Solving Fiber Optics Glass Characteristic Equations with Maple

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Homework Help Overview

The discussion revolves around solving the characteristic equations related to fiber optics glass, specifically focusing on the propagation angles in the context of optical physics. The original poster presents two equations that need to be solved separately using Maple, which involve trigonometric and inverse trigonometric functions.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the similarity of the left-hand expressions in the equations and question how they can equate to both Pi and 0. There is also a suggestion to improve the presentation of the equations for clarity.
  • The original poster clarifies that the equations represent different modes of propagation and that they need to solve for each angle individually.
  • One participant proposes a substitution involving trigonometric identities to potentially simplify the problem.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the equations and suggesting methods to approach the problem. There is no explicit consensus yet, but some guidance has been offered regarding potential substitutions and clarifications on the nature of the equations.

Contextual Notes

The original poster notes that they are working within the constraints of using Maple and a TI calculator, and they mention specific values for constants involved in the equations. There is an emphasis on the need to solve for angles corresponding to different modes of propagation.

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Homework Statement


They ask me to solve the characteristic equation of a fiber optics glass. The equations I need to solve (separately) are:


Homework Equations



solve(2000000*Pi*sin(x)-2*arctan(.6666666667*(2.25*cos(x)^2-2.1904)^.5/sin(x)) = Pi, x):

solve(2000000*Pi*sin(x)-2*arctan(.6666666667*(2.25*cos(x)^2-2.1904)^.5/sin(x)) = 0, x)

That's how I tried to solve them in MAPLE.



The Attempt at a Solution



Maple is not giving me answers, I used my TI and it gave me some results but they were extremely similar and I don't think they are the real ones. How can I solve for the propagation angle x?
 
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The expressions on the left look the same. How can the same expression be both Pi and 0 at the same time?

And please try using tex or at least 2*10^6 and 2/3 instead of 2000000 and .6666666667
More people may decide to help if you put effort in presenting your problem neatly.
 
Well I'm Maple I used fractions, when I copy pasted them here they appeared as decimals. The equations are supposed to be different modes or propagation, so I need to solve for each angle individually.

The equation (general) is:

m*PI=2*PI*d/lambda -2 arctan(SQRT((n12Cos(x))^2 -n2^2)/n1*sin(x))

Where N1=1.5
n2=1.48
lambda=1(10^-6)
d=(3.192(10^-6)
And m=0,1,2,3... Where I only need the first (0 and 1)
 
If you define
[tex]\alpha^2 =1-(n_2/n_1)^2[/tex]
and let
[tex]y=sin(x)/\alpha[/tex], then I think you can write the arctan as:
[tex]arctan\left(\frac{\sqrt{1-y^2}}{y}\right)[/tex]

Try some trig identities from there.
 

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