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A dipole anisotropy of a radiance field is a directional first-order departure from a constant radiance:Here $\mathbf{\omega }$ is a unit viewing direction, and the bound ensures nonnegative radiance. Writing $\mathbf{b}=ϵ\mathbf{a}$ with unit $\mathbf{a}$ gives the amplitude/direction convention in the title. The angular isotropy test with planar irradiance measurements derives its receiver response.

An original inversion uses opposite receiver orientations. Suppose measured irradiances are $E(+x)=3.4$, $E(-x)=2.6$, and all four $y,z$ readings equal $3.0\mathrm{W}{\mathrm{m}}^{-2}$. The pair means give $\pi L_0=3.0\mathrm{W}{\mathrm{m}}^{-2}$. Sincethe fitted vector is $\mathbf{b}=(0.20,0,0)$. Agreement of the three pair sums is an additional check on this restricted model and a stable measurement chain.

The first angular moment of a radiance field is $\int \mathbf{\omega }Ld\Omega =(4\pi /3)L_0\mathbf{b}$. By contrast, integrating $L$ over all directions gives $4\pi L_0$, independent of $\mathbf{b}$. Thus this example's projected-area-normalized spherical irradiance is $12\mathrm{W}{\mathrm{m}}^{-2}$, exactly the value for an isotropic field with the same $L_0$. One all-direction scalar cannot determine its dipole.^[ref] Receiver angular response must also be characterized rather than assumed ideal.

An isotropic radiance field has the same radiance in every direction at the specified point: $L_{\mathrm{e}}(\mathbf{\omega })=L_0$. Angular isotropy is a local property; it does not by itself assert that $L_0$ is the same at every spatial position. Integrating this constant field gives the orientation-independent one-sided irradiance $\pi L_0$. The spherical irradiance article derives the corresponding all-direction value using its projected-area convention.^[ref]

An original counterexample shows why a few equal detector readings do not establish isotropy. Send six ideal narrow beams toward a point along the positive and negative Cartesian axes. Give every beam a normal-incidence irradiance $q=1\mathrm{W}{\mathrm{m}}^{-2}$. A one-sided planar receiver with unit normal $\mathbf{n}$ then recordsThe three axis-aligned receiver orientations all give $q$. However, a normal along $(1,1,1)/\sqrt{3}$ gives $\sqrt{3}q\approx 1.732\mathrm{W}{\mathrm{m}}^{-2}$. Three mutually perpendicular measurements therefore agree even though the idealized field is strongly directional. Finite narrow beams approximate this construction.

An angular isotropy test with planar irradiance measurements must sample enough orientations to test its stated angular model. A collimated light source and a rotating receiver can check angular response before such a test. Detector-area projection and the instrument's own angular sensitivity must be distinguished from variation in the incident field.

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 $E_{\mathrm{e}}$:The area belongs to the receiving surface: its orientation and location matter. An irradiance in $\mathrm{W}{\mathrm{m}}^{-2}$ specifies a rate, while radiant exposure in $\mathrm{J}{\mathrm{m}}^{-2}$ specifies the accumulated arrival of energy.^[ref]

For a uniform rectangular pulse, $H_{\mathrm{e}}=E_0\tau$, with constant irradiance $E_0$ and duration $\tau$. Stipulate one pulse at $20\mathrm{W}{\mathrm{m}}^{-2}$ for $1\mathrm{s}$, and another at $80\mathrm{W}{\mathrm{m}}^{-2}$ for $0.25\mathrm{s}$. Both deliver $20\mathrm{J}{\mathrm{m}}^{-2}$, 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.

Integrating over a receiving patch gives its incident radiant energy: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 $1\mathrm{c}{\mathrm{m}}^2=10^{-4}{\mathrm{m}}^2$ patch receives $20\times 10^{-4}=2\mathrm{mJ}$.

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:The normalization follows by integrating $H(r)2\pi rdr$ 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 $U=1\mu \mathrm{J}$ and $w=1\mathrm{mm}$, the peak is $0.6366\mathrm{J}{\mathrm{m}}^{-2}$, while the average inside that disk is $0.2752\mathrm{J}{\mathrm{m}}^{-2}$. 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 $i_{\mathrm{ph}}(t)=R{P}_{\det }(t)$ has constant spectral responsivity $R$ at the chosen wavelength. Integrating the electric current gives signal electric charge $Q_{\mathrm{sig}}=R{U}_{\det }$. If a known patch of area $A$ is uniformly illuminated and all of its incident energy is collected, thenAs a separate stipulated example, $Q_{\mathrm{sig}}=0.40\mathrm{nC}$ and $R=0.40\mathrm{A}{\mathrm{W}}^{-1}$ correspond to $U_{\det }=1.0\mathrm{nJ}$. Uniform illumination over $A=1.0\mathrm{m}{\mathrm{m}}^2$ gives $H_{\mathrm{e}}=1.0\mathrm{mJ}{\mathrm{m}}^{-2}$. 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.^[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.^[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.^[ref]