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

• helpcometk
In summary, the condition for standing waves is that the wavelength must be within the gain bandwidth, which is 1 GHz around the central wavelength of 632.8 nm. However, to tune the frequency of the He-Ne laser over its entire gain bandwidth, the length of the resonator must be changed by ΔL = -1/2λ.
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.

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 ?

helpcometk said:
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.

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?

helpcometk said:
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.

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

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?

Last edited:

## 1. What is the gain bandwidth of a Helium-Neon laser?

The gain bandwidth of a Helium-Neon laser typically ranges from 1-3 MHz.

## 2. How does the gain bandwidth affect the performance of a Helium-Neon laser?

The gain bandwidth determines the range of frequencies that the laser can emit, which directly affects its power and coherence.

## 3. How can the gain bandwidth of a Helium-Neon laser be increased?

The gain bandwidth can be increased by using a longer resonator or by incorporating a gas mixture of Helium and Neon with a higher pressure.

## 4. What is the importance of the gain bandwidth in laser applications?

The gain bandwidth is crucial in determining the spectral purity and stability of the laser output, making it a key factor in various scientific and industrial applications.

## 5. Are there any limitations to the gain bandwidth of a Helium-Neon laser?

Yes, the gain bandwidth is limited by the gain medium properties, such as the energy level structure and relaxation rates of the Helium-Neon gas mixture.

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