How to Tune a Helium-Neon Laser's Frequency Over Its Gain Bandwidth?

[helpcometk]

Homework Statement

A helium-neon (He-Ne) laser has a gain bandwidth (denoted here as the frequency interval
over which the laser gain equals or exceeds the minimum threshold gain) given by ΔG= 1.0 GHz, centred on the λ = 632.8 nm emission wavelength.

Show that in order to tune the frequency of this He-Ne laser over its entire gain
bandwidth, the length of the resonator should be changed by ΔL = -1/2λ.

Homework Equations

L=mλ/2

The Attempt at a Solution

The length and frequency changes are both small compared to the length and fre-
quency, respectively so differentiation of the condition for standing waves in a
resonator must be applied.

The condition for standing waves is :L=mλ/2

but dL/dλ= m/2 , how can i get ΔL = -1/2λ ?

 

[Cthugha]

The condition for standing waves is :L=mλ/2

You want the fundamental mode, so m=1. The calculation is pretty straightforward. Calculate the longest and shortest wavelength still within the gain bandwidth and calculate the corrsponding resonator lengths needed to get a standing wave for these two wavelengths.

 

[helpcometk]

thanks for the reply but i can't understand what it means:
longest and shortest wavelength still within the gain bandwidth ,how can you see if wavelength is within the bandwidth?

 

[Cthugha]

how can you see if wavelength is within the bandwidth?

The problem statement says you have a bandwidth of 1 GHz around the central wavelength of 632.8 nm, so your bandwidth goes from 632.8nm-0.5 GHz to 632.8nm+0.5 GHz. The conversion between wavelength and frequency is pretty much the only math involved here. You should be able to do that yourself.

 

[helpcometk]

this doesn't make sense because 0.5 Ghz corresponds to 0.6 m wavelength, so 632.8*10^-9 -0.6 gives negative wavelength which cannot be acceptable

 

[Cthugha]

Ok...let me start at the very beginning.

Just do me the favour and calculate the wavelengths corresponding to 0nm +0.5 Ghz (as you did), 100nm+0.5 GHz, 632nm+0.5 GHz and 2000 nm+0.5 GHz by converting wavelength to frequency FIRST and then adding the 0.5 GHz.

And while you are at it please just plot a graph of wavelength vs. corresponding frequency and have a look at whether it is linear or not. Does that help you understand why one cannot just convert 0.5 GHz into a wavelength and subtract it, but have to convert the desired wavelength into a frequency before and then add or subtract the bandwidth from that converted value?