Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

I am following my university course notes on amplification in laser media, and have come across expressions for the gain of a medium, but the notes are not exactly rigorous... The expression given for the gain $G$ in units of dB m$^{-1}$ is:

$$G = 10 \log_{10}(e^\gamma) \mathrm{dB} \mathrm{m}^{-1}$$ where $\gamma$ is also the gain coefficient for the media with units m$^{-1}$. However later in the notes it gives the expression $$\gamma = \Delta N \sigma$$ where $\Delta N$ is the "number of atoms in population inversion" and $\sigma = [\mathrm{m}]^{-2}$ is the "cross-section for stimulated emission". This clearly shows that $\Delta N = [\mathrm{m}]$, which doesn't make much sense to me. Also the first expression implies that $\gamma$ is unitless, so I am mighty confused.

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Bonus points for checking your dimensions. Your forumlas omit a lot of constants and assumes no atoms in the ground state, but that does not really matter for getting the units right. For that you need to realize that:

  1. The unit for cross section is meter squared [m]x[m], not the reciprocal.
  2. When you take e to the power of something, the "something" has to be dimensionless. Your formula for the gain omits the length traversed by the light. It should be (coefficient x length) which is unitless.
  3. Finally you have forgotten the volume in the number of atoms. It should be number of atoms per volume (otherwise a twice as big laser would have a twice as high gain coefficient).

If you do all that the dimensions add up.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.