Editing Mode volume (section)

==In electromagnetic cavities==

The mode volume (or modal volume) of an optical or microwave cavity is a measure of how concentrated the [[Electromagnetic energy density|electromagnetic energy]] of a single [[Mode (electromagnetism)|cavity mode]] is in space, expressed as an effective volume in which most of the energy associated with an electromagentic mode is confined. Various expressions may be used to estimate this volume:<ref>{{Cite web |title=Calculating the modal volume of a cavity mode |url=https://optics.ansys.com/hc/en-us/articles/360034395374-Calculating-the-modal-volume-of-a-cavity-mode |archive-url=https://web.archive.org/web/20220817165833/https://optics.ansys.com/hc/en-us/articles/360034395374-Calculating-the-modal-volume-of-a-cavity-mode |archive-date=17 August 2022 |access-date=13 September 2024 |website=[[Ansys|Ansys Optics]]}}</ref><ref>{{Cite thesis |last=Kippenberg |first=Tobias Jan August |title=Nonlinear Optics in Ultra-High-Q Whispering-Gallery Optical Microcavities |date=2004 |degree=phd |publisher=California Institute of Technology |url=https://thesis.library.caltech.edu/2487/ |doi=10.7907/t5b6-9r14 |language=en}}</ref>

* The volume that would be occupied by the mode if its electromagnetic energy density was constant and equal to its maximum value 
<math display="block">
V_{m} = \frac{\int \epsilon |E|^{2} dV}{\rm{max}(\epsilon |E|^{2})} \;\;\; \rm{or} \;\;\;
V_{m} = \frac{\int (|B|^{2}/\mu) \; dV}{\rm{max}(|B|^{2}/\mu )}
</math>
* The volume over which the electromagnetic energy density exceeds some threshold (e.g., half the maximum energy density)
<math display="block">
V_{m} = \int \left(|E|^{2} > \frac{|E_{max}|^{2}}{2}\right) dV
</math>
* The volume that would be occupied by the mode if its electromagnetic energy density was constant and equal to a weighted average value that emphasises higher energy densities.
<math display="block">
V_{m} = \frac{(\int |E|^{2} dV)^{2}}{\int |E|^{4} dV} \;\;\; \rm{or} \;\;\;
V_{m} = \frac{(\int |B|^{2} dV)^{2}}{\int |B|^{4} dV}
</math>

where <math>E</math> is the [[Electric field|electric field strength]], <math>B</math> is the [[magnetic flux density]], <math>\epsilon</math> is the [[electric permittivity]], <math>\mu</math> denotes the [[Permeability (electromagnetism)|magnetic permeability]], and <math>\max(\cdots)</math> denotes the maximum value of its functional argument. In each definition the integral is over all space and may diverge in leaky cavities where the electromagnetic energy can radiate out to infinity and is thus not is not confined within the cavity volume.<ref>{{Cite web |last=Meldrum |first=A |title=Lesson 5: Whispering Gallery Modes |url=https://sites.ualberta.ca/~ameldrum/science/science5a.html |access-date=2024-12-19 |website=sites.ualberta.ca}}</ref> In this case modifications to the expressions above may be required to give an effective mode volume.<ref>{{Cite journal |last=Kristensen |first=P. T. |last2=Van Vlack |first2=C. |last3=Hughes |first3=S. |date=2012-05-15 |title=Generalized effective mode volume for leaky optical cavities |url=https://opg.optica.org/ol/abstract.cfm?uri=ol-37-10-1649 |journal=Optics Letters |language=en |volume=37 |issue=10 |pages=1649 |doi=10.1364/OL.37.001649 |issn=0146-9592|arxiv=1107.4601 }}</ref>

The mode volume of a cavity or resonator is of particular importance in [[cavity quantum electrodynamics]]<ref>{{Cite journal |last=Kimble |first=H. J. |date=1998 |title=Strong Interactions of Single Atoms and Photons in Cavity QED |url=https://iopscience.iop.org/article/10.1238/Physica.Topical.076a00127 |journal=Physica Scripta |language=en |volume=T76 |issue=1 |pages=127 |doi=10.1238/Physica.Topical.076a00127 |issn=0031-8949}}</ref> where it determines the magnitude<ref>{{Cite journal |last=Purcell |first=E. M. |author-link=Edward Mills Purcell |date=1946-06-01 |title=Proceedings of the American Physical Society: B10. Spontaneous Emission Probabilities at Radio Frequencies |url=https://link.aps.org/doi/10.1103/PhysRev.69.674.2 |journal=Physical Review |language=en |volume=69 |issue=11-12 |pages=674–674 |doi=10.1103/PhysRev.69.674.2 |issn=0031-899X}}</ref><ref>{{Cite journal |last=Boroditsky |first=M. |last2=Coccioli |first2=R. |last3=Yablonovitch |first3=E. |last4=Rahmat-Samii |first4=Y. |last5=Kim |first5=K.W. |date=1998-12-01 |title=Smallest possible electromagnetic mode volume in a dielectric cavity |url=https://digital-library.theiet.org/content/journals/10.1049/ip-opt_19982468 |journal=IEE Proceedings - Optoelectronics |language=en |volume=145 |issue=6 |pages=391–397 |doi=10.1049/ip-opt:19982468 |issn=1350-2433}}</ref><ref>{{Cite web |last=Chen |first=Tom |title=Calculating cavity quality factor, effective mode volume, and Purcell factor in Tidy3D Flexcompute |url=https://www.flexcompute.com/tidy3d/examples/notebooks/CavityFOM/ |access-date=2024-12-19 |website=www.flexcompute.com |language=en}}</ref> of the [[Purcell effect]] and coupling strength between cavity photons and atoms in the cavity.<ref>{{Cite journal |last=Srinivasan |first=Kartik |last2=Borselli |first2=Matthew |last3=Painter |first3=Oskar |last4=Stintz |first4=Andreas |last5=Krishna |first5=Sanjay |date=2006 |title=Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots |url=https://opg.optica.org/oe/viewmedia.cfm?uri=oe-14-3-1094&html=true |journal=Optics Express |language=en |volume=14 |issue=3 |pages=1094 |doi=10.1364/OE.14.001094 |issn=1094-4087|arxiv=physics/0511153 }}</ref><ref>{{Cite journal |last=Yoshie |first=T. |last2=Scherer |first2=A. |last3=Hendrickson |first3=J. |last4=Khitrova |first4=G. |last5=Gibbs |first5=H. M. |last6=Rupper |first6=G. |last7=Ell |first7=C. |last8=Shchekin |first8=O. B. |last9=Deppe |first9=D. G. |date= |title=Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity |url=https://www.nature.com/articles/nature03119 |journal=Nature |language=en |volume=432 |issue=7014 |pages=200–203 |doi=10.1038/nature03119 |issn=0028-0836}}</ref> In particular, the Purcell factor is given by

: <math>F_{\rm P} = \frac{3}{4\pi^2}\left(\frac{\lambda_{\rm free}}{n}\right)^3 \frac{Q}{V_m}\,,</math>
where <math>\lambda_{\rm free}</math> is the [[vacuum]] [[wavelength]], <math>n</math> is the [[refractive index]] of the cavity material (so <math>\lambda_{\rm free}/n</math> is the wavelength inside the cavity), and <math>Q</math> and <math>V_m</math> are the cavity [[quality factor]] and mode volume, respectively.