A local departure from periodicity can support a discrete mode inside a bulk photonic band gap. An isolated defect makes a cavity; a line of defects may guide light. Finite surrounding crystals allow leakage.

Subtracting a reproducible sky, dark or bias mean removes its offset but does not undo random arrivals or electronic fluctuations already present in a science exposure. An independently estimated calibration adds its own uncertainty. Longer integration reduces relative independent counting noise; correlated backgrounds and systematic residuals need separate models and may set an accuracy floor.

Ultraviolet photons can eject electrons from dust grains in interstellar gas. The electron's excess kinetic energy heats the gas through collisions. The efficiency depends on the grain population and electric charge, so absorbed ultraviolet power is not automatically equal to gas-heating power.

Subtracting two photocurrents can suppress a shared classical intensity fluctuation when the optical responses and electronic gains are matched. Independent shot noise contributions do not cancel: their variances add in the difference. With equal charge $q$ and mean currents $I_1,I_2$, the independent-arrival one-sided current power spectral density is $2q(I_1+I_2)$. Correlated optical inputs require their covariance as well.

A regime in which additional nitrogen oxides promote net tropospheric ozone production because NO availability limits peroxy-radical oxidation of NO. Reducing NOx then generally lowers ozone production, subject to transport and chemical history. At much larger NOx burdens, OH removal by nitric-acid formation can instead produce a VOC-limited ozone-production regime.

A regime with abundant nitrogen oxides in which volatile-organic oxidation and radical supply limit net tropospheric ozone production. Reducing volatile organic compounds lowers production. A small NOx reduction can initially increase ozone by weakening OH termination and NO titration; sufficiently large reductions can cross into an NOx-limited ozone-production regime. Local concentration and regional emission sensitivities need not coincide.

A contour plot of a specified ozone outcome against the loadings of nitrogen oxides and volatile organic compounds. The ridge separating opposite NOx sensitivities distinguishes an NOx-limited ozone-production regime from a VOC-limited ozone-production regime. The outcome may be peak concentration, a time-integrated measure or a model production rate; axes and averaging conditions must be stated. A schematic diagram is not a calibrated emissions-response prediction.

A regime in which additional available nitrogen oxides tends to increase net photochemical ozone production when other factors are fixed. Reducing nitrogen oxides generally suppresses that production, but local ozone concentration also depends on transport, deposition and NO titration.

A regime in which precursor/radical supply and NOx-dependent radical termination make net ozone production more responsive to additional reactive organic compounds than to additional nitrogen oxides. Reducing nitrogen oxides can initially raise ozone in some such conditions. Diagnose the chemical regime before extrapolating an emission reduction to every location.

A single acoustic branch $\omega =v|k|$ has thermal energy per length $U_T/\ell =\int dk\hslash v|k|/[2\pi (e^{\beta \hslash v|k|}-1)]=\pi (k_{B}T{)}^2/(6\hslash v)$. Differentiating gives the displayed linear heat capacity. The regime requires temperatures below the microscopic phonon cutoff but above the finite-size level spacing, with the translational zero mode treated separately.

For the nearest-neighbour monatomic-chain dispersion relation, ${\omega }_m=2\sqrt{\alpha /M}$ and $d\omega /dq=(a/2)\sqrt{{\omega }_m^2-{\omega }^2}$ on the positive half-zone. The folded wave-number count gives the displayed density of states, including both propagation directions. Its upper-edge Van Hove singularity is integrable: ${\int }_0^{{\omega }_m}g(\omega )d\omega =N$. A finite chain has discrete modes; the continuous density describes its large-size limit.

The residual wave vector difference in a wave-conversion process. For collinear sum-frequency generation, choose $\Delta k=k_1+k_2-k_3$. Contributions separated by $z$ then differ by wave phase $\Delta kz$. At phase matching this residual is zero. The sign convention changes some amplitude phases but not the unseeded conversion factor ${\mathrm{sinc}}^2(\Delta kL/2)$.

Compensation of phase mismatch by a spatial modulation of the coupling coefficient. Write $d(z)={\sum }_m{d}_m{e}^{imKz}$. The driven amplitude contains $e^{i(\Delta k+mK)z}$, so the selected Fourier component accumulates coherently when $\Delta k+mK=0$. For first-order compensation the full spatial period is $2\pi /|\Delta k|$, twice the nonlinear optical coherence length.

An ideal equal-duty pattern alternating $+d$ and $-d$ has first Fourier coefficient of magnitude $2d/\pi$. Hence its conversion intensity is $4/{\pi }^2$ of a uniformly phase-matched crystal using that same coefficient, under the same undepleted plane-wave assumptions. The modulation can nevertheless allow a larger tensor coefficient or an otherwise inaccessible wavelength. Periodically poled lithium niobate implements such sign reversals using ferroelectric domains.

In a common birefringent down-conversion arrangement, the two daughter photons have the same polarization class, ordinarily both ordinary or both extraordinary relative to the crystal. The pump has a suitable different index for phase matching. This classification alone does not establish polarization entanglement.

The daughter photons occupy orthogonal ordinary and extraordinary polarization classes. Alternative assignments to two collected paths can interfere if timing, spectrum and spatial mode do not distinguish them. spatial walk-off and unequal group delays often require compensation.

A diagram of straight segments joining two parallel lines, with distinct endpoints on each line. Ordering the endpoints along the two lines encodes a permutation; crossing segments define the associated permutation graph.

For $T>0$, values of a function $f$ on a half-open interval $[a,a+T)$ determine its periodic extension to the real line: choose the unique integer $k$ with $x-kT\in [a,a+T)$ and set $F(x)=f(x-kT)$. The resulting periodic function has period $T$. If the endpoint limits of $f$ differ, the extension has a jump discontinuity at every period boundary. Under the usual piecewise-smooth convergence hypotheses, its Fourier series takes the average of the two one-sided limits there; an assigned value at an isolated endpoint does not change the Fourier coefficients.

The linear carrier acceleration equation $\dot{\mathbf{j}}={ϵ}_0{\omega }_p^2\mathbf{E}$ and Faraday's law imply this conserved local quantity. Setting its initial value to zero gives the second London equation and London screening. An undamped response alone does not select that initial value, so it must not be used to infer expulsion of a pre-existing magnetic field for every magnetic history.^[ref]

A value $k(3k-1)/2$ with $k$ any integer, including negative values and zero. For $k=\pm 1,\pm 2,\dots$, positive values occur as pairs $j(3j-1)/2$ and $j(3j+1)/2$. These are the indices in the Euler pentagonal identity.