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Detector linearity describes how consistently a detector's corrected output remains proportional to input over a stated range and under specified conditions. For optical power $P$ and output $y$, an affine measurement model is $y=y_0+SP$, where $y_0$ is background and $S$ is constant. After background subtraction, doubling power should double output. A change in detector responsivity with input level is evidence of departure from that proportional relation.

For an original two-level test, suppose independently established powers are $P_1=0.1\mathrm{mW}$ and $P_2=0.2\mathrm{mW}$. A background reading is $y_0=0.020\mathrm{V}$, while illuminated readings are $y_1=0.220\mathrm{V}$ and $y_2=0.412\mathrm{V}$. Define the response ratioThe second corrected output is $2\%$ below the doubling expected from the first. Ignoring the background would instead give $0.412/(2\cdot 0.220)\simeq 0.936$, confusing offset with nonlinearity. These are stipulated teaching readings, not measured performance of a named detector.

A primary apparatus example is NIST's characterization of a photodiode at $633\mathrm{nm}$. It reports responsivity relative to the low-power value, using a beam splitter to illuminate two photodiodes at about a $10:1$ intensity ratio, with the detector aperture uniformly filled. The report explicitly notes that the observed linearity depends on irradiation geometry.^[ref]

A useful test therefore controls wavelength, beam position and size, bias, temperature, readout gain and source stability while changing input level. Reference-detector nonlinearity or a changing beam ratio can imitate nonlinearity of the test detector. Quantify the uncertainty in the comparison before interpreting a small response-ratio departure.

Detector saturation is a strong limiting case, but detectable nonlinearity can arise before the output becomes flat. A satisfactory response ratio at two levels also does not establish detector linearity over every intermediate level or different operating conditions. Record the tested range and the reference relation with the measurement result.

Detector responsivity relates a detector's output to the incident radiant input under stated conditions. For a photodiode illuminated at a specified wavelength, current spectral responsivity iswhere $I_0$ is the background current and $P_{\lambda }$ is incident optical power. Its units are $\mathrm{A}{\mathrm{W}}^{-1}$. NIST's detector-calibration publication develops spectral radiant-power responsivity as the quantity transferred from optical standards to photodetectors.^[ref]

For an original example, $20\mu \mathrm{W}$ incident power produces $10\mu \mathrm{A}$ net photocurrent, giving $R_{\lambda }=0.5\mathrm{A}{\mathrm{W}}^{-1}$. If readout electronics convert that current to $1\mathrm{V}$, the complete system's voltage responsivity is $50,000\mathrm{V}{\mathrm{W}}^{-1}$. A value without its output quantity and units can therefore be misleading.

For a photodiode with no internal multiplication, let quantum efficiency $\eta$ be collected electrons per incident photon. A photon has energy $hc/\lambda$, with Planck constant $h$, speed of light $c$ and elementary charge $e$. Thus a photon arrival rate $\Phi$ gives $P=\Phi hc/\lambda$ and net current $I=\eta e\Phi$. ThereforeAt $800\mathrm{nm}$, the original $0.5\mathrm{A}{\mathrm{W}}^{-1}$ example corresponds to $\eta \simeq 0.775$. This relation assumes the stated charge-collection model; internal gain changes the interpretation.

Responsivity can depend on wavelength, irradiation geometry, bias and temperature. Detector linearity tests whether a response ratio stays constant as input level changes; detector saturation can make it strongly level-dependent. In a nonlinear response, the ratio of corrected output to input and the local transducer sensitivity are different quantities. Retain the wavelength, beam conditions, readout convention and measurement uncertainty when reporting a responsivity measurement.

Detector saturation occurs when increasing the input no longer produces the response expected from the detector's lower-input calibration. The limitation may arise in the sensing element, its load circuit, an amplifier or the digitizing stage of the measuring chain. A nearly flat output then represents many different inputs, so the lost amplitude information cannot be recovered by applying the original linear conversion.

A primary physical example is Thorlabs' silicon-photodiode saturation study. It varies reverse bias and load resistance and examines the resulting saturation limit and noise floor. Its stated limitations include testing one FDS100 photodiode with a continuous-wave source; those observations do not establish a universal limit for every detector or pulsed illumination condition.^[ref]

For an original idealized output-limit model, let incident optical power be $P\ge 0$ andThe linear branch reaches its limit at $P=0.5\mathrm{mW}$. Inputs $0.5$, $1$ and $2\mathrm{mW}$ all produce $2.5\mathrm{V}$ in this model. Interpreting that output as $P=V/a$ therefore reports $0.5\mathrm{mW}$ for each, hiding the differences. The abrupt limit is a teaching model; real response can depart from linearity gradually.

An optical attenuator placed before the detector can provide a check while retaining usable input information. For transmission $T=1/4$ and original $P=1\mathrm{mW}$, the detector receives $0.25\mathrm{mW}$ and gives $1.25\mathrm{V}$ on the linear branch. Correcting the known attenuation recovers $P=V/(aT)=1\mathrm{mW}$. Scaling a clipped stored value after detection cannot perform the same recovery.

To test detector linearity, vary calibrated attenuation or source power while monitoring stability and keeping geometry and wavelength fixed. Compare output ratios, including the background contribution, rather than assuming every output plateau is a constant source. Thorlabs also identifies reverse bias, load and bandwidth as relevant system conditions.^[ref] In the flat part of the ideal response, transducer sensitivity is zero, and the inverse measurement model is no longer unique.

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]