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:

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

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

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

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

• 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.

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

where $E$ is the electric field strength, $B$ is the magnetic flux density, $\epsilon$ is the electric permittivity, and $\mu$ denotes the magnetic permeability. For cavities in which the electromagnetic energy is not totally confined within the cavity, modficiations to these expressions may be required.

The mode volume of a cavity or resonator is of particular importance in cavity quantum electrodynamics where it determines the magnitude of the Purcell effect and coupling strength between cavity photons and atoms in the cavity.

In fiber optics

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

The mode volume M is approximately given by $V^{2} \over 2$ and ${V^{2} \over 2}\left({g \over g+2}\right)$, 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

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

References

•  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).