Theory - Saturated Absorption Spectroscopy

As discussed last module, one of the most powerful applications of the laser was absorption spectroscopy. This allowed us to prove and understand te structure of various atoms and molecules. This week, we will look at Saturated absorption spectroscopy, which runs on the same principles of absorption spectroscopy, but fixes a few issues.

Doppler Broadening

The main issue we run into is due to Doppler broadening. When our atoms are in the vapor cell, they are not completely frozen in space. The atoms might be moving with some velocity, $v$ . In this case, if the atom interacts with any sort of radiation (like a laser), in our la frame we see that the light has angular frequency $\omega.$ However, in the atom’s rest frame, it will see the light moving with $\omega \pm kv$ , where $k$ is the magnitude of the radiation’s wavevector. The sign depends on the direction that the radiation is coming from.

What effect does this have on our atoms? For a full mathematical derivation, refer to Foot’s Atomic Physics book. Because every atom in our sample will be moving at a different velocity, the way each atom interacts with the incoming radiation will vary. What we will find is that the Doppler effect will lead to a broadening of the linewidth, and thus limit our resolution.

Saturated Absorption Spectroscopy

Saturated Absorption Spectroscopy aims to remove the effects of Doppler broadening. Atoms will absorb radiation over a range given by the homogenous width of the transition. This means that the atoms do not absorb radiation exactly at the resonance frequency, but rather by a range dictated by the radiative broadening.

The setup has two beams - one probe beam and one pump beam. Both beams are derived from the same laser and thus have the same frequency $\omega$ . The two beams go in opposite directions., so the pump beam only excites the atoms that have velocity $v=(\omega - \omega_0)/k$ , putting them in the excited state. If the laser is far off resonance, the pump and probe beams interact with different atoms. However, when the laser is close to resonance, both beams interact with the atoms that have velocity very close to zero. The pump beam excites these atoms to the excited state. This leads to a signal with a peak at $\omega_0$ , effectively isolating the atoms with zero velocity and picking out just their signal.

The figures above are for two level atoms. In reality, our atomic structure is much more complex. Let’s now consider atoms with a single ground state and two excited states. This is what we should expect! We know from our study of the hyperfine structure and other corrections to the atomic energy levels that multiple excited states exist.

In this case, we have two transition frequencies at play: $\omega_{12}$ for our ground state to the first excited state, and $\omega_{13}$ for our ground state to the higher energy excited state. Since we now have multiple transitions, we should expect our signal to have multiple peaks. We will have two peaks that correspond to each transition. Then, we will have an additional peak called the cross-over resonance. This extra peak comes from atoms that move with velocities such that the pump is in resonance with one transition, and the probe is in resonance with the other.

Pre-lab Questions and Concept Checks

How do you decide whether to use saturated absorption spectroscopy or regular absorption spectroscopy?
For the case of multiple excited states, what are the corresponding velocities that the atoms at the peaks have?
What frequencies should the beams be tuned to?

References

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Doppler Broadening

The main issue we run into is due to Doppler broadening. When our atoms are in the vapor cell, they are not completely frozen in space. The atoms might be moving with some velocity,

v

. In this case, if the atom interacts with any sort of radiation (like a laser), in our la frame we see that the light has angular frequency

\omega.

However, in the atom’s rest frame, it will see the light moving with

\omega \pm kv

, where

k

is the magnitude of the radiation’s wavevector. The sign depends on the direction that the radiation is coming from.

Saturated Absorption Spectroscopy

The setup has two beams - one probe beam and one pump beam. Both beams are derived from the same laser and thus have the same frequency

\omega

. The two beams go in opposite directions., so the pump beam only excites the atoms that have velocity

v=(\omega - \omega_0)/k

, putting them in the excited state. If the laser is far off resonance, the pump and probe beams interact with different atoms. However, when the laser is close to resonance, both beams interact with the atoms that have velocity very close to zero. The pump beam excites these atoms to the excited state. This leads to a signal with a peak at

\omega_0

, effectively isolating the atoms with zero velocity and picking out just their signal.

In this case, we have two transition frequencies at play:

\omega_{12}

for our ground state to the first excited state, and

\omega_{13}

for our ground state to the higher energy excited state. Since we now have multiple transitions, we should expect our signal to have multiple peaks. We will have two peaks that correspond to each transition. Then, we will have an additional peak called the cross-over resonance. This extra peak comes from atoms that move with velocities such that the pump is in resonance with one transition, and the probe is in resonance with the other.