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First intensity minimum. In Fraunhofer diffraction, the path difference between adjacent slits is $d\sin \theta$, so the relative wave phase is $\delta =2\pi d\sin \theta /{\lambda }_1$. Summing the $N$ equal complex amplitudes as a geometric series gives the diffraction grating interference factorThe central maximum has $\delta =0$. Its nearest zero occurs at $N\delta /2=\pi$, where the denominator is nonzero. ThusThere is a symmetric first zero on each side of the incident direction. The narrow slits are treated as having a sufficiently broad single-slit diffraction envelope.

Wavelength resolution and the missing order specification. Adopt the Rayleigh criterion: the principal maximum of one wavelength lies at the adjacent minimum of the other in the same nonzero diffraction order $m$. For ${\lambda }_2>{\lambda }_1$ and $m>0$, those directions obeyConsequently the ideal resolving power of a diffraction grating is ${\lambda }_1/\Delta \lambda =mN$, givingThe usual first-order reading therefore gives $64\mathrm{nm}$. Zero diffraction order cannot separate the two wavelengths, because both central maxima coincide.

The paper does not specify which diffraction order is observed, so $64\mathrm{nm}$ is not an order-independent minimum. In the ideal infinitely narrow-slit model, the largest usable order for this positive wavelength separation must satisfy $(m+1/N){\lambda }_1\le d$. Since $d/{\lambda }_1=125$, this permits $m\le 124$, giving the formal all-order resultThat pair meets the criterion at $\theta \simeq 83.1^{\circ }$. Order $125$ of ${\lambda }_1$ is at grazing incidence and the corresponding longer-wavelength peak cannot propagate. In a real device, finite slit width, weak high-order intensity and screen acceptance may prevent such an extreme order from being used; those data are absent. Thus the specified-order expression, and $64\mathrm{nm}$ when first order is intended, are the defensible resolution answers under the stated criterion.

For a diffraction grating with $N$ illuminated lines in order $p$, its ideal spectral resolving power isThe two spectral lines have separation $\Delta \lambda =3.4\mathrm{nm}$ and mean wavelength $\overline{\lambda }=395.1\mathrm{nm}$. In first order, $p=1$, so the Rayleigh criterion requiresWith line density $500\mathrm{m}{\mathrm{m}}^{-1}$, the ideal continuous estimate for the illuminated width isRounding the required number of illuminated grating lines upward gives $117$ lines, conventionally an illuminated width $W\simeq 117/(500\mathrm{m}{\mathrm{m}}^{-1})=\boxed{0.234\mathrm{mm}}$. Using the longer wavelength rather than the mean also gives the same integer requirement. This is the ideal limit set by the resolving power of a diffraction grating; additional broadening from the illumination or observing apparatus could require a larger width.

The etendue is the area–solid-angle acceptance of a ray bundle, more generally $\mathcal{G}=n_{mopt}^2\int \cos \theta dAd\Omega$. In an ideal lossless optical system it is conserved. Increasing pupil area increases accepted power only together with the appropriate field and detector cone; passive optics cannot arbitrarily increase radiance. For a small nadir pixel here, $A\Omega$ denotes entrance-pupil area times the solid angle of its surface footprint.

Choose $1.00$–$1.03\mu \mathrm{m}$ within the surface-sensitive window. A representative band-mean total radiance read from the original blue curve is ${\bar{L}}_{\lambda }\simeq 4\mu \mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}\mu {\mathrm{m}}^{-1}$, uncertain at roughly the tens-of-per-cent level. Its band integral isThe provided SNR formula depends on $L\Delta \nu$ only through this band integral. If $\nu$ is wavenumber, the selected band spans about $291\mathrm{c}{\mathrm{m}}^{-1}$ and $L_{\nu }d\nu =L_{\lambda }|d\lambda |$; numerical radiance densities in the two coordinates must not be mixed. Convert the specific detectivity to SI: $D^{\ast }=5\times 10^9\mathrm{m}\mathrm{H}{\mathrm{z}}^{1/2}{\mathrm{W}}^{-1}$ and $\sqrt{a}=2\times 10^{-4}\mathrm{m}$. Using the footprint dwell time from part (c),A square $200\mathrm{m}$ footprint at altitude $259\mathrm{km}$ subtends $\Omega \simeq (200/259000{)}^2=5.96\times 10^{-7}\mathrm{sr}$. Thus the collecting area and circular aperture diameter areThe Rayleigh criterion independently requires only $D\gtrsim 1.22\lambda /(200/h)\simeq 1.6\mathrm{mm}$ at $1.015\mu \mathrm{m}$, so the noise requirement dominates diffraction. The $200\mu \mathrm{m}$ detector size corresponds to focal length about $0.259\mathrm{m}$ for this pixel field. This assumes one detector footprint per stated feature size; stricter spatial sampling would need smaller footprints and a revised photon budget.

The estimate assumes a clear, non-scattering atmosphere as in the graph, a night-side or otherwise isolated thermal signal, negligible extra stray-light and read-noise penalties beyond the supplied detectivity, and transmission $0.7$ for the instrument alone. Atmospheric attenuation is already included in the plotted incoming radiance. The selected band has little atmospheric-only signal, but actual hot-spot detection concerns contrast relative to the background: SNR $50$ on total radiance is not a guarantee for an arbitrarily small temperature anomaly. An illustrative radiance range $3$–$5$ in the plotted units would give diameters about $0.20$–$0.15\mathrm{m}$.

A calibrated multispectral filter radiometer is a natural choice for rapid thermal mapping: a few selected bands provide higher photon throughput and faster imaging than spreading the light over many spectrometer channels. Several bands can help separate temperature, dust emissivity and atmospheric effects. A spectrometer would be justified if identifying composition or resolving spectral structure were a central additional objective; one broadband radiance alone cannot generally determine both temperature and unknown emissivity.

Thermal infrared radiation is preferable to reflected visible light for measuring temperature. The Rayleigh criterion setsA band near $10\mu \mathrm{m}$ is therefore a reasonable compromise between useful planetary thermal emission and the required diffraction-limited resolution. At that wavelength, the nominal surface Rayleigh criterion separation is about $415\mathrm{m}$; at $15\mu \mathrm{m}$ it is about $623\mathrm{m}$ and fails the target. The exact bands must avoid or model strong atmospheric absorption and account for the dust spectrum. The $500\mathrm{m}$ sampling pitch is still not a guarantee of faithful reconstruction of arbitrary $500\mathrm{m}$ features: detector sampling and the point spread function must be considered together.

Ignoring refraction, the line of sight is perpendicular to the radius at its tangent point. For a surface tangent, Pythagorean theorem gives the satellite-to-tangent distanceFor tangent altitude $z_t$, replace $R^2$ by $(R+z_t{)}^2$; lower atmospheric scale heights change the estimate little. The required angular resolution isUsing the Rayleigh criterion for a circular telescope aperture, $\theta =1.22\lambda /d$. The longest wavelength imposes the strongest condition, soThe corresponding requirement at $7\mu \mathrm{m}$ is $3.3$ mm. This is a diffraction limit, without allowance for scan sampling, aberrations or instrumental field of view.

For an ideal circular pupil, the point spread function along a central line isHere $x$ is image-plane position and the central ratio is interpreted by its limit at $q=0$. The Airy disk has a bright central maximum, a first zero at object-space radius $0.61\lambda /\mathrm{NA}$ and weaker side lobes. The Rayleigh criterion gives two-source optical resolution $d\simeq 0.61\lambda /\mathrm{NA}$. At an illustrative $\lambda =550\mathrm{nm}$ this is $258\mathrm{nm}$ in object space or $25.8\mu \mathrm{m}$ in the image. No unique dimensional width follows without a wavelength. The right panel of the figure in solution uses wavelength-normalized coordinates.

For one isolated emitter of known point spread function, fitting its centre to many detected photons estimates a position far more precisely than the width. For approximately Gaussian width $s$ and $N$ independent signal photons, the ideal localization standard deviation scales as $s/\sqrt{N}$. Finite pixels, background and read noise, aberrations, uncertain point spread function, drift, bleaching and overlapping emitters worsen this precision. Single-source localization precision is a different task from resolving two unknown nearby sources.

At maximum projected separation, an Earth analogue at $10\mathrm{pc}$ is separated from its star by $\vartheta =a/D_{\mathrm{sys}}=0.1\mathrm{arcsec}=4.85\times 10^{-7}\mathrm{rad}$. The Rayleigh criterion for a circular pupil givesThe relevant quantity is the star–planet separation, not the tiny angular diameter of the planet itself. Simple angular optical resolution is insufficient: the planet lies in the star's diffraction pattern at reflected contrast of order $10^{-10}$, or thermal contrast of order $10^{-6}$. Residual speckles, wavefront errors, scattered light, detector/background noise and, in the infrared, strong telescope and sky thermal emission obstruct detection. A coronagraph or other stellar suppression method generally has an inner working angle of several $\lambda /D$, making practical aperture requirements larger than the bare Rayleigh criterion. The question's instrumentation claim is interpreted in its 2019 context; the physical requirements remain the basis of the estimate. Maximum reflected brightness near full phase also coincides with small projected separation, so maximum brightness and maximum resolvable separation cannot generally be obtained simultaneously.

The circumstellar habitable zone is the range of stellar irradiation at which a suitable rocky planet could maintain liquid water at its surface under specified atmospheric conditions. It is not a guarantee of life or even of liquid water on every planet at that distance. Its boundaries depend on greenhouse gases, clouds, humidity, pressure, albedo, rotation and the stellar spectrum; stellar evolution, ultraviolet activity, atmospheric loss and tidal locking can complicate applications to cool stars.

Approximating both stars as blackbodies, the host luminosity relative to the Sun isThe Earth-irradiance orbit therefore hasUse the largest projected separation and a visible reference wavelength of $550\mathrm{nm}$. The Rayleigh criterion for a circular aperture givesThis is only a diffraction-based minimum: detecting a faint planet beside its star additionally requires contrast suppression and adequate photon counts. An interferometer baseline, rather than its collecting-aperture diameter, sets its angular resolution.

For reflected light the reflected-light planetary phase curve gives the flux ratio $ϵ=A_g\Phi (\alpha )(R_{\oplus }/a_{\mathrm{E}}{)}^2$, where $A_g$ is geometric albedo and $\Phi (0)=1$. Numerically,A unit-geometric-albedo reference gives about $6\times 10^{-7}$; an Earth-like chosen $A_g=0.3$ gives $1.7\times 10^{-7}$. Albedo and phase function were not specified, so an exact contrast cannot be inferred just from the radius. Full-phase brightness occurs close to zero projected separation for an edge-on orbit; at the best-resolved quadrature, a Lambert sphere has $\Phi (\pi /2)=1/\pi$, reducing the Earth-albedo example to $5.4\times 10^{-8}$. Thus the best angular separation and maximum reflected contrast are not generally achieved at the same phase.

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.

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.

For $N$ coherently illuminated grating lines used in diffraction order $m$, the Rayleigh criterion yields the displayed ideal resolving power of a diffraction grating. The angular maximum-to-first-zero separation is $\lambda /(ND\cos {\theta }_m)$, while nearby wavelengths are separated by $|m|\delta \lambda /(D\cos {\theta }_m)$. The order must exist, carry appreciable light intensity and remain free from dominant instrumental broadening. Line count alone therefore does not determine whether a particular instrument resolves a specified doublet.