How Is Path Length Calculated for Light in an Optical Fiber?

[roam]

Homework Statement

A glass optical fibre of length L = 3.2 m is in a medium of glycerine with a refractive index n0 = 1.47 . The fibre has a core of refractive index, n₁ = 1.58 and diameter, d = 100μm surrounded by a thin cladding of refractive index, n₂ = 1.53. The end of the fibre is cut square (see diagram).

[PLAIN]http://img69.imageshack.us/img69/4976/imagenz.gif

For the aforementioned fibre, what is the total path length of a ray within the fibre that enters the fibre at the acceptance angle?

The Attempt at a Solution

Here are the stuff that I have already calculated:

* The minimum angle, θ₂, for total internal reflection at the core/cladding boundary: 75.5 degrees.

* The angle, θ₁, to the axis of the fibre that corresponds to the minimum angle, θ₂: 14.5 degrees.

* The acceptance angle θ₀: 15.6 degrees.

So, now to find the path length of a ray within the fibre that enters the fibre at the acceptance angle I tried using trigonometry:

Since d=100.0 μm = 100 x 10⁶ m. We have d/2 = 50 x 10⁶.

$$sin (14.5) = \frac{50 \times 10^6}{x}$$

x= a very huge number!

But the answer must be 3.305 m! What did I do wrong?

 

[ehild]

1 μm is not 10⁶ m.

ehild

 

[roam]

1 μm is not 10⁶ m.

ehild

Oh, I meant 10^-6. But why is it that I end up with 1.9 x 10^-4? This is not the right answer...

 

[ehild]

Oh, I meant 10^-6. But why is it that I end up with 1.9 x 10^-4? This is not the right answer...

There are a lot of reflections in that 3.2 m long fibre. The light goes through the whole length of the fibre, but trawels along a zigzag path which is longer than the fibre length. ehild

 

[rwisz]

I would suggest double checking your calculations and equations to ensure they are correct. It is possible that you may have made a mistake in your calculations or used the wrong equation. Additionally, it may be helpful to draw a diagram to visualize the path of the light ray and use the law of reflection and Snell's law to accurately determine the path length. It may also be helpful to consult with a colleague or a textbook to confirm your approach and calculations.