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|Rene Reimann, [email protected] www.nano-optics.org
Cavity Optomechanics
- Interaction of Light with Mechanical StructuresNano-Optics Lecture
2017-12-08 1
Image: PhD thesis of Albert Schließer, LMU München
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 2
Cavity Optomechanics
From M. Aspelmeyer et al., Physics Today 65, 29-35 (2012)
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 4
Cavity Basics
Fabry-Perot Cavity – Input-Output Formalism
▪ 𝐸in = 𝐸0 exp[𝑖 𝜔𝑡 − 𝑘𝑧 ], for simplicity we set 𝐸0 =1
▪ 𝑟 2 + 𝑡 2 + 𝑎 2 = 𝑅 + 𝑇 + 𝐴 = 1, for simplicity we keep 𝑎 = 0
▪ Solve for 𝐸𝑥 as a function of 𝐸in
▪ Write field as 𝐸𝑥 = 𝑢𝑥 exp 𝑖𝜙𝑥 × 𝐸in
▪ Intensity 𝐼𝑥 = 𝐸𝑥2 = 𝑢𝑥
2 = 𝑇𝑥, as 𝐼in = 1
𝐸in
𝐸ref
𝐸1
𝐸2
𝐸out
𝑟, 𝑡 𝑟, 𝑡
𝐸1 = 𝑖𝑡𝐸in + 𝑟𝐸2
𝐿
𝐸2 = exp 𝑖2𝑘𝐿 𝑟𝐸1𝐸ref = 𝑖𝑡𝐸2 + 𝑟𝐸in𝐸out = exp 𝑖𝑘𝐿 𝑖𝑡𝐸1
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 5
Cavity Basics
Fabry-Perot Cavity – Output Field
𝐸in
𝐸ref
𝐸1
𝐸2
𝐸out
𝑅, 𝑇 𝑅, 𝑇
→ 𝑇out=𝑇2
1 + 𝑅2 − 2𝑅 cos 2𝑘𝐿
𝐿
𝐸1 = 𝑖𝑡𝐸in + 𝑟𝐸2𝐸2 = exp 𝑖2𝑘𝐿 𝑟𝐸1𝐸ref = 𝑖𝑡𝐸2 + 𝑟𝐸in𝐸out = exp 𝑖𝑘𝐿 𝑖𝑡𝐸1
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 6
Cavity Basics
Fabry-Perot Cavity – Output Field
𝑇out =𝑇2
1 + 𝑅2 − 2𝑅 cos 2𝜔𝑐 𝐿
𝜔FSR = 𝜋𝑐/𝐿
𝜆/2
in real space
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 7
Cavity Basics
Fabry-Perot Cavity – Cavity Language
𝑇out =𝑇2
1 + 𝑅2 − 2𝑅 cos 2𝜔𝑐 𝐿
with cos 𝑥 ≈ 1 − 𝑥2/2 and 𝑇 = 1 − 𝑅one finds for 𝑅 ≈ 1 and close to a resonance (e.g 𝜔 ≈ 0)
𝑇out ≈ Lorentzian =𝛾02
𝛾02 + 𝜔2
𝛾0 =(1 − 𝑅)
2
𝑐
𝐿where
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 8
Cavity Basics
Fabry-Perot Cavity – Cavity Language
2𝛾0 = 𝜔FWHM
𝑇out ≈ Lorentzian =𝛾02
𝛾02 + 𝜔2
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 9
Cavity Basics
Fabry-Perot Cavity – Cavity Language
2𝛾0
𝜔FSR
Define cavity finesse 𝐹 =𝜔FSR
2𝛾0
𝛾0 =1 − 𝑅
2
𝑐
𝐿𝜔FSR = 𝜋𝑐/𝐿
=𝜋
1 − 𝑅𝑅 𝐹 = 105 = 0.99997
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 10
Equations of Motion
Cavity:
Oscillator:
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 12
Effective Equation of Motion + Solutions
Detailed treatment: PhD thesis of Albert Schließer, LMU München
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 13
Effective Temperature
With Wiener Khintchine theorem:
With Equipartition theorem:
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 15
Sideband Cooling
ω
Ωmech Ωmech
𝜔𝐿 𝜔cav
𝑚𝑚 − 1
𝑚 + 1
Ground-state cooling
works in the resolved
sideband regime:
2𝛾0 ≪ Ω0
2𝛾0
|Rene Reimann, [email protected] www.nano-optics.org
Optomechanical Systems in the Novotny group
2017-12-08 17
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 18
An Optical Tweezer for a Dielectric Particle
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 20
Cavity Optomechanics
𝑚𝑚 − 1
𝑚 + 1𝑉cav
Ωmech
𝐿
𝛾0 =𝜋𝑐
2𝐹𝐿
Sensitivity
Sensitivity to particle motion 𝑆 = 𝑔/(2𝛾0).With 𝑔 ∝ 1/𝑉cav and 𝑉cav ∝ 𝐿2 one finds 𝑺 ∝ 𝑭/𝑳.
Bandwidth
For 𝐿 = 0.5 mm and 𝐹 = 300 × 103 one finds 2𝛾0 = 2𝜋 × 1 MHz.2𝛾0 > Ωmech guarantees a fast information retrieval rate.
2𝛾0
|Rene Reimann, [email protected] www.nano-optics.org 2017-12-08 21
References
▪ Recent Review: Aspelmeyer, M., Kippenberg, T. J., & Marquardt, F. (2014). Cavity
optomechanics. Reviews of Modern Physics, 86(4), 1391–1452.
http://doi.org/10.1103/RevModPhys.86.1391
▪ In the language of the course: Chapter 11.4: Novotny, L., & Hecht, B. (2006). Principles of Nano-
Optics. Cambridge University Press.
▪ Reflection from vibrating mirror: Van Bladel, J., & De Zutter, D. (1981). Reflections from linearly
vibrating objects: Plane mirror at normal incidence. IEEE Transactions on Antennas and
Propagation, 29(4), 629–637. http://doi.org/10.1109/TAP.1981.1142645
▪ 𝛿𝑇 and 𝛿Γ equations: Schließer, A. (2009). Cavity optomechanics and optical frequency comb
generation with silica whispering-gallery-mode microresonators. Thesis LMU München.
http://edoc.ub.uni-muenchen.de/10940/1/Schliesser_Albert.pdf
▪ For fluctuating force: Kubo, R. (1966). The fluctuation-dissipation theorem. Reports on Progress
in Physics, 29(1), 306. http://doi.org/10.1088/0034-4885/29/1/306
▪ Very nice lectures (lectures 18 to 21) by Florian Marquardt alvailable as videos on
http://theorie2.physik.uni-erlangen.de/index.php/Lecture_Quantum-
optical_phenomena_in_nanophysics#Videos
