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Photocurrent is the electric current attributable to illumination. Here $I_{\mathrm{ph}}\ge 0$ denotes its magnitude after background subtraction. For a photodiode without internal multiplication, let $P$ be incident radiant flux concentrated at wavelength $\lambda$, and let $\eta$ be external quantum efficiency, including losses before collection. An incident photon has energy $E_{\gamma }=hc/\lambda$, with Planck constant $h$ and speed of light $c$. Dividing energy per second by energy per photon gives the incident photon rate ${\Phi }_{\gamma }$. Multiplying by $\eta$ gives the mean collected pair rate ${\dot{N}}_{\mathrm{pair}}$; multiplying by elementary charge $q$ givesThe factor is $q$: each collected pair replenishes an electron at one contact and removes an electron at the other. A cross-section of the external wire therefore counts one transferred electron per completed pair, consistent with charge conservation. The electron and electron hole are two internal charge carriers, not two independent transfers through that wire.

Choose an illustrative photodiode model with $\lambda =650\mathrm{nm}$, $\eta =0.75$ and $P=1.00\mu \mathrm{W}$. Using the exact SI values $h=6.62607015\times 10^{-34}\mathrm{J}\mathrm{s}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ and $q=1.602176634\times 10^{-19}\mathrm{C}$ givesA mean fraction $0.75$ does not require each fourth photon to be lost in a repeating pattern. It specifies a mean conversion rate. Nor does $0.75$ mean $0.75\mathrm{A}{\mathrm{W}}^{-1}$: quantum efficiency is dimensionless, whereas spectral responsivity compares electric current with power.

At fixed radiant flux, longer-wavelength photons have less energy, so more arrive per second. This explains the rising straight line when quantum efficiency is held constant. A real photodiode's quantum efficiency varies with wavelength; the plotted assumption cannot be extended past its band gap absorption cutoff. At fixed incident photon rate instead, $I_{\mathrm{ph}}=\eta q{\Phi }_{\gamma }$ has no additional factor of $\lambda$.

A photodiode is a semiconductor diode producing photocurrent from photon absorption. Exciting an electron across the band gap leaves an electron hole. The depletion region's electric field separates the charge carriers; others can arrive by diffusion before electron-hole recombination. Contacts complete the electrical circuit.^[ref]

The useful bridge is from light to a measurable electric current: spectral responsivity converts incident radiant flux into photocurrent, while quantum efficiency counts collected signal charge carriers per incident photon. These answer different questions: how much electric current one watt produces, and how many incident photons contribute to the signal. The worked calculation below makes their relationship explicit.

Thorlabs' silicon FDS100 experiment sends a collimated, fibre-coupled laser beam through a beam splitter to the photodiode and a reference optical power meter. Varying reverse bias and electrical load tests linearity. The measured readout approaches saturation when its electric potential difference consumes the bias.^[ref] A separate low-signal setup uses a $637\mathrm{nm}$ laser, polarizers, an optical chopper, optical attenuators and a lock-in amplifier inside a light-tight enclosure. These are actual apparatus choices, not the parameters of the constructed example above.^[ref]

For an independent dark current check, Thorlabs enclosed a photodiode in nested metal boxes to exclude light and suppress interference, used a Keithley 6487 ammeter supplying reverse bias, and monitored temperature. Only one specimen per type was tested; the nearby temperature sensor did not directly contact the semiconductor. Such a dark control measures the baseline but cannot certify spectral responsivity.^[ref]

To infer spectral responsivity at a selected wavelength, measure the radiant flux actually reaching the test photodiode, rather than just the reference arm. A reference-arm measurement calibration must include the beam splitter ratio and path losses. With signed readouts in one fixed orientation, subtract a beam-blocked baseline and usefor direct electric current readout or a calibrated linear transimpedance amplifier, respectively. Repeating at several incident powers tests whether this ratio is constant. Blocking only the source estimates ambient light plus electronic baseline; covering the detector excludes ambient light too. Their difference distinguishes optical background from dark current and readout offset. For a broad optical spectrum, the ratio is a spectrum-weighted response, not generally $R$ at one wavelength.

For independent Poisson distribution collection, the root mean square shot noise in a one-sided rectangular frequency band of signal bandwidth $B>0$ isHere $I_d$ is reverse dark current; thermal electrical noise and amplifier noise also contribute.^[ref]

Subtracting a mean dark current removes its offset, not its fluctuations: subtracting a constant leaves the variance unchanged. Consequently $I_{\mathrm{ph}}\gg I_d$ makes dark current shot noise small relative to signal shot noise; it does not by itself establish high signal-to-noise ratio. With an independently measured baseline, the variance of that estimate must also be propagated. Increasing the measurement signal bandwidth raises the admitted noise. Calibration of wavelength, reference radiant flux, readout gain and collection geometry, together with drift and linearity, determines what measurement uncertainty can be claimed.

The photoelectric saturation current is the photocurrent reached when increasing the collecting electric potential difference no longer increases the fraction of available emitted electrons collected. It is saturation with collecting voltage at fixed illumination. It does not mean that increasing incident light intensity cannot increase emission.

If $P$ is incident radiant flux on the emitting area and $\eta$ is the external quantum efficiency for producing escaping electrons, then the incident photon arrival rate is $P/(h\nu )$ and complete collection gives $I_{\mathrm{sat}}=e\eta P/(h\nu )$. For the example's $\nu =8.00\times 10^{14}\mathrm{Hz}$, choose $\eta =0.10$ and $P=1.00\mu \mathrm{W}$. The calculated photon arrival rate is $1.886488\times 10^{12}{\mathrm{s}}^{-1}$ and $I_{\mathrm{sat}}=30.224866\mathrm{nA}$. Doubling $P$ doubles this current to $60.449731\mathrm{nA}$ while leaving $V_s$ fixed. At fixed illuminated area this doubles the irradiance. Changing frequency at fixed $P$ changes the arrival rate as well as the photon energy; $\eta$ can also change, so there is no general rule that a higher frequency gives a larger photocurrent.

A missing plateau can indicate incomplete collection or an electric field altered by the emitted electric charge. Heating, surface changes and additional emission channels can invalidate a constant $\eta$. Even a small amount of unwanted higher-frequency radiation can move an apparent endpoint. A stopping potential inferred from zero net current additionally needs background subtraction and a check for emission from the collector itself.

Photometry weights visible electromagnetic radiation by standardized visual response functions; radiometry measures its energy and power without visual weighting.^[ref]

The question is how much light a source supplies under a stated visual weighting. A beam containing $1\mathrm{mW}$ at the defining frequency $540\mathrm{THz}$ has luminous flux $0.683\mathrm{lm}$. Moving the same radiant flux to a region of smaller spectral luminous efficacy reduces its luminous flux. Changing the optical spectrum changes this conversion even when its total power stays fixed. This makes photometry useful for comparing lighting, while radiometry remains necessary for an energy budget. The lumen counts luminous flux; the candela additionally specifies its distribution over solid angle.

Two measurement routes implement the weighting: measure an optical spectrum and evaluate a weighted integral, or use a photometer whose spectral responsivity approximates the prescribed weighting. Both require a specified collection geometry. Neither is a complete prediction of an individual's perception of a scene.^[ref]

The photopic luminous efficiency function $V(\lambda )$ is dimensionless and describes standardized relative sensitivity for photopic vision. It is normalized to $V(555\mathrm{nm})=1$ and tabulated over $360$–$830\mathrm{nm}$. The wavelength coordinate is in standard air, rather than vacuum; changing coordinates requires converting the spectral radiant flux density as well. The official prescription uses linear interpolation between the specified 1 nm entries.^[ref]

Write ${\lambda }_a$ for the standard-air wavelength at $540\mathrm{THz}$. With spectral radiant flux ${\Phi }_{e,\lambda }$ in $\mathrm{W}\mathrm{n}{\mathrm{m}}^{-1}$ and $d\lambda$ in nanometres, the luminous flux isThe integral has units of watts, and $K_m$ converts them to lumens. For radiation at a single wavelength ${\lambda }_0$ with total radiant flux $P$, this reduces to ${\Phi }_v=K_{m}V({\lambda }_0)P$. For a sufficiently narrow finite band, this is an approximation requiring nearly constant $V$ across the band.^[ref]

An ideal point emitter emits radiant flux from one location. It approximates a physical source when its emitting dimensions are small relative to the distances relevant to the calculation. With isotropy, its radiant intensity is $I=\Phi /(4\pi )$ for total radiant flux $\Phi$. An extended source generally requires integrating contributions from different emitting positions.

A point spread function is an imaging system's response to an optical point source. Here it means the distribution of image light intensity: an ideal point object becomes a finite spot, often with faint rings, because of diffraction. An unresolved star and a sufficiently small bead emitting fluorescence are practical approximations to an optical point source. Their image tells us how nearby object details will overlap, and reveals blurring from optical aberrations, defocus, and, for an optical telescope, astronomical seeing.

A useful convention separates shape from throughput. At a fixed image plane, let $p(\mathbf{x})\ge 0$ be the point spread function with ${\int }_{{\mathbb{R}}^2}p(\mathbf{x})d^2\mathbf{x}=1$. Its units are inverse area. A point delivering detected radiant flux $F$ at image position ${\mathbf{x}}_0$ produces light intensity $Fp(\mathbf{x}-{\mathbf{x}}_0)$; this point spread function normalization makes the integral over the image equal to $F$. A profile divided by its own peak is convenient for comparing widths, but is a different point spread function normalization and does not by itself compare total radiant flux.

In incoherent imaging, different object points have negligible mutual spatial optical coherence over the measurement, so their image light intensities add. If the response also obeys translation invariance over the chosen field of view, the expected continuous image is the convolutionHere $O$ is the object brightness distribution expressed in image-scaled coordinates, with magnification already included, and $\tau$ is constant transmission. For $O(\mathbf{u})=F_0\delta (\mathbf{u}-{\mathbf{u}}_0)$, where $\delta$ is the two-dimensional Dirac delta function, this gives $I=\tau F_{0}p(\mathbf{x}-{\mathbf{u}}_0)$. A changing response across the field of view instead requires a position-dependent integral kernel $p(\mathbf{x};\mathbf{u})$; one measured point spread function is then only a local calibration.

In coherent imaging, the superposition principle applies to the complex wave amplitudes. With amplitude spread function $h$, the corresponding ideal model isThus $|h\ast E{|}^2$ generally differs from $|h{|}^2\ast |E{|}^2$: coherent contributions produce wave interference terms that depend on their relative wave phases. Their time average vanishes for mutually incoherent emitters, which explains the intensity convolution. Partial optical coherence requires the correlations between object points as well.^[ref]

The figure is an original diffraction model, not measured data. All panels have the same total radiant flux: each member of a pair contributes half the single-source radiant flux. Dashed curves show the separate light intensity contributions; the solid curve is their sum. The point spread functions have tails, so even sources satisfying the Rayleigh criterion overlap. As separation decreases, the image changes continuously; there is no abrupt disappearance of all information at one separation.

A motivating measured example is Measurement of point-spread function (PSF) for confocal fluorescence microscopy by InCheon Song and others (2005). A $543.8\mathrm{nm}$ laser, delivered through an optical fiber, excited fluorescent beads on a slide through a $100\times$, numerical aperture $1.3$ oil-immersion objective. A moving stage scanned the bead; emitted light returned through the objective, an emission filter and a pinhole to a photodiode. The readout was fluorescence light intensity against scanned position. Under the same setup, nominal bead diameters $0.17$, $0.5$, and $1\mu \mathrm{m}$ produced progressively broader measured spots. The size comparison controls for the finite object: the measured response is the convolution of bead emission with the effective point spread function, rather than a pure point response. Inverting this convolution model gave a response compared with the theoretical one. The smaller bead gave a weaker signal relative to noise; uncertainty in actual bead diameter also affected the restored width. These are limits of the calibration. In confocal microscopy, excitation and pinhole detection jointly determine the effective point spread function; this experiment does not measure the ideal widefield Airy disk plotted above.^[ref]

For a new calibration, useful additional controls are to measure detector background, avoid saturating the photodetector, repeat the scan to check stability, and compare isolated beads at several positions in the field of view. Record wavelength, numerical aperture, pinhole setting, sampling interval and optical focus. These are practical recommendations, not extra measurements claimed for the published experiment.

Point spread function normalization specifies how the response is scaled. Unit-integral scaling separates image shape from total radiant flux; unit-peak scaling compares shapes but does not make their total radiant fluxes equal.

A grey atmosphere model with mass absorption coefficient proportional to pressure. Under hydrostatic balance, $\kappa ={\kappa }_{s}p/p_s$ gives downward optical depth $t={\kappa }_s{p}^2/(2g{p}_s)$. For $T=T_s(p/p_s{)}^a$, a non-scattering optically thick column emits hemispheric radiant flux $F\sim \sigma T_s^4{t}_s^{-2a}\Gamma (2a+1)/(a+1)$. The angular factor follows from integrating ray attenuation over the upward hemisphere; a vertical-ray approximation omits it. The extension of the emitting-depth integral to infinity requires fixed $a$ as $t_s\to \infty$.

Radiance resolves both position and direction: it is radiant flux per projected area and per solid angle. For a surface element of area $dA$, and a small outgoing cone $d\Omega$ at angle $0\le \theta <\pi /2$ to its outward unit normal,Here $\mathbf{x}$ locates the patch and $\mathbf{s}$ is a unit vector along the outgoing direction. The factor $dA\cos \theta$ is the patch's apparent area viewed along that direction. The same local definition applies to transmitted or received radiation, choosing the positive projected area for the crossing direction. ^[ref].

For a planar patch of area $A$ with spatially uniform, direction-independent radiance $L$, integrating over the patch givesTake $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$ and $A=0.020{\mathrm{m}}^2$. The radiant intensities are $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and $I(60^{\circ })=0.10\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$. A sufficiently narrow $0.010\mathrm{sr}$ cone therefore carries approximately $2.0\mathrm{mW}$ or $1.0\mathrm{mW}$, respectively. Both views have the same radiance $L$: the oblique view presents half the projected area. This is Lambert's cosine law, not a loss of radiance with viewing angle.

Radiance is useful because it separates a source's local directional output from the geometry of an observation. Integrating over outgoing directions gives radiant exitance; integrating incoming radiance at a receiver gives irradiance. In the stationary vacuum setting of vacuum conservation of specific intensity, a resolved uniform source retains its radiance along each light ray, while the radiant flux received from the whole source can decrease as its apparent solid angle shrinks.

Radiant exitance is total outward emitted radiant flux per actual emitting area, measured in $\mathrm{W}{\mathrm{m}}^{-2}$. Integrate radiance over the outward hemisphere, keeping the projected area factor:If the radiance is direction-independent, the spherical coordinates element is $d\Omega =\sin \theta d\theta d\phi$, soThe weighted solid angle is $\pi \mathrm{sr}$, although the whole hemisphere covers $2\pi \mathrm{sr}$. Directions near grazing contribute less because of the projected area. This geometric factor is the bridge between Lambert's cosine law and the total output of a diffuse patch. ^[ref].

For the uniform patch with $L=10\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$, radiant exitance is $M=10\pi \mathrm{W}{\mathrm{m}}^{-2}$. Its area $0.020{\mathrm{m}}^2$ gives total radiant flux $\Phi =MA=0.20\pi \mathrm{W}\approx 0.628319\mathrm{W}$ over the outward hemisphere. This total differs from its radiant intensity $I(0)=0.20\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$, which describes only one direction.

For an ideal blackbody, integrate Planck's law over frequency as well: $M=M_e=j^{\ast }=\sigma T^4$ by the Stefan–Boltzmann law. An ideal $6000\mathrm{K}$ emitting patch of area $1\mathrm{m}{\mathrm{m}}^2$ emits about $73.4881\mathrm{W}$ outward. Irradiance counts incoming radiation separately; for purely emitted output the net outward exchange per area is $M-E$. Reflected output must also be included if the surface reflects incident radiation.

Radiant flux is the power carried by electromagnetic radiation: radiation energy transferred per unit time, measured in watts, $\mathrm{W}=\mathrm{J}{\mathrm{s}}^{-1}$. Specify what is counted: all radiation emitted by a source, radiation within one beam, or radiation intercepted by a photodetector. Writing $Q(t)$ for the cumulative energy crossing the specified boundary givesThe first expression is an instantaneous derivative; the second is an average over the stated time interval. If $0.020\mathrm{J}$ arrives during $0.010\mathrm{s}$, the mean radiant flux is $2\mathrm{W}$. The photodetector's absorbed power equals this incident radiant flux only if it absorbs all of the incident radiation.

Keeping radiant flux fixed while widening a beam reduces its irradiance, the incident power per receiving area. Radiance additionally resolves the propagation direction and has units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}$; an optical camera can collect different radiant fluxes from sources with the same radiance but different apparent solid angles. Luminous flux, measured in lumens, weights the optical spectrum by the human eye's standardized response instead of counting energy equally at every wavelength. Equal radiant fluxes need not give equal luminous fluxes.^[ref]

Radiant power also differs from the optical power of an optical lens, which is reciprocal focal length in dioptres. Likewise, astronomical radiative flux is power per area. Stating the units prevents these shared words from hiding different physical quantities.

In geometrical optics, suppose radiation has a nonnegative radiance $L(\mathbf{x},\mathbf{s})$ whose directional contributions add as power. Here $\mathbf{s}$ is the actual propagation unit vector, $d\Omega$ is a solid angle element, and the collecting oriented surface $A$ has chosen unit normal $\mathbf{n}(\mathbf{x})$. Assume the following integrals exist. The outward and inward radiant fluxes areBoth quantities are nonnegative; their difference is the signed net radiant flux. The dot product supplies the cosine of the angle with the unit normal, hence the projected area. For incoming radiation the outward unit normal gives a negative dot product; the minus sign makes the collected power positive. Measuring the incoming angle from a source-facing unit normal instead gives the familiar positive $\cos \theta$. The units check is $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\times {\mathrm{m}}^2\times \mathrm{sr}=\mathrm{W}$.

Equivalently, the incoming surface integral is ${\Phi }_{-}={\int }_{A}EdA$ with local irradiance $E$; the emitted part of ${\Phi }_{+}$ is ${\int }_{A}MdA$ with radiant exitance $M$. For a small flat patch and a narrow accepted cone, nearly constant radiance and angle give $\Phi \simeq LA\cos \theta \Delta \Omega$. This approximation requires uniformity across the patch and cone. Direction-independent outgoing radiance over a whole hemisphere gives the different exact result ${\Phi }_{+}=\pi LA$, by Lambert's cosine law and the spherical coordinates calculation in radiant exitance. Replacing the cosine-weighted integral by the hemisphere's $2\pi$ solid angle would double the answer.^[ref]

These radiance formulas presume a description in additive directional power; they do not reconstruct arbitrary coherent wave interference from separate ray contributions. For general electromagnetic fields, the signed instantaneous power crossing an oriented surface is ${\int }_A\mathbf{S}\cdot \mathbf{n}dA$, using the Poynting vector $\mathbf{S}$; optical measurements commonly average over many wave periods. A zero net radiant flux can conceal equal nonzero incoming and outgoing radiant fluxes.

Radiant intensity describes how a source distributes its radiant flux over directions. A small cone of solid angle $\Delta \Omega$ carries approximately $I\Delta \Omega$ when $I$ varies little across it; integrating $I$ over all emitting directions recovers the total radiant flux. For a source with isotropy emitting $4\pi \mathrm{W}$ over a full sphere, $I=1\mathrm{W}\mathrm{s}{\mathrm{r}}^{-1}$ and a $0.01\mathrm{sr}$ cone carries $0.01\mathrm{W}$. A planar source satisfying Lambert's cosine law instead has direction-dependent radiant intensity, even though its radiance is direction-independent.

For normal incidence in vacuum, a uniform beam delivers energy $IA\Delta t$ to an illuminated surface area $A$ during time $\Delta t$. A perfect absorber receives momentum $IA\Delta t/c$, so force divided by surface area gives $p_{\mathrm{abs}}=I/c$. A perfect stationary mirror sends the photons back with reversed momentum, so momentum conservation gives twice the normal transfer, $p_{\mathrm{refl}}=2I/c$. These are time-averaged pressures; the microscopic arrivals need not be uniform.

For a stipulated model light intensity $I=1000\mathrm{W}{\mathrm{m}}^{-2}$ and the exact SI speed of light $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$,These are computed predictions, not measured data or a universal sunlight value. If the beam fully covers a normally oriented surface area $A=0.010{\mathrm{m}}^2$, the intercepted radiant flux is $\Phi =IA=10\mathrm{W}$ and the absorber's force is $\Phi /c\simeq 33.356\mathrm{nN}$. Using the total source power would be wrong if only part of its beam reaches the receiver. The absorption prediction assumes complete photon absorption, while the reflection prediction assumes complete specular reflection; transmission, scattering of light and directed thermal photon emission require their own outgoing momentum accounting.

Measure the angle of incidence $\theta$ between the incoming path and the surface's normal line, with $0\le \theta <\pi /2$. There are two separate cosine factors. A fixed actual surface area $A$ intercepts only its projected area $A\cos \theta$, so $\Phi =IA\cos \theta$. Each incoming photon supplies normal momentum $(E_{\gamma }/c)\cos \theta$ on absorption, or twice that on specular reflection. Thus, for radiometric absorptance $a$ and reflected power fraction $R$,This model assumes transmitted light continues straight in the same medium and carries away its original momentum. It omits thermal recoil and assumes a stationary receiver. At $\theta =60^{\circ }$, fixed beam light intensity therefore gives one quarter of the normal-incidence pressure. For fixed intercepted radiant flux $\Phi$, however, the normal force has only one cosine factor. Equivalently, if the given quantity is receiving-surface irradiance $E_{\mathrm{inc}}=I\cos \theta$, then $p_{\perp }=(a+2R)E_{\mathrm{inc}}\cos \theta /c$. The distinction matters when interpreting an experiment's angle scan. ^[ref]

Absorption also transfers a tangential momentum component at oblique incidence. It contributes a tangential force, rather than normal pressure. Ideal specular reflection preserves the tangential momentum component, so it transfers only normal momentum.

Radiometric absorptance is the fraction of incident radiant flux converted to heat in the specified receiver: $P_{\mathrm{abs}}=a{P}_{\mathrm{in}}$, with $0\le a\le 1$ for a passive receiver. Specify the entrance boundary, wavelength, incidence angle and receiver temperature. Radiation reflected or transmitted out of that receiver is not absorbed power. A cavity can improve radiometric absorptance by giving light repeated opportunities for wave absorption; a dark coating alone does not establish $a=1$.

For broadband light, spectral radiant flux matters: $P_{\mathrm{abs}}=\int a(\lambda ){\Phi }_{e,\lambda }(\lambda )d\lambda$. A single effective radiometric absorptance therefore depends on the incident optical spectrum as well as on the receiver. If the stated incident boundary lies before a window of known power transmission ${\tau }_w$, the receiver absorbs $a{\tau }_w{P}_{\mathrm{before}}$ when other optical losses are absent. Window loss and receiver radiometric absorptance are separate corrections.

Radiometry measures the energy and power carried by electromagnetic radiation, including light. It asks how much radiation a source supplies, where it travels, and how much reaches a receiving surface. The measurements are physical rather than weighted by the sensitivity of the human eye: equal radiant fluxes at different wavelengths need not look equally bright. ^[ref].

The five quantities below answer different questions about the same radiation. The International System of Units makes the distinctions visible: a watt is a joule per second, while a steradian measures solid angle. ^[ref].

Resolving wavelength or frequency adds a further “per unit interval”, as in spectral radiance. The notation is consistent with NIST's table of spectral quantities. In particular, radiant flux is power, whereas the existing astronomical radiative flux is power per area. ^[ref].

This bookkeeping makes an optical camera's collected signal calculable from a source's radiance and the geometry of the receiving aperture. It also lets radiation measurements from different instruments be compared. For example, NIST describes separate calibrations of spectral radiance and spectral irradiance for instruments used to observe Earth. ^[ref].

The fraction of incident radiant flux returned by reflection, for specified wavelength, angle of incidence and collection directions.

A dark, opaque circular planet of radius $R_p$ blocks specific intensity over its overlap with the projected star. With $k=R_p/R_{\star }$ and no transit dilution, the instantaneous exoplanet transit depth is exactlyWhen $k\ll 1$, the entire planetary disk is inside the stellar disk, and specific intensity varies little across that disk, small-planet transit weighting reduces toHere $z$ is the projected centre separation divided by $R_{\star }$. The numerator samples the local specific intensity; the denominator normalizes by the whole star's mean specific intensity. The small-size argument and the distinction between full overlap and ingress are explicit in the original analytic treatment. ^[ref]

For a short approximately straight crossing at constant transverse velocity $v_{\perp }$, write $x=v_{\perp }t/R_{\star }$ with $t=0$ at mid-transit, and let $b$ be the transit impact parameter. Then $z^2=x^2+b^2$ and ${\mu }_p=\sqrt{1-b^2-x^2}$. With stipulated $u=0.6$ and $k=0.05$, a central crossing has mid-transit exoplanet transit depth approximately $0.003125$, whereas $b=0.8$ gives $0.002375$. A uniform disk would give $k^2=0.0025$ for either fully overlapping crossing. The brighter central chord produces a deeper dip and the off-centre chord a shorter crossing at the same $v_{\perp }$.

At fixed $z<1$ away from the edge, a two-dimensional Taylor's theorem expansion over the symmetric planetary disk cancels the first derivatives; the first correction to the blocked radiant flux is of order $k^4$ if the second derivatives remain bounded. This error estimate is not uniform at the edge, where the radial derivative of the linear limb-darkening law diverges. Ingress, egress, a grazing exoplanet transit, and a crossing of a starspot require the overlap integral or a suitable more complete model. Inferring $k=\sqrt{\delta }$ without the local-to-mean specific intensity ratio can bias the estimated planet radius.