Total internal reflection inside a fiber optic cable

[aChordate]

Homework Statement

A multi-mode fiber optic cable has a core diameter d = 115 mm, n_core = 1.42, and n_cladding = 1.17. What is the largest possible angle θ_in relative to the surface normal that light can enter the fiber (i.e. pass from air into the fiber core) and experience total internal reflection inside the fiber?

Homework Equations

sinθ_c=n₂/n₁

n_claddingθ₁=n_coreθ₂

The Attempt at a Solution

d = 115 mm
n_core = 1.42
n_cladding = 1.17
θ_in=?

Do I have the correct equations?

 

[PeterO]

Homework Statement

A multi-mode fiber optic cable has a core diameter d = 115 mm, n_core = 1.42, and n_cladding = 1.17. What is the largest possible angle θ_in relative to the surface normal that light can enter the fiber (i.e. pass from air into the fiber core) and experience total internal reflection inside the fiber?

Homework Equations

sinθ_c=n₂/n₁

n_claddingθ₁=n_coreθ₂

The Attempt at a Solution

d = 115 mm
n_core = 1.42
n_cladding = 1.17
θ_in=?

Do I have the correct equations?

Is the end of this optical fibre squared off , like a cylinder, or basically hemispherical, like many of these fires are when created?

Also I like to match subscripts, to avoid losing track of which angle you are calculating, so I would write either:

n_claddingsinθ_cladding=n_coresinθ_core

or

n₁sinθ₁=n₂sinθ₂

Having established [written down] that medium 1 was the cladding and material 2 was the core.

(and you seem to have left the sin out - though if the angles involved are small you can approximate sinθ to θ

 

[aChordate]

n_claddingsinθ_cladding=n_coresinθ_core

1.17sinθ_cladding=1.42sinθ_core

How do I find the angles?

 

[PeterO]

n_claddingsinθ_cladding=n_coresinθ_core

1.17sinθ_cladding=1.42sinθ_core

How do I find the angles?

For T.I.R. the angle of refraction is set at 90^o so one of those angles is 90^o.

Which one do you think that might be.

Hint: if you pick the wrong one, the problem gives no real solution.

 

[Nickie]

Yes, you have the correct equations. To find the largest possible angle θin, you can use the equation sinθc = n2/n1, where n1 is the refractive index of air (approximately 1) and n2 is the refractive index of the fiber core. In this case, n2 = 1.42. Plugging in these values, we get sinθc = 1.42/1 = 1.42. However, the maximum value for sinθc is 1, so θc = sin^-1(1) = 90°. This means that the largest possible angle θin is 90°, which is the critical angle for total internal reflection. Any angle larger than 90° would result in the light passing through the fiber core and not experiencing total internal reflection.