Wavelength of a laser within an optical cavity

AI Thread Summary
The discussion centers on calculating the wavelength of a laser within an optical cavity, focusing on the relationship between cavity length, mode number, and spectral range. The user initially miscalculated the cavity length, assuming it to be 0.5m, which led to an incorrect mode number of 0.3 when applying the equations provided. After recognizing the need to account for the entire path length at index n, the user corrected the cavity length to 0.9m. This adjustment aims to verify the expected wavelength of 600nm using the formula λ=2×Lc/N. The conversation highlights the importance of accurate cavity length determination in laser mode calculations.
Taylor_1989
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1. The problem statement, all variables and given/known date
optic1.png
optic2.png

Homework Equations


$$\delta v=\frac{c}{2nL} \:[1]$$
$$N=\frac{\Delta v}{\delta v}=\frac{2nL\Delta v}{c} \:[2]$$

The Attempt at a Solution


I am having trouble with question 5, but have come to realize I think my cavity length is wrong but I can't see how.

Here my working for question 1

Assuming that the medium is in the middle at equal distance from each mirror, then I could assume that ##L=L_{m}+2x## so subsituiting this into equation 1.

$$\delta v=\frac{c}{2nL}=\frac{c}{2n(L_{m}+2x)}$$

by rearranging the equation for x

$$x=\left(\frac{c}{\delta \:v}\cdot \:\frac{1}{2n}-Lm\right)\cdot \frac{1}{2}$$

so subbing ##x=0.1## I make the cavity length ##L=L_{m}+2(0.1)=0.5m##.

Which seem to reasonable to me. But if I plug this length of the cavity into [2] and using the spectral range of ##3\times 10^{7}## I make the number of modes 0.3 which can't be correct.

So the reason I want to workout the number of modes is so I can verify the ##600nm## via the equation ##\lambda=2\times L_{c}/N##, which dose not give the 600nm so either my cavity is length is wrong or my understanding is wrong.

Any advice would be much appreciated.
 

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Taylor_1989 said:
δv=c2nL=c2n(Lm+2x)δv=c2nL=c2n(Lm+2x)​
\delta v=\frac{c}{2nL}=\frac{c}{2n(L_{m}+2x)}
Only Lm is at index n. Here you have the whole path at index n
 
Cutter Ketch said:
Only Lm is at index n. Here you have the whole path at index n

Yes I realized this after a while, and then re corrected my ans for the path length being 0.9m
 
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