1. Aislinn Daniels Spectrum Lab Seminar Fall 2015
Optical Nutation with Gaussian Beam in an Inhomogeneous Broadened Medium and the Effects from Coherence Time
Aislinn Daniels
Spectrum Lab Seminar Fall 2015
Spectrum Lab Montana State University

2. Summary What is Nutation? Brief Review of Bloch Vectors
Theory Behind Nutation
Nutation of:
Homogeneous Media, “Top Hat” Beam
Inhomogeneous Media, “Top Hat” Beam
Inhomogeneous Media, Gaussian Beam
Effects from Coherence Time
Usefulness of Nutation (Examples)
References
Spectrum Lab Montana State University

3. What is Nutation? According to Oxford English Dictionary:
2b. “Rotation of an axis (of a radar beam, aerial, etc.) so as to describe a cone”
Also:
1a. “The action of nodding the head, esp. as a sign of drowsiness”
Spectrum Lab Montana State University

4. What is Nutation? According to Oxford English Dictionary:
2b. “Rotation of an axis (of a radar beam, aerial, etc.) so as to describe a cone”
Also:
1a. “The action of nodding the head, esp. as a sign of drowsiness”
According to R.L. Shoemaker:
Nutation refers to the “alternating absorption and emission of radiation” when a sample is incident with a resonant optical field
This nutation is readily described using Bloch vectors
Motion of vector can be described as oscillation superimposed on precession, similar to spinning top
Figure 1: Precession and Nutation of a Planet
Spectrum Lab Montana State University

5. Brief Review of Bloch Vectors
Energy Levels, 2 Level System
2, E2
Absorption
Emission
 𝑅 =2  0
 0 =𝑒 𝑧 21 𝐸 0 / 2ħ
In Atom frame (or Lab frame) Rabi vector rotates if detuned
This motion gives nutation
But, to simplify, use laser frame
1, E1
Figure 2: Two-Level System
Spectrum Lab Montana State University

6. Brief Review of Bloch Vectors (cont.)
 𝐺 = ∆ 2 +  𝑅 2
∆=𝜔− 𝜔 21
Frequency of Nutation is magnitude of Generalized Rabi frequency
Spectrum Lab Montana State University

7. Nutation in Homogeneous Media, “Top Hat” Beam
One intensity (E-field)
Material homogeneous, so all atoms have same frequency
Therefore 1 Rabi frequency and 1 detuning
Process:
Material will absorb radiation, dropping intensity
Bloch vector will rotate about Rabi frequency
Eventually get emission (vector begins to swing back down)
Material capable to absorb radiation again
Signal will oscillate sinusoidally
Spectrum Lab Montana State University

8. Nutation in Inhomogeneous Media, “Top Hat” Beam
Now, because media inhomogeneous, atoms have “spread” of frequencies, which varies delta, and thus gives “spread” of generalized Rabi frequencies
So, absorption and emission cycle at different frequencies
Then total intensity (power) signal I is the summation of all of these cycles, which gives
𝐼(𝑡)∝ 𝐼 0 1−𝐾  𝑅 2 𝐽 0 (  𝑅 𝑡)
(technically intensity calculated from summation of polarizations)
Spectrum Lab Montana State University

9. Nutation in Inhomogeneous Media, Gaussian Beam
Now with Gaussian beam, must include variation in beam’s E-field, and thus Rabi frequency
If this field varies slowly over radial distance, so over a few wavelengths of light the amplitude is about constant, beam can be approximated as summation of “rings” of top hat beams
Thus solution from previous derivation can be integrated over radius, with proper modulation of intensity from Gaussian function
Spectrum Lab Montana State University

10. Nutation in Inhomogeneous Media, Gaussian Beam (cont.)
Resulting intensity
𝐼(𝑡)∝ 𝐼 0 (1−𝑀 𝐽 1  𝑅 𝑡  𝑅 𝑡 )
Note this function (in general) is lower in magnitude and “dampens” more quickly
In a crystal with different crystallographic sites (and thus may have different frequencies  𝑅,𝑖 ), solutions at these sites would be added together, usually resulting in irregular shape
Spectrum Lab Montana State University

11. Effects from Coherence Time
These systems will “relax” in phase overtime
Represented by multiplying sinusoidal or Bessel function by exponential decay
𝑒 −𝑡 𝑇 2
T2 coherence time
 𝑅 >1/T2 to see nutation, otherwise decay will dampen before initial peak
T2 can be decreased though many methods
Material properties
Temperature
Etc.
Spectrum Lab Montana State University

12. Usefulness of Nutation (Examples)
Calculating  𝑅,𝑚𝑎𝑥 , maximum Rabi frequency; in Gaussian beam, related to E-field at the center of the beam
Finding beam focus; initial rise of Bessel function will be steepest when beam is focused in material
Spectrum Lab Montana State University

13. References
Allen, Leslie, and Joseph H. Eberly. "Optical Nutation." Optical Resonance and Two-Level Atoms. New York: John Wiley, Print. "nutation, n." OED Online. Oxford University Press, September Web. 13 October "Praezession" by User Herbye (German Wikipedia). Designed by Dr. H. Sulzer - Original. Licensed under CC BY-SA 3.0 via Commons - Shoemaker, R. L. "Coherent Transient Infrared Spectroscopy." Laser and Coherence Spectroscopy. Ed. Jeffrey I. Steinfeld. New York: Plenum, Print. Sun, Y., G. M. Wang, R. L. Cone, R. W. Equall, and M. J. M. Leask. "Symmetry Considerations Regarding Light Propagation and Light Polarization for Coherent Interactions with Ions in Crystals." Physical Review B (2000): APS Physics. Web. 7 Oct Special thanks to Randy Babbitt for providing Bloch Vector notes and for answering my questions.
Spectrum Lab Montana State University