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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.

Background subtraction must account for uncertainty in the background measurement itself. In an original idealized experiment, an on-source exposure contains source and background, while an independent off-source exposure contains background alone. Assume stationary rates and identical background acceptance per unit live time. With $\alpha =t_{\mathrm{on}}/t_{\mathrm{off}}$, let independent counts obey Poisson distributions:Then linearity of expectation and addition of variances for independent random variables giveThe result concerns accepted source counts $S$, after all efficiencies. For large counts the normal approximation gives the descriptive signal-to-noise ratio in photon countingThis plug-in ratio is not an exact null-hypothesis significance. Small counts require the discrete Poisson model; searching many sky directions also changes a detection claim's p-value.

Choose $t_{\mathrm{on}}=100\mathrm{s}$ and $t_{\mathrm{off}}=200\mathrm{s}$, with observed counts $N_{\mathrm{on}}=1000$ and $N_{\mathrm{off}}=800$. Thus $\alpha =1/2$, $\hat{S}=600$, and the estimated standard deviation is $\sqrt{1200}=34.64$ counts. The inferred $F=\hat{S}/(A_{\mathrm{eff}}{t}_{\mathrm{on}})=0.300\pm 0.0173\mathrm{photons}\mathrm{c}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$ uses the preceding $A_{\mathrm{eff}}=20\mathrm{c}{\mathrm{m}}^2$ and includes counting uncertainty only. The descriptive ratio is $17.32$. Treating the scaled off-source background as exact would omit 200 counts squared from the estimated variance.

For a selection-design comparison, suppose before a veto the expected source contribution is $S_0=750$, the on-source background is $B_0=2000$, and $\alpha =1/2$. Let source acceptance be ${ϵ}_s=0.8$ and background acceptance be ${ϵ}_b=0.2$. Then $S=600$, $B=400$, and $S/\sqrt{S+(1+\alpha )B}=17.32$, whereas omitting the veto gives $750/\sqrt{750+3000}=12.25$. Rejection helps because the background is reduced enough to compensate for lost source events. A different veto can make matters worse; source efficiency and background rejection are separate probabilities. This comparison is an independent teaching example, not the efficiency calibration used above.

If both exposure times grow with fixed $\alpha$ and stationary rates, $S=s{t}_{\mathrm{on}}$ and $B=b{t}_{\mathrm{on}}$ imply $Z\simeq s\sqrt{t_{\mathrm{on}}}/\sqrt{s+(1+\alpha )b}$. Four times the live time doubles this ideal ratio. Misestimated background rates, uncertain X-ray effective area, and source variability can defeat that scaling; longer exposure does not remove systematic error.

With constant detected rates, $Q=R_{Q}T$, $B=R_{B}T$. Squaring the previous signal-to-noise ratio in photon counting givesFor $R_Q>0$, nonnegative background and target $Z>0$, choose the positive root:The other root is nonpositive. With no read noise, this reduces to $T=Z^2(R_Q+2R_B)/R_Q^2$; in a read-dominated short exposure it approaches $Z\sqrt{2n}r/R_Q$. If $R_Q=0$, no positive target stellar signal-to-noise is attainable. A zero target can be assigned zero exposure, with its limiting interpretation when all noise terms also vanish. Real quantum efficiency or multiple reads modifies the rates or read-variance term before solving the quadratic.

Put $Q=R_{Q}T$ and $B=R_{B}T$ in the root-mean-square accuracy measure:For a fixed uncorrected $f\ne 1$ and $R_B>0$, the squared bias eventually dominates the shot-noise terms. ThereforeThe PDF omits the absolute value. Its expression is the positive accuracy ratio only if $f<1$; for $f>1$ the residual background changes sign, but an error magnitude and a signal-to-noise ratio in photon counting remain nonnegative. The printed signed formula can instead be read as source divided by signed bias.

For exact background matching $f=1$, there is no systematic-error signal-to-noise ceiling: $Z=R_Q\sqrt{T/(R_Q+2R_B)}$ grows without bound in this idealized model. With $R_B=0$ the same conclusion holds. Limits in which $f$ itself approaches one with exposure time are different from the fixed-mismatch limit used here.

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.

For field of view, fiber-fed spectrographs commonly cover wider sky areas: fibers can pick targets across a broad focal plane while feeding a compact spectrograph with a fixed output slit. A multi-slit spectrograph must image its field through the spectrograph optics, and spectra must fit on the detector without overlap, which restricts both field and target layout.

For spectral resolution, fibers can feed an optimized, stable high-dispersion instrument, including an echelle grating. The fiber image acts as its entrance width. In a multi-slit spectrograph, slit width and dispersion similarly determine the spectral resolving power. Neither feed type alone imposes a universal resolution ranking: narrower fibers or slits improve resolution at the cost of losing source light, and both can be designed for high or low resolution.

For faintness limit, slit masks often have an advantage for individual faint objects because they avoid fiber coupling and transmission losses, permit a slit width matched to astronomical seeing, and sample local sky along the slit. Fibers can admit more sky through a fixed circular aperture and require sky subtraction from separate locations; focal-ratio degradation can also reduce throughput. The actual limit depends on throughput, aperture size, background stability, and detector noise through the signal-to-noise ratio in photon counting. Well-designed fiber instruments can nevertheless be very efficient for wide-field surveys, so the comparison is conditional on the optical design and observing conditions.

Sky brightness is the diffuse radiance received away from an astronomical target. Ground-based observations include atmospheric airglow, scattered light, and thermal radiation; extraterrestrial diffuse light also contributes. Its photon fluctuations contribute to the signal-to-noise ratio in photon counting.