For a homogeneous bosonic condensate, the one-body density matrix retains a nonzero long-distance component. More generally, macroscopic occupation is identified by an eigenvalue of that density matrix proportional to particle number. This gives a number-conserving description even when the one-point annihilation-field expectation vanishes. In a paired fermionic system the corresponding macroscopic order belongs to the two-particle density matrix, rather than an electron single-particle condensate.

A localized temporal wave envelope supported by a balance of group velocity dispersion and self-phase modulation. Bright solitons decay to zero background; dark solitons are dips on a nonzero background. This temporal balance differs from the diffraction balance of a spatial optical soliton.

Lateral resolution describes distinguishable separation in a plane perpendicular to an imaging system's axis. For incoherent point objects imaged through a circular aperture, the Rayleigh criterion gives the specimen-space scale $d\simeq 0.61\lambda /\mathrm{NA}$, where $\lambda$ is the vacuum wavelength and $\mathrm{NA}$ the numerical aperture.^[ref]

For example, $\lambda =550\mathrm{nm}$ and $\mathrm{NA}=1.0$ give $d\simeq 336\mathrm{nm}$ under that criterion. Sampling the image with more pixels can represent this blur more accurately, but cannot by itself distinguish closer features. Comparisons between instruments must specify both the object and the criterion.

Axial resolution describes the ability to distinguish features separated along an imaging system's axis. In optical microscopy, the three-dimensional point spread function is usually longer along that axis than across it, so two layers can overlap even when comparable sideways separations are distinguishable.

A small fluorescent bead moved through focus provides an experimental response curve: record the detected intensity at successive axial positions and measure a stated width, such as its full width at half maximum. The result depends on numerical aperture, wavelength, specimen refractive index and the detection method. Confocal microscopy can reject much of the signal from outside the focal plane, improving optical sectioning.

The resolution scale imposed by a finite-aperture wave-imaging system under a stated criterion. For conventional optical microscopy, a common lateral scale is $\lambda /(2\mathrm{NA})$. Changing the illumination, measured signal or reconstruction assumptions can change the applicable criterion.

For real $ϵ$ with $|ϵ|\le 1$, let $|\psi ⟩=\sqrt{1-{ϵ}^2}|0⟩+ϵ|1⟩$. The Fock state matrix elements give $⟨a⟩=ϵ\sqrt{1-{ϵ}^2}$, $⟨a^2⟩=0$ and $⟨a^{†}a⟩={ϵ}^2$. ThereforeOnly the first of these optical quadratures is squeezed for this real superposition, with minimum variance $3/16$ at ${ϵ}^2=1/4$. The second is never below vacuum noise. This non-Gaussian superposition is distinct from an optical squeezed vacuum.

With $X_{\vartheta }=(a{e}^{-i\vartheta }+a^{†}{e}^{i\vartheta })/2$, the coherent-state and vacuum variances are $1/4$. A state is squeezed in the chosen quadrature when its variance about its own mean is smaller. A second moment about zero is not the variance when that mean is nonzero. balanced homodyne detection measures a phase-selected quadrature distribution.

For a single-mode Gaussian state with no quadrature covariance, the covariance contour has semiaxes $\sqrt{\mathrm{Var}X}$ and $\sqrt{\mathrm{Var}P}$. A coherent state has both equal to $1/2$ in the convention $[X,P]=i/2$. A pure squeezed vacuum state has semiaxes $e^{-r}/2$ and $e^r/2$; the uncertainty area remains fixed while one noise width decreases. An ellipse of means alone does not specify a general non-Gaussian state.^[ref]

For one optical mode, $E=E_0(a+a^{†})$ and $[a,a^{†}]=1$. A coherent state obeys $a|\alpha ⟩=\alpha |\alpha ⟩$, so normal ordering gives $⟨E^2⟩=E_0^2[(\alpha +{\alpha }^{\ast }{)}^2+1]$. Subtracting the squared mean leaves $E_0^2$, independent of the coherent displacement. The extra one is the canonical commutation relation, not noise from a random coherent amplitude. A rotated optical quadrature has the same coherent-state variance with the same amplitude normalization.

A Gaussian pulse has a field or intensity envelope specified by a Gaussian function; the distinction must be stated. For the displayed intensity, the temporal full width at half maximum is $2\sqrt{2\ln 2}{\sigma }_t$. A transform-limited pulse with this shape has intensity spectral and temporal widths satisfying $\Delta \nu \Delta t=2\ln 2/\pi$. A quadratic spectral phase can lengthen the pulse without changing its spectral intensity.

Measure the spectrum of a nonlinear optical gate signal as its relative delay is varied. A retrieval algorithm seeks a complex pulse consistent with the resulting two-dimensional intensity record and the particular gate law. Different gate laws have different ambiguities and response limitations; a small fitting residual alone does not guarantee stable pulses or correct calibration.

A microscope that uses controlled incident and analysed polarization of an electromagnetic wave to study specimens. Crossed polarizers reveal orientation-dependent phase retardation in a birefringent specimen.

Mix a signal mode $a$ and an independent coherent reference mode $b$ on a balanced beam splitter, detect both outputs, and subtract the currents. For $c=(a+ib)/\sqrt{2}$ and $d=(ia+b)/\sqrt{2}$, the difference is $I=c^{†}c-d^{†}d=i(a^{†}b-a{b}^{†})$. A reference $\beta =|\beta |e^{i\varphi }$ gives mean $2|\beta |⟨X_{\vartheta }⟩$ with $\vartheta =\varphi +\pi /2$. Other beam-splitter conventions shift this phase calibration. The exact finite-reference balanced homodyne variance quantifies the strong-reference approximation.

Matched subtraction cancels the shared total-power term. It therefore suppresses common reference intensity noise, while imbalance and phase fluctuations reduce this advantage. Repeated samples measure a distribution, not just its mean; scanning reference phase supports optical homodyne tomography.

Optical homodyne tomography reconstructs an optical quantum state from repeated optical quadrature measurements at different phases. A strong coherent local oscillator interferes with the signal on a beam splitter; balanced homodyne detection subtracts the two detector outputs to estimate the quadrature selected by the reference phase.

Many identically prepared signals give a distribution for that quadrature. Repeating at several phases supplies complementary distributions from which a density matrix or optical Wigner function can be estimated. One quadrature distribution alone generally does not determine the state, just as a single projection of a three-dimensional object does not reveal its full shape. Detector efficiency, mode matching and phase noise must be included in the measurement model; otherwise apparent loss or excess noise can be assigned incorrectly to the signal state.

A coherent reference field that selects the measured optical quadrature through its relative phase. Mode matching and phase stability determine how faithfully the desired signal optical quadrature is sampled.

The common optical halo has a radius of approximately $22^{\circ }$ around the Sun or Moon. Refraction through a $60^{\circ }$ ice-crystal prism supplies a minimum-deviation edge. Nearby incident directions are concentrated near that edge, forming a caustic.^[ref]

Broadband sunlight has a short longitudinal coherence length, controlled by spectral bandwidth, but a wider transverse coherence region because the solar disk has a small angular size. The scales are ${\ell }_c\sim c/\Delta \nu$ and $w_c\sim \lambda /\delta$ for angular diameter $\delta$. Filtering changes the comparison.

For an unresolved incoherent luminous disc of angular diameter $\alpha$, transverse optical coherence has scale $\lambda /\alpha$; a uniform circular brightness profile puts its first zero at $1.21967\lambda /\alpha$. A receiver transmitting a continuous frequency band of width $\Delta \nu$ instead has temporal coherence scale $1/\Delta \nu$ and longitudinal length $c/\Delta \nu$. Angular source size and spectral filtering control different correlations. Atmospheric phase fluctuations and instrument averaging can further reduce measured fringe visibility.

For allowed direct transitions between three-dimensional parabolic bands with a slowly varying dipole matrix element, the joint state density grows as $\sqrt{\hslash \omega -E_g}$. Linear absorption is proportional to that factor divided by $\omega$, before Coulomb enhancement and broadening are included.