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

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

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

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

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

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

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

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

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

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

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

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

References Allen, Leslie, and Joseph H. Eberly. "Optical Nutation." Optical Resonance and Two-Level Atoms. New York: John Wiley, 1975. 67-69. Print. "nutation, n." OED Online. Oxford University Press, September 2015. Web. 13 October 2015. "Praezession" by User Herbye (German Wikipedia). Designed by Dr. H. Sulzer - Original. Licensed under CC BY-SA 3.0 via Commons - https://commons.wikimedia.org/wiki/File:Praezession.svg#/media/File:Praezession.svg Shoemaker, R. L. "Coherent Transient Infrared Spectroscopy." Laser and Coherence Spectroscopy. Ed. Jeffrey I. Steinfeld. New York: Plenum, 1978. 197-369. 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 62.23 (2000): 443-51. APS Physics. Web. 7 Oct. 2015. Special thanks to Randy Babbitt for providing Bloch Vector notes and for answering my questions. Spectrum Lab Montana State University