Light in an optical fiber - variational calculus

In summary, I'm having trouble setting this one up. If I let the functional beF(x,x',y) = n(y)\sqrt{1+(x')^2}Applying the LE equation I obtain:\frac{d}{dy}\frac{x'}{\sqrt{1+(x')^2}}n_0(1-\frac{y^2}{a^2})=0And this is where I stopped (I haven't used either of the hints). Any hints?Any ideas please? If p/a is small then I could write n(y)=n_0; but then I'm minimizing path length... so the answer is a straight line. Obviously
  • #1
physman55
13
0

Homework Statement



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Homework Equations



[tex]\frac{\delta{F}}{\delta{y}} - \frac{d}{dx}\frac{\delta{F}}{\delta{y'}} = 0[/tex]

The Attempt at a Solution



I'm having trouble setting this one up. If I let the functional be

[tex]F(x,x',y) = n(y)\sqrt{1+(x')^2}[/tex]

Applying the LE equation I obtain:

[tex]\frac{d}{dy}\frac{x'}{\sqrt{1+(x')^2}}n_0(1-\frac{y^2}{a^2})=0[/tex]

And this is where I stopped (I haven't used either of the hints). Any hints?
 
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  • #2
Any ideas please? If p/a is small then I could write n(y)=n_0; but then I'm minimizing path length... so the answer is a straight line. Obviously this problem couldn't be that easy.
 
  • #3
it tells you y should be the principal variable so your functional should read

[tex]F(y,y')[/tex]= [tex]n(y)[/tex][tex]\sqrt{1+y'^{2}}[/tex] and then calculate the EL equations

by my calculations you should get

[tex]\frac{\partial}{\partial x}[/tex][tex](\frac{y'}{\sqrt{1+y'^{2}}})[/tex]= [tex]-2\frac{y}{a^{2}}\sqrt{1+y'^{2}}[/tex]

disregarding the (y/a)^2 term
 
Last edited:
  • #4
By "y" being the principal value my prof means that "y" is the independent variable.

For the last line, why did you ignore n(y) when you took the partial with respect to y' (on the left)?
 
  • #5
yeah Y is the degree of freedom and the Lagrangian are always expressed in term of the degrees of freedom L(q,q') and this makes more sense since you are asked to find y(x)

I didn't ignore the n(y) its just that y is comparible to [tex]\rho[/tex] and since [tex]\frac{\rho}{a}[/tex] is small n(y) = n0
 
  • #6
Ok thanks for the explanation, that makes sense. But shouldn't the d/dx be a total derivative and not a partial; and secondly; how the hell do you solve that ODE?
 
  • #7
i'm not sure about the entire thing but since dy is comparable to rho you can deduce that y' is approximate to alpha and thus all the y'^2 terms are negligible and you get the resulting ODE

[tex]y''=\frac{-2y}{a^{2}}[/tex]

which has solution

[tex]Acos(\frac{\sqrt{2}x}{a}) +Bsin(\frac{\sqrt{2}x}{a})[/tex] which kind of makes sense since you expect it to bounce off the walls in a periodic fashion
 

What is an optical fiber?

An optical fiber is a thin, flexible strand of glass or plastic that is used to transmit light signals over long distances. It works by using the principle of total internal reflection to keep the light contained within the fiber.

How does light travel through an optical fiber?

Light travels through an optical fiber by bouncing off the walls of the fiber at a specific angle. This angle, known as the critical angle, ensures that the light remains within the core of the fiber and does not escape.

What is variational calculus and how is it related to light in an optical fiber?

Variational calculus is a mathematical method used to find the optimal path or shape for a given system. In the case of light in an optical fiber, variational calculus is used to determine the path that the light takes through the fiber in order to minimize the time it takes to travel from one end to the other.

What are the main advantages of using optical fibers for communication?

There are several advantages to using optical fibers for communication. These include high bandwidth, low signal loss, and immunity to electromagnetic interference. Additionally, optical fibers are lightweight, flexible, and inexpensive to manufacture.

How is the quality of light in an optical fiber measured?

The quality of light in an optical fiber is measured by several factors, including the amount of light that is lost over a given distance, the signal-to-noise ratio, and the bandwidth of the fiber. Other factors such as dispersion and attenuation also play a role in determining the overall quality of the light signal.

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