The propagation of electromagnetic waves in various media obeys the laws of reflection and refraction. Under certain conditions, one interesting effect follows from these laws, which in physics is called the total internal reflection of light. Let's take a closer look at what this effect is.
Reflection and refraction
Before proceeding directly to the consideration of internal total reflection of light, it is necessary to explain the processes of reflection and refraction.
Reflection is understood to mean a change in the direction of motion of a light beam in the same medium when it encounters any interface. For example, if you direct a light beam from a laser pointer onto a mirror, you can observe the described effect.
Refraction is, like reflection, a change in the direction of the movement of light, but not in the first, but in the second medium. The result of this phenomenon will be a distortion of the outlines of objects and their spatial location. A common example of refraction is the breaking of a pencil or pen if he / she is placed in a glass of water.
Refraction and reflection are interconnected. They are almost always present together: part of the energy of the beam is reflected, and the other part is refracted.
Both phenomena are the result of applying the Fermat principle. He claims that the light moves along such a path between two points that will take him the least time.
Since reflection is an effect that occurs in one medium, and refraction in two media, it is important for the latter that both media are transparent to electromagnetic waves.
Concept of refractive index
The refractive index is an important quantity for the mathematical description of the phenomena under consideration. The refractive index of a particular medium is defined as follows:
n = c / v.
Where c and v are the speeds of light in vacuum and matter, respectively. The value of v is always less than c, so the exponent n will be greater than unity. The dimensionless coefficient n shows how much light in a substance (medium) will lag behind light in a vacuum. The difference in these speeds leads to the occurrence of a refraction phenomenon.
The speed of light in matter correlates with the density of the latter. The denser the medium, the harder the light to move in it. For example, for air n = 1,00029, that is, almost like for a vacuum, for water n = 1,333.
Reflections, refraction and their laws
The basic laws of light refraction and reflection can be written as follows:
- If we restore the normal to the point of incidence of the light beam on the boundary between two media, then this normal, along with the incident, reflected and refracted rays, will lie in the same plane.
- If the angles of incidence, reflection and refraction are designated as θ 1 , θ 2 , and θ 3 , and the refractive indices of the 1st and 2nd medium are n 1 and n 2 , then the following two formulas are valid:
- for reflection θ 1 = θ 2 ;
- for refraction sin (θ 1 ) * n 1 = sin (θ 3 ) * n 2 .
Analysis of the formula for the 2nd law of refraction
To understand when the internal total reflection of light will begin, one should consider the law of refraction, which is also called Snell's law (the Dutch scientist who discovered it at the beginning of the XVII century). We rewrite the formula:
sin (θ 1 ) * n 1 = sin (θ 3 ) * n 2 .
It can be seen that the product of the sine of the angle of the beam to the normal and the refractive index of the medium in which this beam propagates is unchanged. This means that if n 1 > n 2 , then for the equality to hold it is necessary that sin (θ 1 ) <sin (θ 3 ). That is, when moving from a denser medium to a less dense one (meaning optical density), the beam deviates from the normal (the sine function increases for angles from 0 o to 90 o ). Such a transition occurs, for example, when a beam of light crosses a water-air boundary.
The refraction phenomenon is reversible, that is, when passing from less dense to more dense (n 1 <n 2 ), the ray will approach the normal (sin (θ 1 )> sin (θ 3 )).
Internal total reflection of light
Now let's move on to the most interesting. Consider the situation when a light beam passes from a denser medium, that is, n 1 > n 2 . In this case, θ 1 <θ 3 . Now we will gradually increase the angle of incidence θ 1 . The angle of refraction θ 3 will also increase, but since it is larger than θ 1 , it will earlier become equal to 90 o . What does θ 3 = 90 o mean from a physical point of view? This means that all the energy of the beam, when it hits the interface, will propagate along it. In other words, the refracting ray will not exist.
A further increase in θ 1 will cause the entire beam to be reflected from the surface back to the first medium. This is the phenomenon of internal total reflection of light (refraction is completely absent).
The angle θ 1 at which θ 3 = 90 o is called critical for a given pair of media. It is calculated by the following formula:
θ c = arcsin (n 2 / n 1 ).
This equality follows directly from the 2nd law of refraction.
If the propagation velocities v 1 and v 2 of electromagnetic radiation in both transparent media are known, then the critical angle is calculated by the following formula:
θ c = arcsin (v 1 / v 2 ).
It should be understood that the main condition for internal total reflection is that it exists only in an optically denser medium surrounded by a less dense one. So, at certain angles, light coming from the seabed can be completely reflected from the surface of the water, but at any angle of incidence from the air, the beam will always penetrate the water column.
Where is observed and where is the effect of total reflection applied?
The most famous example of using the phenomenon of internal total reflection is fiber optics. The idea is that due to 100% reflection of light from the surface of the media, electromagnetic energy can be transmitted over arbitrarily large distances without loss. The working material of the fiber optic cable, from which its inner part is made, has a higher optical density than the peripheral material. Such a composition is enough to successfully use the effect of full reflection for a wide range of angles of incidence.
A striking example of the result of total reflection are the shiny surfaces of diamond. The refractive index for diamond is 2.43, so many rays of light, once in a gemstone, experience multiple full reflection before leaving it.
The problem of determining the critical angle θc for diamond
Consider a simple problem, where we show how to use the above formulas. It is necessary to calculate how much the critical angle of total reflection will change if a diamond from air is placed in water.
After looking in the table the values for the refractive indices of these media, we write them:
- for air: n 1 = 1,00029;
- for water: n 2 = 1.333;
- for diamond: n 3 = 2.43.
The critical angle for a diamond-air pair is:
θ c1 = arcsin (n 1 / n 3 ) = arcsin (1,00029 / 2.43) ≈ 24.31 o .
As you can see, the critical angle for this pair of media is quite small, that is, only those rays can leave the diamond into the air, which will be closer to the normal than 24.31 o .
For the case of diamond in water, we get:
θ c2 = arcsin (n 2 / n 3 ) = arcsin (1.333 / 2.43) ≈ 33.27 o .
The increase in the critical angle was:
Δθ c = θ c2 - θ c1 ≈ 33.27 o - 24.31 o = 8.96 o .
This slight increase in the critical angle for the complete reflection of light in diamond leads to the fact that it shines in water almost the same as in air.