How Does Refraction Affect Laser Beam Diameter?

[pinkerpikachu]

Homework Statement

A laser beam of diameter = 3.1mm in air has an incident angle = 26 degrees at a flat air-glass surface.If the index of refraction of the glass is n=1.5, determine the diameter of the beam after it enters the glass.

Homework Equations

n₁sinθ₁=n₂sinθ₂

The Attempt at a Solution

okay, well the initial application of snell's law is obvious. The laser is going from air, where n = 1 into glass where n = 1.5

So, (1)sin(26) = (1.5)sinθ₂

θ₂ = (approx) 17 degrees or 16.99

I know how to find the diameter of the laser, the solutions have this equation:

d1/cosθ1 = d2/cosθ2

and simple plug and chug would yield the answer 3.3 mm

But I just don't understand where this second equation came from. Can some one walk me through the sense behind it? My book doesn't even mention it, so perhaps its obvious (just not to me).

 

[ehild]

A picture always helps.


ehild

 

[pinkerpikachu]

I see, the two diameters are related through the common interface (the hypothenuse of the two triangles). Just some geometry from there.

It makes sense, thank you :)

 

[ehild]

You are welcome. And start drawing!

ehild

 

[asteriatic]

I would first clarify that the second equation is not Snell's Law, but rather a trigonometric relationship between the angle of incidence and the angle of refraction, as well as the diameter of the laser beam in air and in the glass. This equation can be derived from Snell's Law, but it is not the same thing.

To understand where this equation comes from, we can use the concept of the conservation of energy. When light enters a medium with a different refractive index, some of its energy is transmitted through the medium, while some is reflected back. In this case, we are assuming that all of the light is transmitted through the glass, so the energy of the beam must remain constant.

We know that the energy of a beam is proportional to its intensity, which is related to its diameter. Therefore, if we assume that the intensity of the laser beam remains constant before and after entering the glass, we can use the relationship between intensity and diameter to set up an equation:

I1 = I2 (since the energy remains constant)
d1^2/4 = d2^2/4
d1^2 = d2^2

This equation tells us that the diameter of the beam in air (d1) is equal to the diameter of the beam in glass (d2). However, we need to take into account the fact that the beam will be refracted at the air-glass interface, which will change the angle of the beam. This is where the trigonometric relationship between the angles comes in.

Using basic trigonometry, we can relate the diameter of the beam to the angle of incidence and the angle of refraction:

d1/cosθ1 = d2/cosθ2

This equation is essentially saying that the diameter of the beam in air (d1) is equal to the diameter of the beam in glass (d2) divided by the cosine of the angle of refraction (cosθ2). This relationship allows us to find the diameter of the beam in glass, knowing the diameter of the beam in air and the angle of refraction.

In summary, the second equation is a result of the conservation of energy and basic trigonometry, and it allows us to find the diameter of the beam in glass using the diameter of the beam in air and the angle of refraction.