Fluence

In physics, fluence is the flux (either particle or radiative flux) integrated over time. For particles, it is defined as the total number of particles that intersect a unit area in a specific time interval of interest, and has units of m–2 (number of particles per meter squared). Fluence can also be used to describe the energy delivered per unit area, in which case it has units of J/m2.

Definition

It has two equivalent definitions:

1. Suppose N particles pass through an area A. The particle fluence for the area A is defined as:

$\Phi = \frac{N}{A}$.

In the limit of infinitesimal area, this is:

$\Phi = \frac{{\rm d} N}{{\rm d} A}$.

2. Imagine an infinitesimal volume dV with particles passing through it. The particle fluence can be defined as

$\Phi = \frac{\sum {\rm d \ell} }{{\rm d} V}$,

where $\sum {\rm d \ell}$ is the sum of all the path lengths of the particles that traverse the volume. These definitions are equivalent as can be seen by multiplying the first definition by ${\rm d} x/{\rm d} x$ where dx is the typical path length of a particle in the volume. The numerator (${\rm d}N {\rm d} x$) then gives the total path length traced out by the dN particles in the volume (${\rm d}N {\rm d}x \simeq \sum {\rm d \ell}$) while the denominator (${\rm d} A {\rm d} x$) gives the volume dV.

Uses

In visible light based medicine and dentistry, fluence, which may be more properly referred to as radiant exposure, is a measurement of energy over area. The area is usually the spot size of the light device.

It is considered one of the fundamental units in dosimetry. The attached diagram shows its relationship to other radiation quantities.

References

• ICRU 33: Radiation Quantities and Units, April, 1980. (www.icru.org)

Quantity Unit Dimension Notes
Name Symbol[nb 1] Name Symbol Symbol
Radiant energy Qe[nb 2] joule J ML2T−2 energy
Radiant flux Φe[nb 2] watt W or J/s ML2T−3 radiant energy per unit time, also called radiant power.
Spectral power Φ[nb 2][nb 3] watt per metre W⋅m−1 MLT−3 radiant power per wavelength.
Radiant intensity Ie watt per steradian W⋅sr−1 ML2T−3 power per unit solid angle.
Spectral intensity I[nb 3] watt per steradian per metre W⋅sr−1⋅m−1 MLT−3 radiant intensity per wavelength.
Radiance Le watt per steradian per square metre W⋅sr−1m−2 MT−3 power per unit solid angle per unit projected source area.

confusingly called "intensity" in some other fields of study.

or
L[nb 4]
or

metre per hertz

W⋅sr−1m−3
or
W⋅sr−1⋅m−2Hz−1
ML−1T−3
or
MT−2
commonly measured in W⋅sr−1⋅m−2⋅nm−1 with surface area and either wavelength or frequency.

Irradiance Ee[nb 2] watt per square metre W⋅m−2 MT−3 power incident on a surface, also called radiant flux density.

sometimes confusingly called "intensity" as well.

or
E[nb 4]
watt per metre3
or
watt per square metre per hertz
W⋅m−3
or
W⋅m−2⋅Hz−1
ML−1T−3
or
MT−2
commonly measured in W⋅m−2nm−1
or 10−22 W⋅m−2⋅Hz−1, known as solar flux unit.[nb 5]

Me[nb 2] watt per square metre W⋅m−2 MT−3 power emitted from a surface.
M[nb 3]
or
M[nb 4]
watt per metre3
or

watt per square
metre per hertz

W⋅m−3
or
W⋅m−2⋅Hz−1
ML−1T−3
or
MT−2
power emitted from a surface per unit wavelength or frequency.

Radiosity Je watt per square metre W⋅m−2 MT−3 emitted plus reflected power leaving a surface.
Spectral radiosity J[nb 3] watt per metre3 W⋅m−3 ML−1T−3 emitted plus reflected power leaving a surface per unit wavelength
Radiant exposure He joule per square metre J⋅m−2 MT−2 also referred to as fluence
Radiant energy density ωe joule per metre3 J⋅m−3 ML−1T−2