Mode volume

Mode volume may refer to figures of merit used either to characterise optical and microwave cavities or optical fibers.

In electromagnetic cavities

The mode volume (or modal volume) of an optical or microwave cavity is a measure of how concentrated the electromagnetic energy of a single 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: ^{[1] [2]}

• The volume that would be occupied by the mode if its electromagnetic energy density was constant and equal to its maximum value

$${\displaystyle V_{m}={\frac {\int \epsilon |E|^{2}dV}{\rm {{max}(\epsilon |E|^{2})}}}\;\;\;{\rm {{or}\;\;\;V_{m}={\frac {\int (|B|^{2}/\mu )\;dV}{\rm {{max}(|B|^{2}/\mu )}}}}}}$$

• The volume over which the electromagnetic energy density exceeds some threshold (e.g., half the maximum energy density)

$${\displaystyle V_{m}=\int \left(|E|^{2}>{\frac {|E_{max}|^{2}}{2}}\right)dV}$$

• 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.

$${\displaystyle V_{m}={\frac {(\int |E|^{2}dV)^{2}}{\int |E|^{4}dV}}\;\;\;{\rm {{or}\;\;\;V_{m}={\frac {(\int |B|^{2}dV)^{2}}{\int |B|^{4}dV}}}}}$$

where ${\displaystyle E}$ is the electric field strength, ${\displaystyle B}$ is the magnetic flux density, ${\displaystyle \epsilon }$ is the electric permittivity, ${\displaystyle \mu }$ denotes the magnetic permeability, and ${\displaystyle \max(\cdots )}$ 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. ^[3] In this case modifications to the expressions above may be required to give an effective mode volume. ^[4]

The mode volume of a cavity or resonator is of particular importance in cavity quantum electrodynamics ^[5] where it determines the magnitude ^{[6] [7] [8]} of the Purcell effect and coupling strength between cavity photons and atoms in the cavity. ^{[9] [10]} In particular, the Purcell factor is given by

${\displaystyle F_{\rm {P}}={\frac {3}{4\pi ^{2}}}\left({\frac {\lambda _{\rm {free}}}{n}}\right)^{3}{\frac {Q}{V_{m}}}\,,}$

where ${\displaystyle \lambda _{\rm {free}}}$ is the vacuum wavelength, ${\displaystyle n}$ is the refractive index of the cavity material (so ${\displaystyle \lambda _{\rm {free}}/n}$ is the wavelength inside the cavity), and ${\displaystyle Q}$ and ${\displaystyle V_{m}}$ are the cavity quality factor and mode volume, respectively.

In fiber optics

In fiber optics, mode volume is the number of bound modes that an optical fiber is capable of supporting. ^[11]

The mode volume M is approximately given by ${\displaystyle V^{2} \over 2}$ and ${\displaystyle {V^{2} \over 2}\left({g \over g+2}\right)}$, respectively for step-index and power-law index profile fibers, where g is the profile parameter, and V is the normalized frequency, which must be greater than 5 for this approximation to be valid.

See also

• Electromagnetic resonator
• Purcell effect
• Cavity quantum electrodynamics
• Cavity perturbation theory
• Quantum optics
• Equilibrium mode distribution
• Mode scrambler
• Mandrel wrapping

References

1. "Calculating the modal volume of a cavity mode". Ansys Optics . Archived from the original on 17 August 2022. Retrieved 13 September 2024.
2. Kippenberg, Tobias Jan August (2004). Nonlinear Optics in Ultra-High-Q Whispering-Gallery Optical Microcavities (phd thesis). California Institute of Technology. doi:10.7907/t5b6-9r14.
3. Meldrum, A. "Lesson 5: Whispering Gallery Modes". sites.ualberta.ca. Retrieved 2024-12-19.
4. Kristensen, P. T.; Van Vlack, C.; Hughes, S. (2012-05-15). "Generalized effective mode volume for leaky optical cavities". Optics Letters. 37 (10): 1649. arXiv: 1107.4601 . doi:10.1364/OL.37.001649. ISSN   0146-9592.
5. Kimble, H. J. (1998). "Strong Interactions of Single Atoms and Photons in Cavity QED". Physica Scripta. T76 (1): 127. doi:10.1238/Physica.Topical.076a00127. ISSN   0031-8949.
6. Purcell, E. M. (1946-06-01). "Proceedings of the American Physical Society: B10. Spontaneous Emission Probabilities at Radio Frequencies". Physical Review. 69 (11–12): 674–674. doi:10.1103/PhysRev.69.674.2. ISSN   0031-899X.
7. Boroditsky, M.; Coccioli, R.; Yablonovitch, E.; Rahmat-Samii, Y.; Kim, K.W. (1998-12-01). "Smallest possible electromagnetic mode volume in a dielectric cavity". IEE Proceedings - Optoelectronics. 145 (6): 391–397. doi:10.1049/ip-opt:19982468. ISSN   1350-2433.
8. Chen, Tom. "Calculating cavity quality factor, effective mode volume, and Purcell factor in Tidy3D Flexcompute". www.flexcompute.com. Retrieved 2024-12-19.
9. Srinivasan, Kartik; Borselli, Matthew; Painter, Oskar; Stintz, Andreas; Krishna, Sanjay (2006). "Cavity Q, mode volume, and lasing threshold in small diameter AlGaAs microdisks with embedded quantum dots". Optics Express. 14 (3): 1094. arXiv: physics/0511153 . doi:10.1364/OE.14.001094. ISSN   1094-4087.
10. Yoshie, T.; Scherer, A.; Hendrickson, J.; Khitrova, G.; Gibbs, H. M.; Rupper, G.; Ell, C.; Shchekin, O. B.; Deppe, D. G. "Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity". Nature. 432 (7014): 200–203. doi:10.1038/nature03119. ISSN   0028-0836.
11. Weik, Martin H. (2000), "mode volume", Computer Science and Communications Dictionary, Boston, MA: Springer US, pp. 1033–1033, doi:10.1007/1-4020-0613-6_11695, ISBN   978-0-7923-8425-0 , retrieved 2024-09-13

•  This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).