# 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 received, emitted, reflected, or transmitted by a system in form of electromagnetic radiation.
Radiant energy density we joule per cubic metre J/m3 ML−1T−2 Radiant energy of a system per unit volume at a given location.
Radiant flux / Radiant power Φe[nb 2] watt W or J/s ML2T−3 Radiant energy of a system per unit time at a given time.
Spectral flux / Spectral power Φe,ν[nb 3]
or
Φe,λ[nb 4]
watt per hertz
or
watt per metre
W/Hz
or
W/m
ML2T−2
or
MLT−3
Radiant power of a system per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1.
Radiant intensity Ie,Ω[nb 5] watt per steradian W/sr ML2T−3 Radiant power of a system per unit solid angle around a given direction. It is a directional quantity.
Spectral intensity Ie,Ω,ν[nb 3]
or
Ie,Ω,λ[nb 4]
or
W⋅sr−1⋅Hz−1
or
W⋅sr−1⋅m−1
ML2T−2
or
MLT−3
Radiant intensity of a system per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is a directional quantity.
Radiance Le,Ω[nb 5] watt per steradian per square metre W⋅sr−1⋅m−2 MT−3 Radiant power of a surface per unit solid angle around a given direction per unit projected area of that surface along that direction. It is a directional quantity. It is sometimes also confusingly called "intensity".
or
Le,Ω,λ[nb 4]
watt per steradian per square metre per hertz
or
watt per steradian per square metre, per metre
W⋅sr−1⋅m−2⋅Hz−1
or
W⋅sr−1⋅m−3
MT−2
or
ML−1T−3
Radiance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is a directional quantity. It is sometimes also confusingly called "spectral intensity".
Irradiance Ee[nb 2] watt per square metre W/m2 MT−3 Radiant power received by a surface per unit area. It is sometimes also confusingly called "intensity".
or
Ee,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Irradiance of a surface per unit frequency or wavelength. The former is commonly measured in 10−22 W⋅m−2⋅Hz−1, known as solar flux unit, and the latter in W⋅m−2⋅nm−1.[nb 6] It is sometimes also confusingly called "spectral intensity".
Radiosity Je[nb 2] watt per square metre W/m2 MT−3 Radiant power leaving (emitted, reflected and transmitted by) a surface per unit area. It is sometimes also confusingly called "intensity".
or
Je,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Radiosity of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. It is sometimes also confusingly called "spectral intensity".
Radiant exitance Me[nb 2] watt per square metre W/m2 MT−3 Radiant power emitted by a surface per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. It is sometimes also confusingly called "intensity".
Spectral exitance Me,ν[nb 3]
or
Me,λ[nb 4]
watt per square metre per hertz
or
watt per square metre, per metre
W⋅m−2⋅Hz−1
or
W/m3
MT−2
or
ML−1T−3
Radiant exitance of a surface per unit frequency or wavelength. The latter is commonly measured in W⋅sr−1⋅m−2⋅nm−1. "Spectral emittance" is an old term for this quantity. It is sometimes also confusingly called "spectral intensity".
Radiant exposure He joule per square metre J/m2 MT−2 Irradiance of a surface times exposure time. It is sometimes also called fluence.