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For a nonrelativistic point charge, let $C_{ij}=⟨a_i{a}_j⟩$ be the time-averaged second moment of acceleration, without subtracting its mean. The outgoing radiation zone fields satisfy $\mathbf{B}=\mathbf{n}\times \mathbf{E}/c$ and $\mathbf{E}=q\mathbf{n}\times (\mathbf{n}\times \mathbf{a})/(4\pi {ϵ}_0{c}^{2}r)$. Their Poynting vector gives the displayed power per solid angle. Its integral is $⟨P⟩=q^2\mathrm{tr}C/(6\pi {ϵ}_0{c}^3)$, the averaged Larmor formula.

For periodic motion the mean acceleration vanishes and $C$ is also its covariance matrix. More generally the second moment, rather than the centered covariance matrix, is the relevant quantity. The angular formula is valid to leading nonrelativistic order and evaluates source motion at retarded time.

The normalized angular average is over the unit sphere, with solid angle measure $d\Omega$. Rotational symmetry gives $⟨e^i{⟩}_{\Omega }=0$, $⟨e^i{e}^j{⟩}_{\Omega }={\delta }^{ij}/3$, and $⟨e^i{e}^j{e}^k{e}^{\ell }{⟩}_{\Omega }=({\delta }^{ij}{\delta }^{k\ell }+{\delta }^{ik}{\delta }^{j\ell }+{\delta }^{i\ell }{\delta }^{jk})/15$. The constants follow by contracting indices and using $e^i{e}_i=1$. These identities extract photon angular temperature moments. For normalized spherical harmonics, $\int Y_{\ell m}^{\ast }{Y}_{{\ell }^{\prime }{m}^{\prime }}d\Omega ={\delta }_{\ell {\ell }^{\prime }}{\delta }_{m{m}^{\prime }}$; their angular average instead has an extra factor $1/(4\pi )$.

If $\theta$ is the angle between the observation direction and $\ddot{p}$, thenIntegrating over solid angle uses $\int {\sin }^2\theta d\Omega =8\pi /3$ and gives the total electric dipole radiation power.

Take nonrelativistic quantum mechanics, a short-range interaction, a fixed heavy target, and a common projectile mass $m$. An incoming plane wave in atomic state $a$ has unit coefficient. In an open scattering channel $i$, the target finishes in a normalized atomic state ${\chi }_i$, and the projectile has positive velocity magnitude $v_i=\hslash k_i/m$. At large distance,Here $\Omega$ denotes direction, $d\Omega$ is a solid angle element, and ${\epsilon }_i$ is an internal atomic energy. Different ${\chi }_i$ are orthogonal. The scattering amplitude $f_{ia}$ has units of length. A closed scattering channel has no propagating outgoing probability current at this total energy.

The incoming probability current density is $j_{\mathrm{in}}=v_a$. The outgoing radial probability current density in channel $i$ is $j_{r,i}\sim v_i|f_{ia}{|}^2/r^2$. Multiplying by the area $r^{2}d\Omega$ and dividing by $j_{\mathrm{in}}$ gives the differential scattering cross-section:The $r^{-2}$ dilution cancels the detector's area growth. The velocity ratio remains: a squared wave coefficient alone is not generally a flux fraction. A differential scattering cross-section has units of area per solid angle, rather than being a normalized directional probability density. For selected scattering events in acceptance $D$, an ideal detector with equal efficiency in all retained channels instead givesThe denominator specifies the event selection; it need not equal one square unit of area. Absolute rates require incident flux, target number and detector efficiency in addition to these relative weights. The next finite-mode model normalizes outgoing probability current to one incoming event and predicts event fractions directly.

When both endpoint bundles have the filled circular-cone geometry of etendue of a circular aperture and cone, conservation of etendue givesFor the same refractive index, quadrupling the area halves the sine of the cone half-angle. In the $10\mathrm{mm}$, $30^{\circ }$ example, an ideal output of diameter $20\mathrm{mm}$ must have $\sin {\theta }_2=1/4$, so ${\theta }_2\approx 14.4775^{\circ }$. The exact result differs from simply halving $30^{\circ }$. Conversely, reducing the diameter to $5\mathrm{mm}$ requires a complete forward hemisphere, ${\theta }_2=90^{\circ }$, as a limiting case.

Equal bounding area and solid angle do not alone prove equal etendue: an actual output can have unoccupied directions or position-direction correlations. The filled-cone equation is an exact model of the specified bundles and a capacity constraint on an optical instrument, not a guarantee of an aberration-free implementation.

A Berry phase is the geometric phase acquired when a nondegenerate energy eigenstate follows a closed cycle of parameters of a Hamiltonian operator in the ideal limit of the adiabatic theorem. The energy eigenstate must stay separated from other energy eigenstates by a nonzero spectral gap, and the parameters must change sufficiently slowly. Returning the Hamiltonian operator to its starting value returns the state to the same ray, but its quantum phase can remember the journey. The total quantum phase contains both the dynamical phase and the Berry phase.

The beauty of the Berry phase is that, within the adiabatic theorem's limit, it depends on the oriented path of the parameters rather than how long the journey takes. For a spin one-half qubit, the Berry phase of a Bloch-sphere latitude turns that memory into half a solid angle. Berry-phase interference with a reference path converts this geometry into a detection probability.

Michael Berry's 1984 paper formulated the geometric phase for cycles governed by the adiabatic theorem and derived the solid angle result for spin. Its bibliographic details are Proceedings of the Royal Society of London A, volume 392, pages 45–57, DOI 10.1098/rspa.1984.0023. ^[ref]; ^[ref]. The wider concept of geometric phase does not require every cycle to obey the adiabatic theorem: Aharonov and Anandan formulated a geometric phase for general cyclic quantum evolution in 1987. ^[ref].

Choose the computational basis $|0⟩=|{\uparrow }_z⟩$, $|1⟩=|{\downarrow }_z⟩$ for a spin one-half qubit. On the Bloch sphere, let $\mathbf{n}=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$ and use the normalized quantum stateThis is the $+1$ eigenstate of $\mathbf{n}\cdot \mathbf{\sigma }$, where $\mathbf{\sigma }$ are the Pauli matrices. To make it an instantaneous ground state, take the model Hamiltonian operator $H=-\Delta \mathbf{n}\cdot \mathbf{\sigma }/2$ with constant $\Delta >0$. Its energy eigenvalues are $-\Delta /2$ and $+\Delta /2$, so its spectral gap is $\Delta$ everywhere. The spin points along $\mathbf{n}$; this explicit convention fixes the sign of the Berry phase.

Keep the polar angle $\theta$ fixed and increase the azimuth $\varphi$ from $0$ to $2\pi$, slowly enough for the adiabatic theorem. The state vector written above is periodic in $\varphi$. Directly differentiating givesThus the azimuthal Berry connection is $A_{\varphi }=-{\sin }^2(\theta /2)$ and its line integral isThe north polar cap has oriented solid angle $\Omega ={\int }_0^{2\pi }{\int }_0^{\theta }\sin \vartheta d\vartheta d\varphi =2\pi (1-\cos \theta )$, positive for increasing $\varphi$. Reversing the loop reverses the Berry phase. The opposite spin eigenstate has the opposite Berry phase modulo $2\pi$.

At $\theta =60^{\circ }$, the solid angle is $\pi$ and the Berry phase is $-\pi /2$. At $\theta =90^{\circ }$, the solid angle is $2\pi$ and the Berry phase is $-\pi$. At $\theta =0$, the loop collapses and $\gamma =0$. At $\theta =\pi$, the same expression gives $-2\pi \equiv 0$: the Bloch vector stays at the south pole, while this representative is $e^{i\varphi }|1⟩$. Its winding is only a choice of global phase, so the physically relevant phase factor is again one. The displayed formula is a continuous, unwrapped representative, not a unique real-valued observable.

Consider a hypothetical star whose emitted specific intensity is exactly that of a uniform $5000\mathrm{K}$ blackbody. Assume transparent propagation with negligible Doppler shift and interstellar extinction, so geometric dilution multiplies every spectral interval by the same factor. The stipulated temperature givesUsing the converted frequency peak as though it maximized $B_{\lambda }$ would return $b/(c/{\nu }_{\max })\simeq 2841.26\mathrm{K}$, a substantial error caused solely by mixing spectral conventions.

Now stipulate a photodetector with constant quantum efficiency $\eta$ over the relevant wavelengths, collecting equal narrow wavelength bins. For fixed accepted projected area times solid angle $G$, with dilution included in $G$, its expected photon count rate density isThis follows by dividing the received power by the photon energy $hc/\lambda$. It peaks where $x=4(1-e^{-x})$, at $x=3.9206903949\dots$, giving $0.73394057\mu \mathrm{m}$ for the same model star. Constant quantum efficiency is not constant energy spectral responsivity: the latter includes the wavelength-dependent photon energy conversion. For example, $G=10^{-12}{\mathrm{m}}^2\mathrm{sr}$, $\eta =0.5$ and a $1\mathrm{nm}$ bin centered at $0.600\mu \mathrm{m}$ give approximately $1.93\times 10^{10}$ detected photons per second in the narrow-bin approximation. These are calculated model values, not observations of a named star.

A more general linear detector channel has outputwhere $S_{i,0}$ is the background reading and $R_i$ includes spectral selection, optical transmission and detector spectral responsivity. A raw channel maximum need not locate either Planck function maximum. Recovering a temperature requires an appropriate calibrated response and an emission model. For a real stellar atmosphere, opacity and radiative transfer can reshape the spectrum; interstellar extinction can reshape it again. A single peak does not establish a unique physical temperature without those assumptions.

Black-body radiation is the electromagnetic radiation in thermal equilibrium with an ideal blackbody. It can be described as a photon gas: a cavity whose walls are all at the same temperature contains radiation with the universal Planck's law spectrum. A small opening in such a cavity approximates a blackbody emitter because incoming radiation undergoes repeated absorption inside. The radiation's equilibrium spectrum depends on temperature, rather than the wall material. ^[ref]

There are three useful quantities to keep distinct: directional spectral radiance $B_{\nu }$ or $B_{\lambda }$, total energy density $u$ in the cavity, and outward radiant exitance $j^{\ast }$ from its opening. Planck's law supplies the first; integrals over frequency and directions supply the others. For isotropy, every direction has the same $B_{\nu }$, so the full $4\pi$ solid angle givesHere $\hslash =h/(2\pi )$ is the reduced Planck constant, and the radiation constant is $a_{\mathrm{rad}}\approx 7.565733250\times 10^{-16}\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-4}$. The explicit Boltzmann constant makes this an SI formula; adopting natural units with $k_B=1$ recovers $u={\pi }^2{T}^4/(15{\hslash }^3{c}^3)$.

For a short integral outline, substitute $x=h\nu /(k_{B}T)$ and useThe expansion is a geometric series with nonnegative terms; repeated integration by parts evaluates each integral, and $\zeta (4)={\pi }^4/90$ is a value of the Riemann zeta function. This gives the coefficient above.

The photon gas also exerts radiation pressure, $p=u/3$. The factor $1/3$ comes from the angular average of the squared directional component of photon momentum, $⟨{\cos }^2\theta ⟩=1/3$. The thermodynamic derivation of blackbody radiation pressure gives the same result. In contrast, outgoing power involves one projected-direction factor over an outward hemisphere, yielding radiant exitanceThis is the Stefan–Boltzmann law with unit emissivity; $\sigma$ is the Stefan-Boltzmann constant. The factor $\pi$ follows from Lambert's cosine law, rather than the $4\pi$ full-sphere factor used for energy density.

For example, an ideal $6000\mathrm{K}$ cavity has $u\approx 0.980519\mathrm{J}{\mathrm{m}}^{-3}$, radiation pressure $p\approx 0.326840\mathrm{Pa}$, and outward radiant exitance $j^{\ast }\approx 7.34881\times 10^7\mathrm{W}{\mathrm{m}}^{-2}$. A stipulated $3000\mathrm{K}$ cavity has one-sixteenth each value because all three scale as $T^4$. These describe stored energy, mechanical pressure, and outgoing power per area, respectively. In a closed equilibrium cavity, equal opposing radiation balances the directional power flow even though $u$ and $p$ are nonzero.

The usefulness of black-body radiation is this common reference: Planck's law, Wien's displacement law, and the Stefan–Boltzmann law consistently connect spectral measurements, temperature, and total emission. The three illustrative curves in Planck's law are ideal equilibrium spectra; applying them to a material requires considering its emissivity and whether thermal equilibrium is a suitable approximation.

For unit three-component classical spins and energy $-J{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ on a lattice of coordination $z_c$, mean-field decoupling gives $E_{\mathrm{MF}}=-z_{c}JS{\sum }_i{S}_{iz}+N{z}_{c}J{S}^2/2$. Integrating each spin over solid angle yieldsThe constant term corrects double counting. A normalized angular measure removes $(4\pi {)}^N$ but does not change the self-consistency equation. The lattice dimension and spin-space dimension are different data.

Normalize the scale factor to $a_0=1$. Identical isotropic sources of bolometric luminosity $L$ and conserved comoving number density $n_0=a^{3}n$ produce all-sky radiative flux ${\mathcal{F}}_0=cL{n}_0\int a(t)dt$ in a transparent FLRW metric. The source counts on an FLRW past light cone and cosmological flux dimming cancel the transverse shell-area factor. For arbitrary proper source density the expression is instead $cL\int n(t)a(t{)}^{4}dt$. The intensity per solid angle is ${\mathcal{F}}_0/(4\pi )$. A detector of collecting area $A$ receives power $A{\mathcal{F}}_0$ under an all-sky effective-area convention.

The cosmological spectral background is received radiant power per detector area, received frequency interval and solid angle, summed over a cosmological source population. Its intensity depends on emitted spectra, their cosmic evolution, the expansion history and absorption. A finite radial comoving distance alone does not guarantee a finite background when high-redshift sources are sampled at increasingly high emitted frequencies.

An eigenstate of the rotating-frame Rabi Hamiltonian returns to the same laboratory ray after a full drive period. Its quantum phase includes both the rotating-frame energy and the sign of a full spin rotation. For spin $s$ and magnetic label $\mu$, the factor is $(-1{)}^{2s}{e}^{-i\mu \Omega T}$. Expanding the rotating-frame splitting to first order in the drive frequency and subtracting the adiabatic dynamical phase leaves $-\mu$ times the oriented solid angle of the field loop. Dropping the full-rotation sign gives a spurious half-integer-spin offset.

The differential scattering cross-section is the effective incident area scattered per unit solid angle. It is defined by the ratio of the outgoing particle rate through $d\Omega$ to the incident flux.

A diffuse thermal emitter has emitted radiance independent of outgoing direction over the outward hemisphere. Its emitted radiant intensity from a planar patch still varies with viewing angle through projected area, as described by Lambert's cosine law. Diffuse emission does not mean equal power per solid angle from the patch. This approximation does not by itself impose a grey body spectrum or specify how incident radiation is reflected.

For constant comoving number density and complete selection in a spatially flat Einstein-de Sitter universe, integrate the present-epoch shell volume up to the population's formation redshift. The radial distance is $D_M=2c[1-(1+z{)}^{-1/2}]/H_0$, so the count per unit solid angle is $n_0{D}_M^3(z_f)/3$. It tends to a finite value as $z_f$ tends to infinity because this model's present comoving particle-horizon radius is $2c/H_0$. Formation, mergers or an incomplete observed sample change this ideal count.

Electromagnetic radiation is a propagating disturbance of the electric field and magnetic field that carries energy and momentum away from its source. It can be a sustained electromagnetic wave or a finite wave packet: periodic oscillation is not required. In vacuum the disturbance travels at the speed of light. A receiver responds to fields arriving now from an earlier source event, as expressed by retarded time. This is how electrical activity at one place can deliver energy and information elsewhere without a connecting wire.

Stored electromagnetic energy alone does not establish outgoing electromagnetic radiation. A stationary electric charge in electrostatics, or a steady localized electric current in magnetostatics, can store electromagnetic energy with zero Poynting vector. Static combinations can have circulating energy flux without net outward power. Near a changing source, reactive energy can return later. The near and far field distinction separates these contributions from outgoing radiation zone fields.^[ref]

A single traveling vacuum plane electromagnetic wave obeying the Maxwell equations has $\mathbf{B}=\hat{\mathbf{k}}\times \mathbf{E}/c$. Its Poynting vector gives the energy density and flux of a plane electromagnetic wave:Here $I$ is wave intensity; ${ϵ}_0$ and ${\mu }_0$ are vacuum permittivity and vacuum permeability. For a monochromatic wave with linear polarization and peak amplitude $E_0$, squared cosine averages to $1/2$. Pulses require the instantaneous Poynting vector; arbitrary incident-and-reflected superpositions need not obey this relation.^[ref]

The electromagnetic momentum density is $\mathbf{g}=\mathbf{S}/c^2$. Completely absorbing a unidirectional vacuum pulse of energy $U$ transfers momentum $U/c$; see radiation pressure on a perfectly absorbing surface. A calculated $1.0\mathrm{mJ}$ pulse transfers $3.34\times 10^{-12}\mathrm{kg}m/s$. Symmetric emission can lose energy without net recoil because opposite momenta cancel.^[ref]

The direction of emission matters as much as its total power. For an oscillating electric dipole moment along $z$, the angular distribution of electric dipole radiation is strongest across the axis and vanishes along it. Its origin is geometric: the electric dipole radiation field is the transverse projection of the second time derivative of the electric dipole moment, so its magnitude contains $\sin \theta$. Squaring this electric field to obtain wave intensity gives ${\sin }^2\theta$. The electric-dipole approximation for radiation requires a source much smaller than its wavelength and observation in the radiation zone. ^[ref]

For example, the power per solid angle at $\theta =30^{\circ }$ is one quarter of its equatorial value, while the electric field amplitude is one half. To normalize the angular distribution to total power, use spherical coordinates with $d\Omega =\sin \theta d\theta d\varphi$:The integral follows by substituting $x=\cos \theta$, giving ${\int }_{-1}^1(1-x^2)dx=4/3$. This is a probability-like normalization over direction, with units of inverse steradian; it does not make the dipole an isotropic radiator. The same angular integral underlies the Larmor formula for nonrelativistic accelerated electric charge.

Detection requires the incident electromagnetic field to do something observable. An antenna develops an electrical response; a thermopile converts absorbed energy into a thermal and then electrical signal; a photodetector can respond through photon absorption. Hertz spark transmission and detection experiment makes this conversion visible as a tiny spark. Detector orientation, sensitivity, and surrounding reflections all matter: failure to see a signal does not prove that no electromagnetic radiation reached the receiver.

The electric current delivered per transverse probe area per solid angle. For a uniform circular probe of diameter $d$ and a small convergence semiangle $\alpha$, one convention is $\beta =4I/({\pi }^2{d}^2{\alpha }^2)$. State the accelerating voltage, angular convention and whether brightness is reduced by a voltage factor before comparing sources. A smaller aperture changes current as well as convergence; finite electron beam brightness prevents arbitrary current in an arbitrarily small probe.

Etendue measures how much position-and-direction space a bundle of light rays occupies. A small spot accepting a wide range of directions can have the same etendue as a large spot accepting a narrow range. This makes etendue useful for deciding whether an optical instrument can deliver an extended source's entire admitted radiant flux to a small receiver. For forward crossings of a surface, the geometrical optics definition iswhere $\mathcal{B}$ is the occupied set of surface positions and directions, $n$ is the local refractive index, $dA$ is actual surface area, $d\Omega$ is solid angle, and $\theta$ is the angle to the forward unit normal. The cosine supplies the projected area. Opposite crossing directions must be counted separately with positive projected area, rather than cancelling them in a signed integral. ^[ref]

The shorthand $G\simeq n^{2}A\Omega$ assumes constant refractive index, a narrow nearly normal bundle, and the same admitted directions across the whole area $A$. At larger angles, use the cosine-weighted integral; a position-dependent angular acceptance cannot generally be replaced by the product of two independent extents.

A proposed optical bench test uses a steady illuminated diffuse patch, a relay made from two optical lens elements, interchangeable centred circular stops, and a calibrated optical power meter large enough to collect the whole output bundle. Start with small cone angles so an ordinary relay approximates the area-angle tradeoff. Measure the patch's image diameter and the accepted angular extent at both conjugate planes; the transverse magnification gives the spatial scale, but does not by itself measure etendue.

Map the local radiance with a baffled photodiode on translation and rotation stages, using two small aligned holes to select a narrow known solid angle. Keep the sampled position fixed while scanning direction, calibrate the sampling footprint and angular response, and repeat at several positions. Numerically integrate $n^2\cos \theta dAd\Omega$ over the occupied samples, and integrate $(L/n^2)dG$ to predict the collected radiant flux. This checks the filled-cone assumption instead of inferring it from two outer dimensions.

Compare the relay without a stop and with a stop of $0.6$ times the illuminated diameter in the same plane. The uniform-bundle prediction is a radiant flux ratio of $0.36$, with unchanged local radiance in the surviving samples. Cover the input for a dark reading, monitor source drift, repeat the open-stop reading, check detector linearity with calibrated optical attenuators, and enlarge the collecting detector or scan it to check for missed light rays. Lens losses, angular response, illumination gradients, alignment errors and diffraction can spoil the ideal prediction; their effects must be measured separately. This is a procedure to try, with predicted readouts, not purported experimental data.