the higher index material.  The incident and refracted angles are related to each other by Snell's law.  This is given by

     n₁ sin θ₁ = n₂ sin θ ₂

where n₁ and θ₁ are for the incident light and n₂ and θ₂ are for the refracted light.  The net result of this is if n₂ > n₁ then sin θ₂ < sin θ₁. As a result, the light bends in towards the normal when it goes from low index (air) to higher index (water) materials.  This bending of light is pretty important for focusing and is used by the cornea and lens of the eye as well as the lenses in a pair of eye glasses.

     The light that enters the material and is refracted may subsequently be absorbed by the material as the light travels through it. This results in less light being transmitted, with transmission decreasing as a function of the thickness of the material.  The amount of light that is transmitted at a particular wavelength can be determined by the attenuation coefficient of the material.  This is given by Beer's law :

     I_trans = I₀ exp(- α l)

where I_trans is the amount of light transmitted after traveling a distance l through the material and I₀ is the initial amount of light entering the material.  Just to reiterate, the attenuation coefficient, α, is a function of wavelength so that different wavelengths will be transmitted to different degrees.  The fraction of light that is transmitted is given by

     f_T = I_trans / I₀ = exp(- α l)

The transmitted light will therefore exponentially decrease with distance through the material.  Because light can not be created or destroyed, the intial amount of light must equal the light that gets divided up between the different processes.  So

     Incident light = Reflected light + transmitted light + absorbed light

     1 = fraction reflected + fraction transmitted + fraction absorbed

     1 = f_R + f_T + f_A