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Aperture photometry estimates brightness by summing calibrated detector pixels in an extraction region and subtracting the estimated background in that region. For $n$ equally weighted pixels, let $S$ be their total accumulated signal in electrons, $b$ the estimated background electrons per pixel over the same exposure, and $\Delta t$ the exposure duration. Then $\hat{r}=(S-nb)/\Delta t$ is a background-subtracted electron rate. Kepler applied aperture photometry to calibrated pixels with a fitted sky background. ^[ref].

This measurement trades source collection against background: enlarging the extraction region can retain more starlight but includes more sky brightness and neighbouring sources. Photon shot noise, read noise, and uncertainty in background subtraction contribute to its uncertainty. Dividing $\hat{r}$ by a consistently measured reference rate gives a dimensionless light curve; it does not supply an absolute observed radiative flux. If contaminating light remains, the reference includes it and the inferred exoplanet transit depth suffers transit dilution.

In the first toy figure, the source's reference rate is $100,000$ electrons per second, the summed sky contribution is $4,000$ electrons per second, and a known multiplicative response is $g(t)=1+0.0003t$ with $t$ in hours from mid-transit. The artificial exoplanet transit depth is $0.010$, its total duration four hours, and its flat-bottom duration $2.8$ hours; the straight ingress and egress are a toy approximation, not an exact exoplanet transit photometry model. Each synthetic sample averages a $0.5$-hour exposure and adds seeded normal distribution noise with relative standard deviation $0.0006$. Subtracting the known sky rate and dividing by the known response and reference rate recovers the dip. Real observations require estimating those corrections and carrying their uncertainty.

Background subtraction estimates a source contribution by removing a sky or detector-background measurement. An independent background estimate adds photon shot noise; a mismatched background adds bias of an estimator.

Opening the pinhole increases accepted in-focus and out-of-focus photons; closing it strengthens confocal optical sectioning while discarding some useful in-focus photons too. In the synthetic model, $\alpha =0.5,1.5,4$ transmits approximately $0.1175,0.6753,0.9997$ of the collected in-focus emission. At $t=2$ the corresponding peak-normalized signals are about $0.0420,0.0597,0.1597$. These are calculated fractions under stated assumptions, not instrument measurements.

An Airy unit conventionally specifies a pinhole diameter equal to the first-zero diameter of the emission Airy disk in that plane. For an ideal circular pupil,where $M$ is the total specimen-to-pinhole magnification, ${\lambda }_{\mathrm{em}}$ the emission wavelength in vacuum, and $\mathrm{NA}$ the objective numerical aperture. A software setting needs its specified wavelength and relay magnification; it is not a universal micrometre diameter. ^[ref]

For ideal independent counting, accepted source and background means $S$ and $B$ give photon shot noise with variance $S+B$ and signal-to-noise ratio in photon counting $S/\sqrt{S+B}$. Closing a pinhole need not improve that ratio: it depends on how both $S$ and $B$ change. Increasing exposure or excitation to replace discarded photons can cause bleaching, an irreversible loss of fluorescent emission capacity, or damage. Detection rejection occurs after excitation and therefore does not undo excitation outside the selected depth. Such material can absorb illumination and emit photons even when their emission is omitted from the recorded image.

Diffraction-limited resolution is a scale for distinguishing details with a specified ideal diffraction response and a specified criterion. For two equally bright, mutually incoherent optical point sources imaged through a uniformly transmitting, unobscured circular optical pupil, the Rayleigh criterion places the centre of one Airy disk at the first zero of the other. At small angles in air this separation isHere $D$ is pupil diameter, $\lambda$ is wavelength in air, and $j_{1,1}$ is the first positive zero of the order-one Bessel function of the first kind. This is the radius to the first zero, not the diameter of the central spot. Fraunhofer diffraction gives the radial peak-normalized light intensityThe removable value at zero follows from $J_1(q)\sim q/2$. The pupil Fourier transform reduces to a radial integral by circular symmetry, which produces the Bessel function of the first kind; its first zero supplies the factor $1.22$.^[ref]

In the usual scalar circular-pupil approximation for a fluorescence microscope, the corresponding object-plane Rayleigh criterion isHere ${\lambda }_0$ is the vacuum emission wavelength, and the object-side numerical aperture already includes the medium's refractive index. Do not divide ${\lambda }_0$ by $n$ again. For small cone angles in air, $\mathrm{NA}\simeq D/(2f)$ and image-plane radius $f{\theta }_{\mathrm{R}}$ agrees with this form. The assumptions include a single wavelength, ideal optical focus, and mutually incoherent emitters. High numerical aperture can require a vector description with polarization of an electromagnetic wave and emitter orientation; confocal microscopy has its own excitation-and-detection response. The same $0.61$ formula therefore does not describe every microscope or illumination method.^[ref]

In this original model, ${\lambda }_0=550\mathrm{nm}$ and numerical apertures $0.25$ and $0.50$ give first-zero radii $1.342\mu \mathrm{m}$ and $0.671\mu \mathrm{m}$. Each point spread function is divided by its own peak to compare shape. The pupil axes are transverse wave vector components multiplied by ${\lambda }_0/(2\pi )$, so pupil radius equals numerical aperture. A wider pupil accepts a wider range of transverse wave vectors, producing a narrower image. Increasing magnification alone enlarges the same spot and does not reduce its object-plane radius.

The Rayleigh criterion is a useful visual convention, not an absolute information barrier. For equal incoherent sources with total detected radiant flux $F$ and separation vector $\mathbf{s}$, the ideal image isBelow the Rayleigh criterion this still differs from $Fp(\mathbf{x})$; for small $\mathbf{s}$, symmetry cancels the linear change and the first change is quadratic in separation. Estimating two unknown source positions is then increasingly sensitive to noise, background, relative brightness and uncertainty in the point spread function. Localizing one isolated source and recovering an arbitrary extended object are different inference tasks. Measurements that separate optical modes can also extract separation information differently from direct image light intensity measurement.^[ref]

There remains a real limit for arbitrary images in this model: the optical transfer function is zero above the transmitted spatial frequency band. Its cutoff for scalar incoherent imaging is ${\nu }_c=2\mathrm{NA}/{\lambda }_0$ in object-plane cycles per unit length. The Nyquist–Shannon sampling theorem therefore suggests object-referred detector pixel pitch $p_{\mathrm{pix}}\le {\lambda }_0/(4\mathrm{NA})$, with finer sampling providing margin. Finite detector pixels also integrate over their area, suppressing contrast; coarse sampling can cause aliasing. Finer sampling cannot alone recover spatial frequencies removed by the optics, and Fourier deconvolution amplifies noise where the optical transfer function is small. Photon shot noise, read noise, optical aberrations and defocus can all worsen practical discrimination. A useful resolution claim therefore reports the object model, criterion, wavelength, numerical aperture, sampling and signal-to-noise ratio in photon counting.

George Elwood Smith reports fabricating MOS capacitors, thermally generating the first packet, and clocking it to an output that pushed it into the substrate for electric current detection. A later integrated eight-bit device demonstrated imaging. Dark or reversed-clock control traces are not specified; feasibility alone does not calibrate charge-transfer efficiency.^[ref]

A discriminating repeat would compare identical cycles with and without initial carrier injection, then suppress or reverse the transfer sequence while retaining output switching. An electric current transient synchronized with gate switching can include capacitor feedthrough; only the difference tied to initial electric charge and the expected transfer delay tests packet transport. These are proposed controls, not controls documented in the lecture. For an optical version, a blocked exposure estimates dark current and read noise; a spatially isolated illuminated optical detector pixel exposes any deferred charge in CCD readout trail. Photon shot noise persists even with ideal transfer.

Flat-field correction compensates stable multiplicative variations in photodetector response using calibrated illumination. Identically illuminated exposure ratios cancel the stable response in their mean, but their photon shot noise remains signal dependent.

The inspected apparatus sends picosecond diode-laser pulses into a sample cuvette. A neutral attenuation filter keeps detected events rare; an emission cut-off filter rejects scattered excitation. Collection optics deliver fluorescence to a photodetector; detector pulses and the laser-driver synchronization signal feed the timing electronics. The illustrated arrangement uses a PDL 800-B driver and TimeHarp 200 electronics. The note's recorded comparison uses two dye solutions and a prompt instrument-response trace; the dye tails have distinct slopes on a logarithmic display. Its Figure 9 measurement specifies an 80 MHz source, separately from the Figure 8 driver-setting example.^[ref]

For a defensible new measurement, separately acquire detector background with excitation blocked, a solvent/container blank under excitation, and a prompt response appropriate to the timing channel. Compare successively attenuated acquisitions to expose pile-up; compare early and late acquisition intervals to expose photobleaching. Choose a repetition period long enough for the relevant decay or include residual population from earlier pulses in the model. These are proposed checks, not additional controls claimed for the published curves. Photon shot noise, background mismatch, and uncertainty in the fluorescence instrument response limit how uniquely multiple decay components can be inferred.

High-contrast imaging seeks faint objects near a bright source. Residual point spread functions and speckle patterns can dominate photon shot noise; angular differential imaging and simultaneous spectral differential imaging use source motion or spectral diversity to distinguish them.

An optical photodetector detector pixel is a spatially sampled photosensitive element. Counts from distinct detector pixels have independent photon shot noise in the ideal model, but charge transfer, electrical coupling and resampling can produce covariance.

A single fluorophore creates a diffraction-broadened point-spread function, but its centre can be fitted much more precisely than that function's width. With Gaussian width $s$ and $N_{\gamma }$ detected independent photons, the shot-noise limit is ${\sigma }_{\mathrm{loc}}\simeq s/\sqrt{N_{\gamma }}$. Pixelation, background photons, dye orientation, motion blur, drift and label-to-target geometry impose additional uncertainty and systematic error. Localization precision is therefore not identical to either optical resolution or position accuracy.^[ref]

In myosin-V tracking, localizing a label on one head or its lever arm distinguishes the paths of individual motor parts from the average motion of the whole motor. A trailing head that becomes the leading head travels roughly twice the stalk advance: about $74\mathrm{nm}$ versus a roughly $37\mathrm{nm}$ motor stride. A near-head label can therefore alternate a large movement with a pause while the partner head advances. Other lever-arm label positions give unequal alternating steps whose sum is approximately twice the stride. Those patterns support hand-over-hand walking rather than a simple inchworm model in which the same head remains in front. The label geometry is needed when interpreting individual step sizes.^[ref]

Super-resolution localization microscopy extends the same principle to a map containing many molecules by separating otherwise overlapping emitters in time. Sparse activation, blinking or transient binding yields individually fitted positions, which are accumulated. Dense sampling, registration and labelling specificity determine the final structural resolution; merely taking more photons from an unresolved simultaneous pair does not make it a single isolated emitter.

In PAINT, fluorescent probes bind transiently so that separated binding events can be localized and accumulated. DNA-PAINT uses reversible complementary-strand recognition at docking sites on a DNA nanostructure. The bound probe supplies a localized spot while freely diffusing probes contribute background; repeated visits build up a positional map, with replenishment rather than permanent reliance on one fluorophore. This is an imaging principle, not a requirement for a particular target sequence or preparation protocol.^[ref]

Measure counts in electrons so that photon and detector noise can be combined directly. For source electrons $S$, $n$ aperture pixels, total time $t$, sky rate $b$ per pixel, dark-current rate $d$ per pixel, $M$ reads with rms read noise $r$ per pixel, and an independently estimated mean background per pixel of variance ${\sigma }_{\hat{B}}^2$, an ideal aperture-sum budget isThe last term represents the propagated calibration uncertainty if it can be treated as a random variance; systematic residuals can instead bias the measurement and need an explicit error model. For a background estimated from $n_b$ independent, equally exposed sky pixels, one approximation is ${\sigma }_{\hat{B}}^2=[(b+d)t+M{r}^2]/n_b$. Correlated resampling, detector cross-talk or a spatially structured sky invalidates the independent-pixel simplification.

Photon counting and sky. Source photon shot noise has variance $S$ and cannot be calibrated away. Longer integration or greater throughput improves its fractional error as $S^{-1/2}$. Sky photons, including moonlight, airglow and scattered light, add their own Poisson fluctuations. A dark site, appropriate observing time and filter, good baffling and a smaller or optimally weighted source aperture reduce sky contributions. Better seeing or adaptive optics can concentrate a point source into fewer noise-bearing pixels. Sky subtraction removes the mean background but not the random sky photons already recorded, and the sky estimate adds uncertainty of its own.

Dark and read noise. Thermal dark current contributes a mean and shot noise. Cooling reduces its rate; well-matched dark frames remove a reproducible mean and map hot pixels, but do not undo the science exposure's random dark-current fluctuations. Each read adds electronic noise. At fixed total integration, many short exposures add more read variance than fewer long exposures, so use exposures long enough to become sky limited when saturation, cosmic rays and temporal changes permit. Multiple non-destructive reads can help on detectors that support them. Subtracting a bias frame corrects an electronic offset, not the stochastic read noise.

Calibration and non-Poisson effects. Pixel sensitivity differences, illumination gradients, fringes, nonlinearity, imperfect charge transfer and electronic pattern noise leave calibration errors. High-count flat fields, stable operating conditions, linearity calibration, dithering and appropriate detector models help, while the finite signal-to-noise of calibration frames propagates into the result. Cosmic-ray hits, hot pixels, satellite trails and scattered-light artefacts are not ordinary stationary Gaussian noise: identify or mask them, often using several exposures. Overaggressive clipping can remove real transient or compact-source signal.

Atmospheric scintillation adds correlated intensity fluctuations, especially in short bright-star measurements; seeing changes and guiding errors move flux between pixels or apertures. Differential photometry using appropriate reference stars and stable apertures reduces common variations but does not guarantee removal of chromatic or position-dependent effects. For a constant flux-limited source in independent stationary noise the final signal-to-noise usually grows as $\sqrt{t}$ once background limited; a systematic floor or correlated sky can stop that improvement. Distinguish improvement of calibration, which removes reproducible bias, from elimination of intrinsically random photon arrivals, which is impossible.

The number operator is $\hat{N}=a^{†}a$. The number state expansion gives $a|\alpha ⟩=\alpha |\alpha ⟩$ by shifting the summation index; taking the adjoint gives $⟨\alpha |a^{†}={\alpha }^{\ast }⟨\alpha |$. Thus the expectation value isFor the second moment, use $a{a}^{†}=a^{†}a+I$:Its expectation value in the coherent state is $⟨{\hat{N}}^2⟩=|\alpha {|}^4+|\alpha {|}^2$. ThereforeThe variance equals the mean, exactly as for the Poisson distribution found in b. For $\mu >0$ the fractional standard deviation is $1/\sqrt{\mu }$, the usual photon shot noise scaling. For the vacuum state, both the mean photon number and its variance vanish; this does not imply zero optical quadrature fluctuations. At $\mu =0$ the fractional fluctuation is undefined.

The fundamental angular floor is the diffraction limit of a telescope, of order $\lambda /D$. The ideal diffraction grating has $R_g=mW/d$, whereas its slit-limited resolving power has $R\theta D=mW\lambda /d$. Requiring $R\le R_g$ givesThe coefficient depends on the adopted line-resolution and aperture conventions. Below this scale, a narrower spaxel oversamples the same spatial mode rather than creating a new independently resolved element. Equivalently a spatial optical mode occupies etendue of order ${\lambda }^2$.

There is also an engineering floor before arbitrary shrinkage: the optical camera focal length, photodetector size and support clearances do not shrink, while the number of photodetectors increases as ${\theta }^{-2}$. Thus the shrinking-beam model's constant combined optical camera volume is eventually overtaken by fixed per-camera overheads. For fixed brightness, smaller spaxels receive fewer photons, so read noise and photon shot noise can constrain useful sampling before the formal optical floor. Diffraction limits independent spatial information; fixed photodetector and mechanical dimensions limit practical volume reduction.

A stable multiplicative sensitivity factor $s_i$ at detector pixel $i$ changes both means to $N_i=s_i{N}_0$. The noise-free ratio is still one, so flat-field correction is not required merely to remove that fixed pattern from the mean ratio. This is the principal advantage of using two matched flats rather than the variance of a single image.

However, the photon shot noise is not cancelled. At high counts the conditional ratio variance is $2/(g{N}_i)$. Pooling detector pixels gives $E^2\simeq (2/g)⟨1/N_i⟩$, where brackets denote a spatial average. If the formula uses $\overline{N}=⟨N_i⟩$, it returnsThe inequality follows from the Jensen inequality for $1/x$ on positive signals. For small fractional response variation, the relative bias is approximately minus its squared coefficient of variation. A locally nearly uniform region or a fit using detector pixel-dependent variances avoids this bias from mixing the harmonic mean with the arithmetic mean. Stable response cancellation also presumes the conversion gain itself is uniform; gain variations, offset errors or changing sensitivity do not obey that simple cancellation. The fixed response pattern cancels in the ratio mean; unequal shot-noise levels can still bias a globally pooled gain estimate.

Both methods reduce structured stellar residuals, rather than removing all noise. In angular differential imaging, changing atmospheric turbulence, imperfect adaptive optics, flexure and thermal drift change the point spread function between frames. The reference then fails to represent the instantaneous stellar field. Small sky rotation makes close companions contaminate their own reference, causing differential-imaging self-subtraction. Extended disks are especially susceptible; subtraction can alter shape as well as total flux. More images help independent photon shot noise, but do not necessarily average away correlated residuals.

Simultaneous spectral differential imaging avoids the time delay, but different channels have non-common-path wavefront errors. Chromatic optical aberrations, wavelength-dependent amplitude errors and out-of-pupil propagation prevent a perfect radial rescaling of speckle patterns. Filter throughput, detector calibration, image registration and atmospheric dispersion also leave subtraction residuals. Nearby bands align the stellar field better but give less positional diversity; wider separation gives more displacement but larger chromatic mismatch. A smooth-spectrum companion, or one too close for appreciable rescaled displacement, can undergo severe differential-imaging self-subtraction.

Artificial-companion injection through the complete processing pipeline and forward modelling can calibrate lost throughput and photometric or astrometric biases. They do not guarantee that every correlated residual is a real source. Independent epochs or spectral evidence remain valuable.

Let $N$ be the input photoelectron count and $\mu =\eta Q$ its mean, with quantum efficiency $\eta$ and mean incident photon count $Q$. Independent arrivals give a Poisson distribution, so $\mathrm{Var}N=\mathbb{E}N=\mu$. Conditional on $N$, the preceding gamma distribution gives $\mathbb{E}[X\mid N]=gN$ and $\mathrm{Var}(X\mid N)=g^{2}N$. The law of total variance therefore yieldsOne contribution is the ordinary photon shot noise; the other is multiplication excess noise. Thus, neglecting read noise and backgrounds,This is an excess-noise factor $\sqrt{2}$ in analog operation: actual photon conversion efficiency has not been halved. At low occupancy, thresholding each pixel as zero or one event can avoid most multiplication noise, but high arrival rates produce coincident events that cannot be counted separately. That is why the high-rate result concerns charge measurement rather than ideal binary photon counting.

Increasing exposure reduces fractional photon shot noise as $T^{-1/2}$, but a fixed background mismatch grows in proportion to the signal. The fractional systematic error is $|1-f|R_B/R_Q$, independent of exposure. The systematic-error signal-to-noise ceiling can therefore be poor for a faint source in a bright sky even when random fluctuations are tiny.

Better background subtraction requires matching sky location and time, correcting detector response, dithering or modelling spatial background variations. If $f$ were known exactly, one could instead use $X-Y/f$, which is unbiased and has variance $Q+B+B/f$. The ceiling arises from an uncorrected or unknown mismatch in the prescribed subtraction, not an unavoidable property of measuring two patches.

To reach an accuracy ratio $Z_{\ast }$ at all, a necessary condition in the fixed-mismatch model isEquality allows only an asymptotic approach; finite exposure adds positive random error. More exposure improves precision, but only better calibration removes the fixed background bias.

Treat $Q$ and $B$ as mean detected counts, or as photon counts with unit quantum efficiency. Let the independent patch measurements be $X\sim \mathrm{Poisson}(Q+B)$ and $Y\sim \mathrm{Poisson}(fB)$. For the specified unweighted background subtraction $\hat{Q}=X-Y$,The first error is a fixed bias of an estimator; the last expression is the sum of independent photon shot noise variances. Since $Q,B$ are already patch totals, no extra factor of the pixel count $n$ is needed, and read noise is neglected.

To obtain the systematic-error ceiling requested in the following clause, define the accuracy measure $Z$ using total root-mean-square error relative to the true source count. By the bias-variance decomposition of mean squared error,For $f\simeq 1$, the shot-noise term is approximately $Q+2B$, but the mismatch term must be retained.

There is a terminology qualification: the usual variance-based signal-to-noise ratio in photon counting is $Q/\sqrt{Q+(1+f)B}$ and does not include a fixed bias as noise. The printed next-part limit requires the root-mean-square accuracy convention above. A deterministic background mismatch contributes $b^2$ to mean squared error, not to the statistical variance.

The optical telescope supplies collecting area and spatial selection, the spectral selector supplies a passband, and the photodetector converts received radiant flux into a measured signal. For an extended source, a useful schematic photon count iswhere $A\Omega$ is the etendue, $\tau$ the integration time, $\eta$ the net transmission and quantum efficiency, and $L_{\lambda }$ the spectral radiance. An illustrative independent-noise model givesHere $N_b$ and $N_d$ denote background and dark counts, and ${\sigma }_{\mathrm{read}}$ is readout standard deviation. This makes the roles of the three components explicit.

A larger optical telescope collects more photons at fixed angular pixel size and improves the diffraction-limited resolution. Its limits include optical aberrations, alignment, launch size, stray light and pointing. Conservation of etendue constrains simultaneous collecting area and angular acceptance. Increasing spatial resolution by narrowing a pixel's solid angle reduces its photons.

A narrower spectral passband or higher spectral resolving power separates diagnostic features and rejects unwanted background, but usually reduces photons per channel and hence signal-to-noise ratio. Selector transmission, leakage, slit width, detector format and the required time cadence limit the trade. A wider band helps only insofar as it adds useful signal rather than mostly background.

The photodetector's quantum efficiency, dark current, read noise, dynamic range and response time determine conversion losses and added noise. Cooling can suppress thermal background and dark current, but cannot remove photon shot noise. Longer integration improves a stationary signal's signal-to-noise ratio until motion, saturation, drift or required cadence becomes limiting.

The Rayleigh criterion calls two equal, mutually incoherent optical point sources just resolved when one circular-pupil Airy disk's centre coincides with the other's first zero. It is a specified discrimination convention, distinct from full width at half maximum and from precision in estimating one source's position.

For this circular optical pupil, the Rayleigh criterion uses angular separation $s_{\mathrm{R}}={\theta }_1\simeq 1.22\lambda /D$, or $s_{\mathrm{R}}=r_1\simeq 1.22\lambda f/D$ in the paraxial approximation to the focal plane. The point spread functions still overlap.^[ref]

For the Airy disk design example with $D=5.00\mathrm{mm}$, $f=100\mathrm{mm}$ and $\lambda =550\mathrm{nm}$, the angular convention gives $s_{\mathrm{R}}=1.34164\times 10^{-4}\mathrm{rad}$, or about $27.67$ arcseconds, and the image-plane separation is $13.4164\mu \mathrm{m}$. Two equally bright, mutually incoherent optical point sources at separations $0.7$, $1$ and $1.5$ times this scale would have image separations $9.391$, $13.416$ and $20.125\mu \mathrm{m}$ respectively. These calculated cases are compared in incoherent two-source Airy profile.

The Rayleigh criterion is useful because it ties an easily recognized two-peak convention to a predictable optical scale. It does not assert that all information vanishes below that separation. Inferring two unknown positions also depends on photon shot noise, background, unequal brightness and uncertainty in the point spread function; locating a single isolated source is a different task. A theoretical analysis shows that alternative optical measurements can estimate the separation of two incoherent optical point sources below this scale.^[ref]

The mutual optical coherence condition is essential. In incoherent imaging, time-averaged light intensities add. In coherent imaging, wave amplitudes add before squaring, leaving a wave interference term controlled by relative wave phase. Two holes illuminated by the same coherent laser therefore do not implement the equal-incoherent-source convention merely because their individual patterns resemble Airy disks. The coherent single-aperture laboratory test establishes the response of each aperture; it does not establish the pair's mutual incoherence.