Laser Attenuation/Propagation Constants

[illidari]

Homework Statement

Extending the Taylor approximation to higher-order terms in high-loss materials. Consider a lossy medium having a finite conductivity a but, for purposes of this problem, no laser susceptibility Xt. First-order expressions for the propagation coefficient ft and the attenuation coefficient a for this case are derived in the text by a Taylor-series approximation in sigma/Wе. Extend this approximation to get the next higher-order corrections to both B and a. How large will the power attenuation have to become (in the first-order approximation) before either of these higher-order corrections amounts to 10% of the first-order expressions? Express your answer in units of dB of power attenuation per wavelength of distance
traveled.

Homework Equations

http://books.google.com/books?id=1BZVwUZLTkAC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false

Pg 276, google gives a huge preview of the book

The Attempt at a Solution

I'm not understanding the question. I don't see the derivation they claim to have made in the book. Is the question asking for a taylor expansion where the a = sigma/еW?

I only see them talking about a taylor expansion of r = jb (1-jsigma/We + Xat) about x =0. However this question said Xat = 0. Wouldn't this make all derivatives taken become 0? Therefore no way to expand?

Anyone familiar with Siegmans laser book and can help me out.

 

[mark726]

Hello,

I am a scientist and I would like to help you with your question. Based on my understanding, the question is asking you to extend the Taylor approximation to higher-order terms for the propagation coefficient and attenuation coefficient in a lossy medium with finite conductivity. The first-order expressions for these coefficients can be derived using a Taylor-series approximation in sigma/Wе, as described in the book you mentioned.

To answer the question, you will need to first understand the Taylor-series approximation for the propagation and attenuation coefficients. This can be found in the book on page 276, as you mentioned. The first-order expressions are derived by expanding r = jb (1-jsigma/We + Xat) about x = 0. However, in this case, Xat is equal to 0, which means that all derivatives taken will become 0. This is a valid concern and it may seem like there is no way to expand further.

However, the question is asking you to extend the approximation to the next higher-order terms. This means that you will need to consider the second-order terms in the Taylor expansion, which will give you a non-zero value for Xat. This will allow you to expand further and derive the next higher-order corrections to both B and a.

To find out how large the power attenuation needs to be in the first-order approximation before the higher-order corrections become significant (i.e. 10% of the first-order expressions), you will need to solve for the value of sigma/Wе that results in a 10% change in the coefficients. This value can then be converted to units of dB of power attenuation per wavelength of distance traveled, as requested in the question.

I hope this helps clarify the question for you. If you have any further questions, please let me know. Good luck with your calculation!