Mode volume

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

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:¹²

• 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 electrodynamics⁵ where it determines the magnitude⁶⁷⁸ of the Purcell effect and coupling strength between cavity photons and atoms in the cavity.⁹¹⁰ 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, 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.