MOOC on Lasers - Lecture Notes

Recap of Previous Lecture

Line Broadening Mechanisms:
- Homogeneous Broadening: Response of atoms centered around a resonance frequency.
- Inhomogeneous Broadening: Different atomic groups centered around different frequencies.
Lifetime Broadening: Discussed in the last lecture, derived expression is Lorentzian.
Expression for Lifetime Broadening:
- Line shape function:
  [ g(\nu) = \frac{4\tau_L}{1 + (4\pi\tau_L(\nu - \nu_0))^2} ]
- Full Width at Half Maximum (FWHM):
  [ \Delta\nu = \frac{1}{2\pi\tau_L} ]

Introduction: Further discussion on line broadening mechanisms, focusing on Doppler broadening.
Doppler Effect:
- Apparent frequency seen by moving atoms:
  [ \nu' = \nu(1 - \frac{V_z}{c}) ]
- Velocity components affect frequency perception:
  - Atoms moving towards radiation experience higher frequencies.
  - Atoms moving perpendicular experience no frequency change.

Maxwellian Distribution: Probability of atomic velocities given by:
[ \rho(v) \propto e^{-\frac{mv^2}{2k_BT}} ]
- Distribution centered around zero velocity.
Interaction with Radiation: Atoms interact with a range of frequencies, both less than and greater than (\nu_0).

Line Shape Function:
[ g(\nu) = \int g(\nu, V_z) dV_z ]
- Gaussian distribution derived for Doppler broadening.
- Normalization condition:
  [ \int g(\nu) d\nu = 1 ]
Peak Value of Line Shape Function:
- At (\nu = \nu_0),
  [ g(\nu_0) = \sqrt{\frac{b}{\pi}} ]
- Full Width at Half Maximum:
  [ \Delta\nu = 2 \sqrt{\frac{\ln 2}{b}} ]
Conclusion:
- Inverse relationship between line width and peak value of the line shape function.

Graphical Representation:
- Gaussian: Rapidly decreases to 0.
- Lorentzian: Higher value at peak, decreases more slowly.

Importance of Line Shape Function: Determines the gain coefficient (\gamma(\nu)).
- (\gamma(\nu)) is proportional to the line shape function.
Next Topic: Achieving population inversion as a necessary condition for amplification by stimulated emission.