Engineering Optical Fibres: Wavelength for Zero net Dispersion

AI Thread Summary
The discussion focuses on calculating the wavelength for zero net dispersion in a step-index multimode optical fiber with specific refractive indices. The material dispersion coefficient D is defined by a quadratic equation, and the initial approach involved setting D to zero to find the roots. Participants noted the necessity of having negative coefficients for certain terms to ensure positive wavelength solutions. After correcting signs in the equation, the roots were simplified, revealing a positive wavelength of 100 nm and a negative root of -50 nm, indicating the need for proper arithmetic checks. The conversation emphasizes the importance of accurate coefficient signs in dispersion calculations.
Master1022
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Homework Statement
An optical fibre transmission system uses a step-index multimode optical fibre which has a core refractive index of 1.49 and a cladding refractive index of 1.48. The fibre is also subject to material dispersion which is a function of wavelength ## \lambda ##. The material dispersion coefficient D is given by:
[tex] D = a \lambda^2 + b\lambda + C \text{ps/km} [/tex]
where ## a = 0.01 ##, ##b = 0.50 ## and ## c = 50## and ## \lambda ## is the wavelength in nanometres. Estimate the wavelength at which the fibre has zero net dispersion.
Relevant Equations
D = 0
Hi,

I was working on the problem below:

Question:
An optical fibre transmission system uses a step-index multimode optical fibre which has a core refractive index of 1.49 and a cladding refractive index of 1.48. The fibre is also subject to material dispersion which is a function of wavelength ## \lambda ##. The material dispersion coefficient D is given by:
D = a \lambda^2 + b\lambda + C \text{ps/km}
where ## a = 0.01 ##, ##b = 0.50 ## and ## c = 50## and ##\lambda ## is the wavelength in nanometres. Estimate the wavelength at which the fibre has zero net dispersion.

Attempt:
Do we just let D = 0 and solve the quadratic equation? Is is that simple...

I was slightly confused as I know ## D = -\frac{\lambda}{c} \frac{d^2 n}{d \lambda^2} ## and perhaps there was a trick that we needed to use this.

Nonetheless, the first method yields the answers 4999 nm and 1 nm. Not sure how to choose between them, but I think the higher value might be correct as it is closer to the order of magnitude that we saw in a graph in the lectures (order of microns).

Any help or guidance would be appreciated
 
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You need at least one of the coefficients of the dispersion function ## D ## to be negative if you are going to have a positive root ## \lambda ##.
 
Charles Link said:
You need at least one of the coefficients of the dispersion function ## D ## to be negative if you are going to have a positive root ## \lambda ##.
Ah yes, you are right! Sorry, I think I forgot the -ve sign on ## b ##.
 
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Master1022 said:
Ah yes, you are right! Sorry, I think I forgot the -ve sign on ## b ##.
I think "c" also might need a minus sign, or you get imaginary roots. Please check your arithmetic=putting in minus signs on "b" and "c", I get roots of ## \lambda=+100 ## nm, and ## \lambda=-50 ## nm.

This one doesn't even need the quadratic formula=it factors: multiplying by 100 we get ## (\lambda-100) (\lambda+50)=0 ##.
 
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