Light

Light is electromagnetic radiation in the visible range; optics also studies neighboring infrared and ultraviolet radiation.

Redirection of incident light into outgoing directions at a material boundary or through scattering near it. Specular reflection concentrates light around a geometric reflection direction; diffuse reflection distributes it more broadly. This physical process is distinct from reflection as a mathematical transformation.

Transfer of energy from an optical electromagnetic wave into matter. Electronic absorption depends on transition matrix elements, occupied and empty states, and the joint density of states. Bound excitons can create resonances below an independent-particle continuum edge.

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.

An optical pulse is a finite-duration envelope of light. A nearly monochromatic pulse has many carrier oscillations within its duration $\tau$: in units with light speed one, its wavelength obeys $\lambda \ll \tau$. A slowly varying propagation-time map stretches the pulse duration by its derivative and divides the observed frequency by the same factor.

// Target: general-relativity.bigb

A flying focus is an optical pulse whose position of maximum focusing varies with time because different frequencies focus at different axial locations and arrive at different times. For a broad incident Gaussian beam, $f(\omega )=f_0+\beta (\omega -{\omega }_0)$ and spectral phase $\psi =\eta (\omega -{\omega }_0{)}^2$ give $z=f(\omega )$ and $t={\psi }^{\prime }(\omega )+z/c$, hence $v_f/c=[1+2\eta c/\beta {]}^{-1}$. This is the velocity of a focusing pattern, which can exceed $c$ or be negative without individual portions of the field propagating faster than light.

The carrier-envelope phase locates the carrier oscillation relative to the envelope of an optical pulse. In $E_{\mathrm{real}}(t)=F(t)\cos ({\omega }_{0}t+{\varphi }_{\mathrm{CE}})$ with envelope maximum at zero, it is the displayed phase modulo $2\pi$. Few-cycle dynamics can depend on it even when cycle-averaged light intensity is unchanged. A pulse train's phase increment is related to its carrier-envelope offset frequency; signs depend on the complex-field convention.

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.

An optical Kerr gate rotates a probe's polarization in proportion to a delayed replica's intensity in the weak-gate regime. Resolving the gated spectrum gives a pulse-characterization record with a third-order gate law, distinct from a second-harmonic gate.

The self-gated second-harmonic spectrogram has the displayed ideal form. Its delay marginal is proportional to the intensity autocorrelation by the Parseval identity. Absolute time and wave phase are undetermined; $A(t)$ and $A^{\ast }(-t)$ also give the same record. Finite crystal acceptance and spatial walk-off modify the ideal gate.

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.

The product of a specified temporal width and spectral frequency width. State whether each width refers to power or field amplitude and whether it is a full width at half maximum or a standard deviation.

A transform-limited pulse needs coherent phases across its contributing frequency range. A laser's usable gain bandwidth therefore sets a duration scale through the time-bandwidth product, as displayed. The wavelength interval must first be converted to frequency; the simple conversion $c\Delta \lambda /{\lambda }^2$ is only a narrow-band approximation. Numerical constants depend on pulse shape and width convention: a Gaussian intensity envelope has $\Delta t_{\mathrm{FWHM}}\Delta {\nu }_{\mathrm{FWHM}}=2\ln 2/\pi \simeq 0.441$. A hard gain interval is not automatically a spectral FWHM. Mode locking can arrange the longitudinal modes' phases, while group velocity dispersion must be controlled to approach the transform limit. The cavity round-trip time sets pulse spacing, rather than this bandwidth-limited width.

A diagnostic proportional to $\int I(t)I(t-\tau )dt$ for delay $\tau$. It does not uniquely reconstruct a complex optical electric field; extracting pulse duration requires a shape model.

A two-replica second-harmonic measurement retaining carrier interference fringes. The formula assumes an instantaneous broad-band second-harmonic response and includes the carrier in $E$. The extra interference information does not in general determine an arbitrary complex pulse uniquely.

Overlap delayed replicas in a sufficiently thin phase-matched second-order crystal and collect their cross-generated sum-frequency signal. Its integrated energy gives the displayed autocorrelation under undepleted, broadband-response assumptions. Temporal resolution depends on gate overlap, group-velocity mismatch and spectral acceptance. Duration inference requires a pulse-shape assumption, and arbitrary temporal phase is not recovered.

Reduction of an optical pulse's duration, for example by compensating its spectral phase.

An optical pulse with constant group delay over its spectrum. At fixed spectral amplitude, its power-weighted temporal variance is minimal.