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For a nonrelativistic ideal gas with $n_{\mathrm{tot}}$ thermal particles per volume, divide its thermal energy $3n_{\mathrm{tot}}{k}_{B}T/2$ by the optically thin volume radiation loss. In the $n_H^2$ cooling convention this becomes $3\chi k_{B}T/(2n_H\Lambda )$, where $\chi =n_{\mathrm{tot}}/n_H$. Density scaling and ionization-dependent particle count must be kept consistent.

An optical oscillator strength is a dimensionless weight assigned to an optical resonance. In an independent-resonance Lorentz oscillator model using a reference electron mass $m_e$ and a specified participating number density $n$,Its normalization must be stated: assigning one unit of total strength per participating electron means ${\sum }_j{f}_j=1$. A one-resonance approximation with $f_1=1$ represents the whole chosen electron population, rather than asserting that the first allowed interband transition carries all optical strength. The smallest band gap therefore need not equal the fitted resonance energy.

An optical parametric amplifier transfers energy from a pump beam into a signal and an idler through nonlinear optics. For the common three-wave mixing process, their angular frequencies obey ${\omega }_p={\omega }_s+{\omega }_i$. A strong pump and a weaker signal enter a nonlinear medium; the signal grows and an idler is generated.

Efficient growth requires phase matching so that contributions generated along the medium reinforce. Changing crystal orientation or temperature can tune the matched frequencies. Measuring the signal with and without the pump tests amplification, while observing the corresponding idler checks the frequency-conversion mechanism. The pump supplies the extra energy; the process does not require a stored population inversion.^[ref]

Optical parametric chirped pulse amplification combines the stretch–amplify–compress sequence of chirped pulse amplification with an optical parametric amplifier. A stretched signal pulse overlaps a strong pump pulse in a nonlinear crystal. Three-wave mixing transfers pump energy into the signal and an idler; subsequent pulse compression brings the amplified signal's frequencies back together. The pump supplies the added energy, rather than a stored population inversion.^[ref]

A. Dubietis, G. Jonušauskas and A. Piskarskas reported chirped and stretched pulse parametric amplification in beta barium borate in 1992. Their paper establishes a concrete early implementation of the parametric route; its published title identifies the nonlinear crystal.^[ref]

For pump, signal and idler angular frequencies, the conversion obeysEfficient amplification also requires phase matching, so contributions generated along the crystal reinforce. For collinear propagation the corresponding wave-vector relation is $k_p=k_s+k_i$; a noncollinear geometry requires the vector relation. Crystal dispersion and the chosen geometry determine which signal bandwidth is efficiently amplified.

An original energy-accounting example uses a 515-nanometre pump and an 800-nanometre signal. Their vacuum wavelengths giveSuppose 40 percent of a 2-millijoule pump is converted into signal–idler pairs, with negligible seed energy and other losses in this chosen model. Each conversion divides photon energy according to frequency, so the added signal energy isThe idler receives the remaining 0.285 millijoule. Assigning all depleted pump energy to the signal would violate this accounting.

If a compressor transmits 80 percent of that signal energy into a 50-femtosecond Gaussian pulse, the surviving energy is 0.412 millijoule. The chirped pulse amplification energy budget givesThese are stipulated teaching values, not a reconstruction of the 1992 experiment. Stretching lowers peak power during amplification; it does not remove the need to manage pump intensity or optical damage.

A useful apparatus check varies pump–signal delay while recording signal and idler spectra and energies. This tests temporal overlap and the frequency-conversion relation. Measuring the compressed pulse separately checks whether amplified bandwidth and residual spectral phase support the claimed pulse duration. Gain alone does not demonstrate successful compression, and an energy measurement alone does not establish peak power.

An optical parametric oscillator places an optical parametric amplifier inside an optical resonator that feeds back at least one generated wave. Pump energy is converted into signal and idler waves satisfying the frequency relation of the underlying three-wave mixing process.

Oscillation begins when round-trip gain overcomes loss at an allowed resonator mode. Below that threshold, weak generated emission can still occur, but sustained oscillation is absent. Tuning the phase matching and resonator together gives a tunable source at wavelengths difficult to obtain directly from a particular laser. Pump power alone does not select the output: cavity resonances, conversion gain and optical losses also matter.

For the shearing-wave energy profile, stationary points satisfy $-2\mathrm{Re}T=(1+T^2{)}^2$. At large Reynolds number, the initial minimum is at $T_{-}\sim -(2\mathrm{Re}{)}^{1/3}$ and the subsequent maximum is at $T_{+}\sim -1/(2\mathrm{Re})$. Their energy ratio is $G_{\max }\sim (2\mathrm{Re}{)}^{2/3}{e}^{-2/3}$. The Reynolds number here uses $A$, half the background shear rate; using $2A$ instead changes the coefficient convention.

// Target: astrophysics.bigb

Optoelectronics combines electronic devices with generation, detection or control of light. A light-emitting diode converts carrier recombination into photons, while a photodiode converts absorbed photons into an electrical response. Semiconductor band structure determines which photon energies interact efficiently, but device geometry, recombination and transport also control speed and efficiency. Optical communication links, image sensors and photovoltaic devices use different parts of this interaction. Measuring an optical signal therefore often requires understanding both the light field and the electronic readout.

Orbital energy is the mechanical energy associated with relative motion and mutual attraction. In the isolated Newtonian two-body problem, let $\mu =m_1{m}_2/(m_1+m_2)$ be the reduced mass, $r$ the separation and $v$ the relative speed. ThenDividing by $\mu$ gives specific orbital energy. With zero potential energy at infinity, negative $E$ gives a bound Kepler orbit, zero gives the parabolic limit and positive $E$ gives an unbound encounter.

For an elliptic orbit, $E=-G{m}_1{m}_2/(2a)$ with semi-major axis $a$. Energy loss can therefore shrink a bound orbit, whereas a sufficiently large gain can unbind it. External forces or interactions with additional bodies invalidate conservation of the isolated pair's orbital energy.

For a self-gravitating isothermal slab with $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ and $H^2={\sigma }^2/(2\pi G{\rho }_0)$, hydrostatic equilibrium gives the displayed Newtonian gravitational potential. A test particle has constant in-plane velocity and conserved vertical energy $E_z={\dot{z}}^2/2+\Phi (z)-\Phi (0)$. Its vertical motion is periodic for every finite $E_z>0$, with turning height $H\mathrm{arcosh}\exp [E_z/(2{\sigma }^2)]$. Near the plane it is a harmonic oscillator of squared frequency $4\pi G{\rho }_0$; at large height the force tends to the constant $2\pi G\Sigma$, where $\Sigma =2{\rho }_{0}H$. The ideal infinite slab has no finite vertical escape energy.

An advected order parameter contributes stress because a velocity gradient changes its free energy. For a polar vector this includes distortion, antisymmetric rotational, and symmetric flow-alignment stresses involving the molecular field.

An orifice is a localized opening through which a fluid passes, often from a vessel into a region of lower pressure. A simple inviscid incompressible estimate for an opening of area $A$ iswhen the reservoir velocity and elevation change are negligible and $\Delta p$ is the driving pressure difference. The relation follows from a Bernoulli equation estimate of exit speed.

Real jets contract and dissipate energy, so a discharge coefficient $C_d$ is often introduced through $Q=C_d{Q}_{\mathrm{ideal}}$. The vena contracta can have a smaller area than the opening itself. Measuring collected fluid per unit time and the driving head tests the coefficient for a specified geometry and regime. For substantial gas compression or choked flow, the incompressible formula must be replaced by an appropriate compressible-flow model.

The probability-density equation for the Ornstein-Uhlenbeck process with drift $-v$ and SDE noise amplitude $\sqrt{2}$, corresponding to diffusion coefficient one in the displayed Fokker-Planck equation. The unit-covariance Gaussian density is stationary because its Fokker-Planck probability current vanishes. This normalization makes mean velocity decay at rate one and energy excess decay at rate two.

The Orr mechanism amplifies a leading two-dimensional shear disturbance as the background flow swings its wavefronts toward alignment. Conserved perturbation vorticity implies velocity amplitude proportional to the inverse wavevector magnitude. This produces algebraic or transient energy amplification without an exponentially growing normal mode. Viscosity limits the benefit of choosing an initially tightly wound disturbance.

// Target: gravitational-instability-of-an-astrophysical-disk.bigb

For hydrogen approximated as a linear rigid rotor, the $\ell =1$ to $\ell =0$ energy decrease is ${\hslash }^2/I$, with moment of inertia $I\simeq m_p{r}^2/2$. A sample of mass $M$ initially containing an orthohydrogen fraction $x_o$ therefore releases $Q\simeq x_{o}M{\hslash }^2/(m_p^2{r}^2)$ on complete conversion to the lowest parahydrogen level. This estimate neglects the electron contribution to molecular mass and changes of the rotational constant with environment. It describes an internal heat load that external insulation cannot remove.

An oscillation is repeated variation around a central state, such as the displacement of a mass on a spring, the pressure near a vibrating source or the electric field in a light signal. In simple harmonic motion,the amplitude $A$, angular frequency $\omega$ and wave phase $\varphi$ specify the motion. The period of an oscillation is $T=2\pi /\omega$ for $\omega >0$.

A mechanical oscillator repeatedly exchanges kinetic energy and potential energy. Damping removes mechanical energy, while a periodic applied force can replenish it and produce resonance. Not all oscillations are sinusoidal or keep constant amplitude; anharmonicity can make the period of an oscillation depend on amplitude. Measuring successive peaks in a recorded trace distinguishes the repetition time from the rate at which the envelope decays.

An oscillator is a system capable of repeated motion or variation about a central state. A mass $m$ on an ideal spring with spring constant $k$ satisfiesThe restoring force accelerates the mass toward zero displacement; inertia carries it through that point, producing simple harmonic motion. The kinetic energy is largest at zero displacement, and the elastic potential energy is largest at the turning points.

Real oscillators can include damping, external driving and anharmonicity. A damped harmonic oscillator distinguishes the free transient from the forced periodic response, and a coupled oscillator adds exchange of energy with another dynamical coordinate. This makes an oscillator useful both as a physical device and as an approximation to small disturbances of much more complicated systems.

For the damped harmonic oscillator, the mechanical energy is the sum of kinetic energy and spring potential energy:Thus $b{\dot{x}}^2$ is the power transferred out of the oscillator's mechanical energy. For $b>0$, the rate vanishes instantaneously at a turning point, despite nonzero stored potential energy; it is not a constant loss per unit time. In eddy-current damping the loss becomes Joule heating. ^[ref]

For an underdamped response, advancing one oscillation period multiplies both displacement and velocity by $e^{-\zeta {\omega }_0{T}_{\mathrm{d}}}$, so $E(t+T_{\mathrm{d}})=e^{-2\zeta {\omega }_0{T}_{\mathrm{d}}}E(t)$. This exact cycle relation does not imply that instantaneous energy is a pure exponential function throughout each cycle.

A free particle whose kinetic energy equals the ground-state total energy $\hslash \omega /2$ has momentum $\hslash /a$ and de Broglie wavelength $\lambda =2\pi a$, where $a=\sqrt{\hslash /(m\omega )}$. Substitution into the normalized bound-state Gaussian yields the title density. The comparison wavelength is a useful length scale; it is not a claim that the bound state has a single free momentum or spatially uniform probability density.

A dimensionless measure of the intensity of a specified optical transition. It combines the transition matrix element and transition energy; selection rules, Polarization and the distribution of electronic states determine which transitions carry appreciable absorption. Peak position alone does not give the absorbed fraction.

For a one-dimensional oscillator eigenstate, parity gives $⟨x⟩=0$ and its coordinate variance is the displayed expression. A classical oscillator with that coordinate energy has turning amplitude $A=\sqrt{2E_x/(m{\omega }^2)}$, so $\Delta x=A/\sqrt{2}$. In several dimensions the total energy does not specify a given coordinate's classical amplitude until its energy allocation is fixed. zero-point energies in the other quantum coordinates must not silently be assigned to that one classical mode.