= Radiant exposure
{title2=$H=\int E_e\,dt$}
{wiki}
<Radiant exposure> measures the <radiant energy> received per unit surface <area> during a specified interval. At a fixed surface point $\mathbf x$, it is the time <integral> of incident <irradiance> $E_{\rm e}$:
$$
\boxed{H_{\rm e}(\mathbf x)=\int_{t_0}^{t_1}E_{\rm e}(\mathbf x,t)\,dt,\qquad
[H_{\rm e}]=\mathrm{J\,m^{-2}}.}
$$
The <area> belongs to the receiving surface: its orientation and location matter. An <irradiance> in $\mathrm{W\,m^{-2}}$ specifies a rate, while <radiant exposure> in $\mathrm{J\,m^{-2}}$ specifies the accumulated arrival of <energy>.<International Lighting Vocabulary by CIE>[CIE S 017:2020, entry 17-21-071, definition, symbols and note 3]{ref}
For a uniform rectangular <pulse>, $H_{\rm e}=E_0\tau$, with constant <irradiance> $E_0$ and duration $\tau$. Stipulate one pulse at $20\,\mathrm{W\,m^{-2}}$ for $1\,\mathrm s$, and another at $80\,\mathrm{W\,m^{-2}}$ for $0.25\,\mathrm s$. \b[Both deliver $20\,\mathrm{J\,m^{-2}}$], but their peak <irradiances> differ by a factor of four. The two <radiant exposures> agree only after each complete pulse has arrived. A shorter observation window can intercept different fractions of them, and a <nonlinear> receiver can respond differently even to their complete pulses.
\Image[/equal-radiant-exposure-pulses.png]
{title=Original rectangular-pulse example: different irradiance histories accumulate the same radiant exposure.}
Integrating over a receiving patch gives its incident <radiant energy>:
$$
Q_A=\int_A H_{\rm e}(\mathbf x)\,dA,\qquad
\overline H_A=\frac{Q_A}{A}.
$$
Here $\overline H_A$ is the <spatially averaged radiant exposure>, rather than necessarily the local maximum. Uniform illumination is the special case where this <spatially averaged radiant exposure> equals the local value everywhere on the patch. In the rectangular-pulse example, a uniformly illuminated $1\,\mathrm{cm^2}=10^{-4}\,\mathrm{m^2}$ patch receives $20\times10^{-4}=2\,\mathrm{mJ}$.
For an original spatial example, consider a circular <Gaussian beam> at a fixed transverse plane, with constant <Gaussian beam radius> $w$ throughout the pulse. Let $P(t)$ be its total <radiant power> and $U=\int P(t)\,dt$ its <laser pulse energy>. Integrating its transverse <irradiance> in time gives the <radiant exposure of a Gaussian beam>:
$$
H(r)=\frac{2U}{\pi w^2}e^{-2r^2/w^2},\qquad
H(0)=\frac{2U}{\pi w^2}.
$$
The normalization follows by integrating $H(r)\,2\pi r\,dr$ from $0$ to $\infty$. The <disk> of <radius> $w$ contains only $U(1-e^{-2})$, so its actual <spatially averaged radiant exposure> is $U(1-e^{-2})/(\pi w^2)$, not $U/(\pi w^2)$. With stipulated $U=1\,\mathrm{\mu J}$ and $w=1\,\mathrm{mm}$, the peak is $0.6366\,\mathrm{J\,m^{-2}}$, while the average inside that disk is $0.2752\,\mathrm{J\,m^{-2}}$. State the <Gaussian beam radius> convention and the receiving <area> with any reported value.
An energy readout alone therefore does not specify a surface's <radiant exposure>. In an ideal linear model for <pulse energy measurement with a photodiode>, dark-subtracted <photocurrent> $i_{\rm ph}(t)=R P_{\rm det}(t)$ has constant <spectral responsivity> $R$ at the chosen <wavelength>. Integrating the <electric current> gives signal <electric charge> $Q_{\rm sig}=R U_{\rm det}$. If a known patch of <area> $A$ is uniformly illuminated and all of its incident <energy> is collected, then
$$
\boxed{H_{\rm e}=\frac{Q_{\rm sig}}{RA}.}
$$
As a separate stipulated example, $Q_{\rm sig}=0.40\,\mathrm{nC}$ and $R=0.40\,\mathrm{A\,W^{-1}}$ correspond to $U_{\rm det}=1.0\,\mathrm{nJ}$. Uniform illumination over $A=1.0\,\mathrm{mm^2}$ gives $H_{\rm e}=1.0\,\mathrm{mJ\,m^{-2}}$. <Background subtraction>, collection geometry, <detector linearity> and an integration window covering the full response are part of this measurement model. Continuous-wave responsivity alone does not establish linearity for a short pulse: silicon-<photodiode> experiments have observed pulse-dependent saturation.<Measuring Pulse Energy With Solid-State Photodiodes by Robert E. Vest and others>[Abstract: pulsed nonlinearity and dependence on reverse bias and illuminated area]{ref}
A historical NIST <measurement calibration> separates these operations experimentally. A modulator produces pulses from a continuous-wave <laser>; a calibrated <beam splitter> sends one beam to a slow thermal reference and another to the pulse detector. The reference records the accumulated <energy>, while the pulse repetition rate and injection duration determine the mean <energy> per pulse. The detector's integrating electronics provide a pulse-energy readout. Converting such a readout to <radiant exposure> additionally requires the illuminated <area> and its spatial distribution.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.4, printed p. 6; §4, printed pp. 9–10: calibrated splitting, thermal reference and pulse-energy response]{ref}
The beam-use term <fluence> denotes the same time integral for a directed beam normal to the chosen surface. For <light> arriving from many directions, specify the angular geometry: planar <irradiance> and <spherical irradiance> are different quantities, so a value defined using one cannot simply be substituted for the other.<International Lighting Vocabulary by CIE>[Entry 17-21-054, notes 1–2: directional radiance and the small-sphere construction]{ref}
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